# Serial Dictatorships and House Allocation

I was recently an invited speaker in a series of STEM talks at Moraine Valley Community College. My talk was called “What can algorithms tell us about life, love, and happiness?” and it’s on Youtube now so you can go watch it. The central theme of the talk was the lens of computation, that algorithms and theoretical computer science can provide new and novel explanations for the natural phenomena we observe in the world.

One of the main stories I told in the talk is about stable marriages and the deferred acceptance algorithm, which we covered previously on this blog. However, one of the examples of the applications I gave was to kidney exchanges and school allocation. I said in the talk that it’s a variant of the stable marriages, but it’s not clear exactly how the two are related. This post will fill that gap and showcase some of the unity in the field of mechanism design.

Mechanism design, which is sometimes called market design, has a grand vision. There is a population of players with individual incentives, and given some central goal the designer wants to come up with a game where the self-interest of the players will lead them to efficiently achieve the designer’s goals. That’s what we’re going to do today with a class of problems called allocation problems.

As usual, all of the code we used in this post is available in a repository on this blog’s Github page.

## Allocating houses with dictators

In stable marriages we had $n$ men and $n$ women and we wanted to pair them off one to one in a way that there were no mutual incentives to cheat. Let’s modify this scenario so that only one side has preferences and the other does not. The analogy here is that we have $n$ people and $n$ houses, but what do we want to guarantee? It doesn’t make sense to say that people will cheat on each other, but it does make sense to ask that there’s no way for people to swap houses and have everyone be at least as happy as before. Let’s formalize this.

Let $A$ be a set of people (agents) and $H$ be a set of houses, and $n = |A| = |H|$. A matching is a one-to-one map from $A \to H$. Each agent is assumed to have a strict preference over houses, and if we’re given two houses $h_1, h_2$ and $a \in A$ prefers $h_1$ over $h_2$, we express that by saying $h_1 >_a h_2$. If we want to include the possibility that $h_1 = h_2$, we would say $h_1 \geq_a h_2$. I.e., either they’re the same house, or $a$ strictly prefers $h_1$ more.

Definition: A matching $M: A \to H$ is called pareto-optimal if there is no other matching $M$ with both of the following properties:

• Every agent is at least as happy in $N$ as in $M$, i.e. for every $a \in A$, $N(a) \geq_a M(a)$.
• Some agent is strictly happier in $N$, i.e. there exists an $a \in A$ with $N(a) >_a M(a)$.

We say a matching $N$ “pareto-dominates” another matching $M$ if these two properties hold. As a side note, if you like abstract algebra you might notice that you can take matchings and form them into a lattice where the comparison is pareto-domination. If you go deep into the theory of lattices, you can use some nice fixed-point theorems to (non-constructively) prove the existences of optimal allocations in this context and for stable marriages. See this paper if you’re interested. Of course, we will give efficient algorithms to achieve our goals, which is how I prefer to live life.

The mechanism we’ll use to find such an optimal matching is extremely simple, and it’s called the serial dictatorship.

First you pick an arbitrary ordering of the agents and all houses are marked “available.” Then the first agent in the ordering picks their top choice, and you remove their choice from the available houses. Continue in this way down the list until you get to the end, and the outcome is guaranteed to be pareto-optimal.

Theorem: Serial dictatorship always produces a pareto-optimal matching.

Proof. Let $M$ be the output of the algorithm. Suppose the theorem is false, that there is some $N$ that pareto-dominates $M$. Let $a$ be the first agent in the chosen ordering who gets a strictly better house in $N$ than in $M$. Whatever house $a$ gets, call it $N(a)$, it has to be a house that was unavailable at the time in the algorithm when $a$ got to pick (otherwise $a$ would have picked $N(a)$ during the algorithm!). This means that $a$ took the house chosen by some agent $b \in A$ whose turn to pick comes before $a$. But by assumption, $a$ was the first agent to get a strictly better house, so $b$ has to end up with a worse house. This contradicts that every agent is at least as happy in $N$ than in $M$, so $N$ cannot pareto-dominate $M$.

$\square$

It’s easy enough to implement this in Python. Each agent will be represented by its list of preferences, each object will be an integer, and the matching will be a dictionary. The only thing we need to do is pick a way to order the agents, and we’ll just pick a random ordering. As usual, all of the code used in this post is available on this blog’s github page.

# serialDictatorship: [[int]], [int] -> {int: int}
# construct a pareto-optimal allocation of objects to agents.
def serialDictatorship(agents, objects, seed=None):
if seed is not None:
random.seed(seed)

agentPreferences = agents[:]
random.shuffle(agentPreferences)
allocation = dict()
availableHouses = set(objects)

for agentIndex, preference in enumerate(agentPreferences):
allocation[agentIndex] = max(availableHouses, key=preference.index)
availableHouses.remove(allocation[agentIndex])

return allocation


And a test

agents = [['d','a','c','b'], # 4th in my chosen seed
['a','d','c','b'], # 3rd
['a','d','b','c'], # 2nd
['d','a','c','b']] # 1st
objects = ['a','b','c','d']
allocation = serialDictatorship(agents, objects, seed=1)
test({0: 'b', 1: 'c', 2: 'd', 3: 'a'}, allocation)


This algorithm is so simple it’s almost hard to believe. But it get’s better, because under some reasonable conditions, it’s the only algorithm that solves this problem.

Theorem [Svensson 98]: Serial dictatorship is the only algorithm that produces a pareto-optimal matching and also has the following three properties:

• Strategy-proof: no agent can improve their outcomes by lying about their preferences at the beginning.
• Neutral: the outcome of the algorithm is unchanged if you permute the items (i.e., does not depend on the index of the item in some list)
• Non-bossy: No agent can change the outcome of the algorithm without also changing the object they receive.

And if we drop any one of these conditions there are other mechanisms that satisfy the rest. This theorem was proved in this paper by Lars-Gunnar Svensson in 1998, and it’s not particularly long or complicated. The proof of the main theorem is about a page. It would be a great exercise in reading mathematics to go through the proof and summarize the main idea (you could even leave a comment with your answer!).

## Allocation with existing ownership

Now we switch to a slightly different problem. There are still $n$ houses and $n$ agents, but now every agent already “owns” a house. The question becomes: can they improve their situation by trading houses? It shouldn’t be immediately obvious whether this is possible, because a trade can happen in a “cycle” like the following:

Here A prefers the house of B, and B prefers the house of C, and C prefers the house of A, so they’d all benefit from doing a three-way cyclic trade. You can easily imagine the generalization to larger cycles.

This model was studied by Shapley and Scarf in 1974 (the same Shapley who did the deferred acceptance algorithm for stable marriages). Just as you’d expect, our goal is to find an optimal (re)-allocation of houses to agents in which there is no cycle the stands to improve. That is, there is no subset of agents that can jointly improve their standing. In formalizing this we call an “optimal” matching a core matching. Again $A$ is a set of agents, and $H$ is a set of houses.

Definition: A matching $M: A \to H$ is called a core matching if there is no subset $B \subset A$ and no matching $N: A \to H$ with the following properties:

• For every $b \in B$, $N(b)$ is owned by some other agent in $B$ (trading only happens within $B$).
• Every agent $b$ in $B$ is at least as happy as before, i.e. $N(b) \geq_b M(b)$ for all $b$.
• Some agent in $B$ strictly improves, i.e. for some $b, N(b) >_b M(b)$.

We also call an algorithm individually rational if it ensures that every agent gets a house that is at least as good as their starting house. It should be clear that an algorithm which produces a core matching is individually rational, because for any agent $a$ we can set $B = \{a\}$, i.e. force $a$ to consider not trading at all, and being a core matching says that’s not better for $a$. Likewise, core matchings are also pareto-optimal by setting $B = A$.

It might seem like the idea of a “core” solution to an allocation problem is more general, and you’re right. You can define it for a very general setting of cooperative games and prove the existence of core matchings in that setting. See Wikipedia for more. As is our prerogative, we’ll achieve the same thing by constructing core matchings with an algorithm.

Indeed, the following theorem is due to Shapley & Scarf.

Theorem [Shapley-Scarf 74]: There is a core matching for every choice of preferences. Moreover, one can be found by an efficient algorithm.

Proof. The mechanism we’ll define is called the top trading cycles algorithm. We operate in rounds, and the first round goes as follows.

Form a directed graph with nodes in $A \cup H$. That is there is one node for each agent and one node for each house. Then we start by having each agent “point” to its most preferred house, and each house “points” to its original owner. That is, we add in directed edges from agents to their top pick, and houses to their owners. For example, say there are five agents $A = \{ a, b, c, d, e, f \}$ and houses $H = \{ 1,2,3,4,5 \}$ with $a$ owning $1$, and $b$ owning $2$, etc. but their favorite picks goes backwards, so that $a$ prefers house $5$ most, and $b$ prefers $4$ most, $c$ prefers $3$ (which $c$ also owns), etc. Then the “pointing picture” in the first round looks like this.

The claim about such a graph is that there is always some directed cycle. In the example above, there are three. And moreover, we claim that no two cycles can share an edge. It’s easy to see there has to be a cycle: you can start at any agent and just follow the single outgoing edge until you find yourself repeating some vertices. By the fact that there is only one edge going out of any vertex, it follows that no two cycles could share an edge (or else in the last edge they share, there’d have to be a fork, i.e. two outgoing edges).

In the example above, you can start from A and follow the only edge and you get the cycle A -> 5 -> E -> 1 -> A. Similarly, starting at 4 would give you 4 -> D -> 2 -> B -> 4.

The point is that when you remove a cycle, you can have the agents in that cycle do the trade indicated by the cycle and remove the entire cycle from the graph. The consequence of this is that you have some agents who were pointing to houses that are removed, and so these agents revise their outgoing edge to point at their next most preferred available house. You can then continue removing cycles in this way until all the agents have been assigned a house.

The proof that this is a core matching is analogous to the proof that serial dictatorships were pareto-optimal. If there were some subset $B$ and some other matching $N$ under which $B$ does better, then one of these agents has to be the first to be removed in a cycle during the algorithm’s run. But that agent got the best possible pick of a house, so by involving with $B$ that agent necessarily gets a worse outcome.

$\square$

This algorithm is commonly called the Top Trading Cycles algorithm, because it splits the set of agents and houses into a disjoint union of cycles, each of which is the best trade possible for every agent involved.

Implementing the Top Trading Cycles algorithm in code requires us to be able to find cycles in graphs, but that isn’t so hard. I implemented a simple data structure for a graph with helper functions that are specific to our kind of graph (i.e., every vertex has outdegree 1, so the algorithm to find cycles is simpler than something like Tarjan’s algorithm). You can see the data structure on this post’s github repository in the file graph.py. An example of using it:

>>> G = Graph([1,'a',2,'b',3,'c',4,'d',5,'e',6,'f'])
>>> G.addEdges([(1,'a'), ('a',2), (2,'b'), ('b',3), (3,'c'), ('c',1),
(4,'d'), ('d',5), (5,'e'), ('e',4), (6,'f'), ('f',6)])
>>> G['d']
Vertex('d')
>>> G['d'].outgoingEdges
{('d', 5)}
>>> G['d'].anyNext() # return the target of any outgoing edge from 'd'
Vertex(5)
>>> G.delete('e')
>>> G[4].incomingEdges
set()


Next we implement a function to find a cycle, and a function to extract the agents from a cycle. For latter we can assume the cycle is just represented by any agent on the cycle (again, because our graphs always have outdegree exactly 1).

# anyCycle: graph -> vertex
# find any vertex involved in a cycle
def anyCycle(G):
visited = set()
v = G.anyVertex()

while v not in visited:
v = v.anyNext()

return v

# getAgents: graph, vertex -> set(vertex)
# get the set of agents on a cycle starting at the given vertex
def getAgents(G, cycle, agents):
# make sure starting vertex is a house
if cycle.vertexId in agents:
cycle = cycle.anyNext()

startingHouse = cycle
currentVertex = startingHouse.anyNext()
theAgents = set()

while currentVertex not in theAgents:
currentVertex = currentVertex.anyNext()
currentVertex = currentVertex.anyNext()

return theAgents


Finally, implementing the algorithm is just bookkeeping. After setting up the initial graph, the core of the routine is

def topTradingCycles(agents, houses, agentPreferences, initialOwnership):
# form the initial graph

...

allocation = dict()
while len(G.vertices) &> 0:
cycle = anyCycle(G)
cycleAgents = getAgents(G, cycle, agents)

# assign agents in the cycle their choice of house
for a in cycleAgents:
h = a.anyNext().vertexId
allocation[a.vertexId] = h
G.delete(a)
G.delete(h)

for a in agents:
if a in G.vertices and G[a].outdegree() == 0:
# update preferences
...

return allocation


This mutates the graph in each round by deleting any cycle that was found, and adding new edges when the top choice of some agent is removed. Finally, to fill in the ellipses we just need to say how we represent the preferences. The input agentPreferences is a dictionary mapping agents to a list of all houses in order of preference. So again we can just represent the “top available pick” by an index and update that index when agents lose their top pick.

# maps agent to an index of the list agentPreferences[agent]
currentPreferenceIndex = dict((a,0) for a in agents)
preferredHouse = lambda a: agentPreferences[a][currentPreferenceIndex[a]]


Then to update we just have to replace the currentPreferenceIndex for each disappointed agent by its next best option.

      for a in agents:
if a in G.vertices and G[a].outdegree() == 0:
while preferredHouse(a) not in G.vertices:
currentPreferenceIndex[a] += 1


And that’s it! We included a small suite of test cases which you can run if you want to play around with it more.

One final nice thing about this algorithm is that it almost generalizes the serial dictatorship algorithm. What you do is rather than have each house point to its original owner, you just have all houses point to the first agent in the pre-specified ordering. Then a cycle will always have length 2, the first agent gets their preferred house, and in the next round the houses now point to the second agent in the ordering, and so on.

## Kidney exchange

We still need one more ingredient to see the bridge from allocation problems to kidney exchanges. The setting is like this: say Manuel needs a kidney transplant, and he’s lucky enough that his sister-in-law Anastasia wants to donate her kidney to Manuel. However, it turns out that Anastasia doesn’t the same right blood/antibody type for a donation, and so even though she has a kidney to give, they can’t give it to Manuel. Now one might say “just sell your kidney and use the money to buy a kidney with the right type!” Turns out that’s illegal; at some point we as a society decided that it’s immoral to sell organs. But it is legal to exchange a kidney for a kidney. So if Manuel and Anastasia can find a pair of people both of whom happen to have the right blood types, they can arrange for a swap.

But finding two people both of whom have the right blood types is unlikely, and we can actually do far better! We can turn this into a housing allocation problem as follows. Anyone with a kidney to donate is a “house,” and anyone who needs a kidney is an “agent.” And to start off with, we say that each agent “owns” the kidney of their willing donor. And the preferences of each agent are determined by which kidney donors have the right blood type (with ties split, say, by geographical distance). Then when you do the top trading cycles algorithm you find these chains where Anastasia, instead of donating to Manuel, donates to another person who has the right blood type. On the other end of the cycle, Manuel receives a kidney from someone with the right blood type.

The big twist is that not everyone who needs a kidney knows someone willing to donate. So there are agents who are “new” to the market and don’t already own a house. Moreover, maybe you have someone who is willing to donate a kidney but isn’t asking for anything in return.

Because of this the algorithm changes slightly. You can no longer guarantee the existence of a cycle (though you can still guarantee that no two cycles will share an edge). But as new people are added to the graph, cycles will eventually form and you can make the trades. There are a few extra details if you want to ensure that everyone is being honest (if you’re thinking about it like a market in the economic sense, where people could be lying about their preferences).

The resulting mechanism is called You Request My House I Get Your Turn (YRMHIGYT). In short, the idea is that you pick an order on the agents, say for kidney exchanges it’s the order in which the patients are diagnosed. And you have them add edges to the graph in that order. At each step you look for a cycle, and when one appears you remove it as usual. The twist, and the source of the name, is that when someone who has no house requests a house which is already owned, the agent who owns the house gets to jump forward in the queue. This turns out to make everything “fair” (in that everyone is guaranteed to get a house at least as good as the one they own) and one can prove analogous optimality theorems to the ones we did for serial dictatorship.

This mechanism was implemented by Alvin Roth in the US hospital system, and by some measure it has saved many lives. If you want to hear more about the process and how successful the kidney exchange program is, you can listen to this Freakonomics podcast episode where they interviewed Al Roth and some of the patients who benefited from this new allocation market.

It would be an excellent exercise to go deeper into the guts of the kidney exchange program (see this paper by Alvin Roth et al.), and implement the matching system in code. At the very least, implementing the YRMHIGYT mechanism is only a minor modification of our existing Top Trading Cycles code.

Until next time!

# One definition of algorithmic fairness: statistical parity

If you haven’t read the first post on fairness, I suggest you go back and read it because it motivates why we’re talking about fairness for algorithms in the first place. In this post I’ll describe one of the existing mathematical definitions of “fairness,” its origin, and discuss its strengths and shortcomings.

Before jumping in I should remark that nobody has found a definition which is widely agreed as a good definition of fairness in the same way we have for, say, the security of a random number generator. So this post is intended to be exploratory rather than dictating The Facts. Rather, it’s an idea with some good intuitive roots which may or may not stand up to full mathematical scrutiny.

## Statistical parity

Here is one way to define fairness.

Your population is a set $X$ and there is some known subset $S \subset X$ that is a “protected” subset of the population. For discussion we’ll say $X$ is people and $S$ is people who dye their hair teal. We are afraid that banks give fewer loans to the teals because of hair-colorism, despite teal-haired people being just as creditworthy as the general population on average.

Now we assume that there is some distribution $D$ over $X$ which represents the probability that any individual will be drawn for evaluation. In other words, some people will just have no reason to apply for a loan (maybe they’re filthy rich, or don’t like homes, cars, or expensive colleges), and so $D$ takes that into account. Generally we impose no restrictions on $D$, and the definition of fairness will have to work no matter what $D$ is.

Now suppose we have a (possibly randomized) classifier $h:X \to \{-1,1\}$ giving labels to $X$. When given a person $x$ as input $h(x)=1$ if $x$ gets a loan and $-1$ otherwise. The bias, or statistical imparity, of $h$ on $S$ with respect to $X,D$ is the following quantity. In words, it is the difference between the probability that a random individual drawn from $S$ is labeled 1 and the probability that a random individual from the complement $S^C$ is labeled 1.

$\textup{bias}_h(X,S,D) = \Pr[h(x) = 1 | x \in S^{C}] - \Pr[h(x) = 1 | x \in S]$

The probability is taken both over the distribution $D$ and the random choices made by the algorithm. This is the statistical equivalent of the legal doctrine of adverse impact. It measures the difference that the majority and protected classes get a particular outcome. When that difference is small, the classifier is said to have “statistical parity,” i.e. to conform to this notion of fairness.

Definition: A hypothesis $h:X \to \{-1,1\}$ is said to have statistical parity on $D$ with respect to $S$ up to bias $\varepsilon$ if $|\textup{bias}_h(X,S,D)| < \varepsilon$.

So if a hypothesis achieves statistical parity, then it treats the general population statistically similarly to the protected class. So if 30% of normal-hair-colored people get loans, statistical parity requires roughly 30% of teals to also get loans.

It’s pretty simple to write a program to compute the bias. First we’ll write a function that computes the bias of a given set of labels. We’ll determine whether a data point $x \in X$ is in the protected class by specifying a specific value of a specific index. I.e., we’re assuming the feature selection has already happened by this point.

# labelBias: [[float]], [int], int, obj -> float
# compute the signed bias of a set of labels on a given dataset
def labelBias(data, labels, protectedIndex, protectedValue):
protectedClass = [(x,l) for (x,l) in zip(data, labels)
if x[protectedIndex] == protectedValue]
elseClass = [(x,l) for (x,l) in zip(data, labels)
if x[protectedIndex] != protectedValue]

if len(protectedClass) == 0 or len(elseClass) == 0:
raise Exception("One of the classes is empty!")
else:
protectedProb = sum(1 for (x,l) in protectedClass if l == 1) / len(protectedClass)
elseProb = sum(1 for (x,l) in elseClass  if l == 1) / len(elseClass)

return elseProb - protectedProb


Then generalizing this to an input hypothesis is a one-liner.

# signedBias: [[float]], int, obj, h -> float
# compute the signed bias of a hypothesis on a given dataset
def signedBias(data, h, protectedIndex, protectedValue):
return labelBias(pts, [h(x) for x in pts], protectedIndex, protectedValue)


Now we can load the census data from the UCI machine learning repository and compute some biases in the labels. The data points in this dataset correspond to demographic features of people from a census survey, and the labels are +1 if the individual’s salary is at least 50k, and -1 otherwise. I wrote some helpers to load the data from a file (which you can see in this post’s Github repo).

if __name__ == "__main__":

# [(test name, (index, value))]
tests = [('gender', (1,0)),
('private employment', (2,1)),
('asian race', (33,1)),
('divorced', (12, 1))]

for (name, (index, value)) in tests:
print("'%s' bias in training data: %.4f" %
(name, labelBias(train[0], train[1], index, value)))


(I chose ‘asian race’ instead of just ‘asian’ because there are various ‘country of origin’ features that are for countries in Asia.)

Running this gives the following.

anti-'female' bias in training data: 0.1963
anti-'private employment' bias in training data: 0.0731
anti-'asian race' bias in training data: -0.0256
anti-'divorced' bias in training data: 0.1582


Here a positive value means it’s biased against the quoted thing, a negative value means it’s biased in favor of the quoted thing.

Now let me define a stupidly trivial classifier that predicts 1 if the country of origin is India and zero otherwise. If I do this and compute the gender bias of this classifier on the training data I get the following.

>>> indian = lambda x: x[47] == 1
>>> len([x for x in train[0] if indian(x)]) / len(train[0]) # fraction of Indians
0.0030711587481956942
>>> signedBias(train[0], indian, 1, 0)
0.0030631816119030884


So this says that predicting based on being of Indian origin (which probably has very low accuracy, since many non-Indians make at least \$50k) does not bias significantly with respect to gender.

We can generalize statistical parity in various ways, such as using some other specified set $T$ in place of $S^C$, or looking at discrepancies among $k$ different sub-populations or with $m$ different outcome labels. In fact, the mathematical name for this measurement (which is a measurement of a set of distributions) is called the total variation distance. The form we sketched here is a simple case that just works for the binary-label two-class scenario.

Now it is important to note that statistical parity says nothing about the truth about the protected class $S$. I mean two things by this. First, you could have some historical data you want to train a classifier $h$ on, and usually you’ll be given training labels for the data that tell you whether $h(x)$ should be $1$ or $-1$. In the absence of discrimination, getting high accuracy with respect to the training data is enough. But if there is some historical discrimination against $S$ then the training labels are not trustworthy. As a consequence, achieving statistical parity for $S$ necessarily reduces the accuracy of $h$. In other words, when there is bias in the data accuracy is measured in favor of encoding the bias. Studying fairness from this perspective means you study the tradeoff between high accuracy and low statistical disparity. However, and this is why statistical parity says nothing about whether the individuals $h$ behaves differently on (differently compared to the training labels) were the correct individuals to behave differently on. If the labels alone are all we have to work with, and we don’t know the true labels, then we’d need to apply domain-specific knowledge, which is suddenly out of scope of machine learning.

Second, nothing says optimizing for statistical parity is the correct thing to do. In other words, it may be that teal-haired people are truly less creditworthy (jokingly, maybe there is a hidden innate characteristic causing both uncreditworthiness and a desire to dye your hair!) and by enforcing statistical parity you are going against a fact of Nature. Though there are serious repercussions for suggesting such things in real life, my point is that statistical parity does not address anything outside the desire for an algorithm to exhibit a certain behavior. The obvious counterargument is that if, as a society, we have decided that teal-hairedness should be protected by law regardless of Nature, then we’re defining statistical parity to be correct. We’re changing our optimization criterion and as algorithm designers we don’t care about anything else. We care about what guarantees we can prove about algorithms, and the utility of the results.

The third side of the coin is that if all we care about is statistical parity, then we’ll have a narrow criterion for success that can be gamed by an actively biased adversary.

## Statistical parity versus targeted bias

Statistical parity has some known pitfalls. In their paper “Fairness Through Awareness” (Section 3.1 and Appendix A), Dwork, et al. argue convincingly that these are primarily issues of individual fairness and targeted discrimination. They give six examples of “evils” including a few that maintain statistical parity while not being fair from the perspective of an individual. Here are my two favorite ones to think about (using teal-haired people and loans again):

1. Self-fulfilling prophecy: The bank intentionally gives a few loans to teal-haired people who are (for unrelated reasons) obviously uncreditworthy, so that in the future they can point to these examples to justify discriminating against teals. This can appear even if the teals are chosen uniformly at random, since the average creditworthiness of a random teal-haired person is lower than a carefully chosen normal-haired person.
2. Reverse tokenism: The bank intentionally does not give loans to some highly creditworthy normal-haired people, let’s call one Martha, so that when a teal complains that they are denied a loan, the bank can point to Martha and say, “Look how qualified she is, and we didn’t even give her a loan! You’re much less qualified.” Here Martha is the “token” example used to justify discrimination against teals.

I like these two examples for two reasons. First, they illustrate how hard coming up with a good definition is: it’s not clear how to encapsulate both statistical parity and resistance to this kind of targeted discrimination. Second, they highlight that discrimination can both be unintentional and intentional. Since computer scientists tend to work with worst-case guarantees, this makes we think the right definition will be resilient to some level of adversarial discrimination. But again, these two examples are not formalized, and it’s not even clear to what extent existing algorithms suffer from manipulations of these kinds. For instance, many learning algorithms are relatively resilient to changing the desired label of a single point.

In any case, the thing to take away from this discussion is that there is not yet an accepted definition of “fairness,” and there seems to be a disconnect between what it means to be fair for an individual versus a population. There are some other proposals in the literature, and I’ll just mention one: Dwork et al. propose that individual fairness mean that “similar individuals are treated similarly.” I will cover this notion (and what’s know about it) in a future post.

Until then!

# The Welch-Berlekamp Algorithm for Correcting Errors in Data

In this post we’ll implement Reed-Solomon error-correcting codes and use them to play with codes. In our last post we defined Reed-Solomon codes rigorously, but in this post we’ll focus on intuition and code. As usual the code and data used in this post is available on this blog’s Github page.

The main intuition behind Reed-Solomon codes (and basically all the historically major codes) is

Error correction is about adding redundancy, and polynomials are a really efficient way to do that.

Here’s an example of what we’ll do in the post. Say you have a space probe flying past Mars taking photographs like this one

Courtesy of NASA’s Viking Orbiter.

Unfortunately you know that if you send the images back to Earth via radio waves, the signal will get corrupted by cosmic something-or-other and you’ll end up with an image like this.

How can you recover from errors like this? You could do something like repeat each pixel twice in the message so that if one is corrupted the other will get through. But still, every now and then both pixels in a row will be corrupted and it’s twice as inefficient.

The idea of error-correcting codes is to find a way to encode a message so that it adds a lot of redundancy without adding too much extra information to the message. The name of the game is to optimize the tradeoff between how much redundancy you get and how much longer the message needs to be, while still being able to efficiently decode the encoded message.

A solid technique turns out to be: use polynomials. Even though you’d think polynomials are too simple (we teach them starting in the 7th grade these days!) they turn out to have remarkable properties. The most important of which is:

if you give me a bunch of points in the plane with different $x$ coordinates, they uniquely define a polynomial of a certain degree.

This fact is called polynomial interpolation. We used it in a previous post to share secrets, if you’re interested.

What makes polynomials great for error correction is that you can take a fixed polynomial (think, the message) and “encode” it as a list of points on that polynomial. If you include enough, then you can get back the original polynomial from the points alone. And the best part, for each two additional points you include above the minimum, you get resilience to one additional error no matter where it happens in the message. Another way to say this is, even if some of the points in your encoded message are wrong (the numbers are modified by an adversary or random noise), as long as there aren’t too many errors there is an algorithm that can recover the errors.

That’s what makes polynomials so much better than the naive idea of repeating every pixel twice: once you allow for three errors you run the risk of losing a pixel, but you had to double your communication costs. With a polynomial-based approach you’d only need to store around six extra pixels worth of data to get resilience to three errors that can happen anywhere. What a bargain!

Here’s the official theorem about Reed-Solomon codes:

Theorem: There is an efficient algorithm which, when given points $(a_1, b_1), \dots, (a_n, b_n)$ with distinct $a_i$ has the following property. If there is a polynomial of degree $d$ that passes through at least $n/2 + d/2$ of the given points, then the algorithm will output the polynomial.

So let’s implement the encoder, decoder, and turn the theorem into code!

## Implementing the encoder

The way you write a message of length $k$ as a polynomial is easy. Pick a large prime integer $p$ and from now on we’ll do all our arithmetic modulo $p$. Then encode each character $c_0, c_1, \dots, c_{k-1}$ in the message as an integer between 0 and $p-1$ (this is why $p$ needs to be large enough), and the polynomial representing the message is

$m(x) = c_0 + c_1x + c_2x^2 + \dots + c_{k-1}x^{k-1}$

If the message has length $k$ then the polynomial will have degree $k-1$.

Now to encode the message we just pick a bunch of $x$ values and plug them into the polynomial, and record the (input, output) pairs as the encoded message. If we want to make things simple we can just require that you always pick the $x$ values $0, 1, \dots, n$ for some choice of $n \leq p$.

A quick skippable side-note: we need $p$ to be prime so that our arithmetic happens in a field. Otherwise, we won’t necessarily get unique decoded messages.

Back when we discussed elliptic curve cryptography (ironically sharing an acronym with error correcting codes), we actually wrote a little library that lets us seamlessly represent polynomials with “modular arithmetic coefficients” in Python, which in math jargon is a “finite field.” Rather than reinvent the wheel we’ll just use that code as a black box (full source in the Github repo). Here are some examples of using it.

>>> from finitefield.finitefield import FiniteField
>>> F13 = FiniteField(p=13)
>>> a = F13(7)
>>> a+9
3 (mod 13)
>>> a*a
10 (mod 13)
>>> 1/a
2 (mod 13)


A programming aside: once you construct an instance of your finite field, all arithmetic operations involving instances of that type will automatically lift integers to the appropriate type. Now to make some polynomials:

>>> from finitefield.polynomial import polynomialsOver
>>> F = FiniteField(p=13)
>>> P = polynomialsOver(F)
>>> g = P([1,3,5])
>>> g
1 + 3 t^1 + 5 t^2
>>> g*g
1 + 6 t^1 + 6 t^2 + 4 t^3 + 12 t^4
>>> g(100)
4 (mod 13)


Now to fix an encoding/decoding scheme we’ll call $k$ the size of the unencoded message, $n$ the size of the encoded message, and $p$ the modulus, and we’ll fix these programmatically when the encoder and decoder are defined so we don’t have to keep carrying these data around.

def makeEncoderDecoder(n, k, p):
Fp = FiniteField(p)
Poly = polynomialsOver(Fp)

def encode(message):
...

def decode(encodedMessage):
...

return encode, decode


Encode is the easier of the two.

def encode(message):
thePoly = Poly(message)
return [(Fp(i), thePoly(Fp(i))) for i in range(n)]


Technically we could remove the leading Fp(i) from each tuple, since the decoder algorithm can assume we’re using the first $n$ integers in order. But we’ll leave it in and define the decode function more generically.

After we define how the decoder should work in theory we’ll run through a simple example step by step. Now on to the decoder.

## The decoding algorithm, Berlekamp-Welch

There are a lot of different decoding algorithms for various error correcting codes. The one we’ll implement is called the Berlekamp-Welch algorithm, but before we get to it we should mention a much simpler algorithm that will work when there are only a few errors.

To remind us of notation, call $k$ the length of the message, so that $k-1$ is the degree of the polynomial we used to encode it. And $n$ is the number of points we used in the encoding. Call the encoded message $M$ as it’s received (as a list of points, possibly with errors).

In the simple method what you do is just randomly pick $k$ points from $M$, do polynomial interpolation on the chosen points to get some polynomial $g$, and see if $g$ agrees with most of the points in $M$. If there really are few errors, then there’s a good chance the randomly chosen points won’t have any errors in them and you’ll win. If you get unlucky and pick some points with errors, then the $g$ you get won’t agree with most of $M$ and you can throw it out and try again. If you get really unlucky and a bad $g$ does agree with most of $M$, then you just run this procedure a few hundred times and take the $g$ you get most often. But again, this only works with a small number of errors and while it could be good enough for many applications, don’t bet your first-born child’s life on it working. Or even your favorite pencil, for that matter. We’re going to implement Berlekamp-Welch so you can win someone else’s favorite pencil. You’re welcome.

Exercise: Implement the simple decoding algorithm and test it on some data.

Suppose we are guaranteed that there are exactly $e < \frac{n-k+1}{2}$ errors in our received message $M = (a_1, b_1, \dots, a_n, b_n)$. Call the polynomial that represents the original message $P$. In other words, we have that $P(a_i) = b_i$ for all but $e$ of the points in $M$.

There are two key ingredients in the algorithm. The first is called the error locator polynomial. We’ll call this polynomial $E(x)$, and it’s just defined by being zero wherever the errors occurred. In symbols, $E(a_i) = 0$ whenever $P(a_i) \neq b_i$. If we knew where the errors occurred, we could write out $E(x)$ explicitly as a product of terms like $(x-a_i)$. And if we knew $E$ we’d also be done, because it would tell us where the errors were and we could do interpolation on all the non-error points in $M$.

So we’re going to have to study $E$ indirectly and use it to get $P$. One nice property of $E(x)$ is the following

$\displaystyle b_i E(a_i) = P(a_i)E(a_i),$

which is true for every pair $(a_i, b_i) \in M$. Indeed, by definition when $P(a_i) \neq b_i$ then $E(a_i) = 0$ so both sides are zero. Now we can use a technique called linearization. It goes like this. The product $P(x) E(x)$, i.e. the right-hand-side of the above equation, is a polynomial, say $Q(x)$, of larger degree ($e + k - 1$). We get the equation for all $i$:

$\displaystyle b_i E(a_i) = Q(a_i)$

Now $E$, $Q$, and $P$ are all unknown, but it turns out that we can actually find $E$ and $Q$ efficiently. Or rather, we can’t guarantee we’ll find $E$ and $Q$ exactly, instead we’ll find two polynomials that have the same quotient as $Q(x)/E(x) = P(x)$. Here’s how that works.

Say we wrote out $E(x)$ as a generic polynomial of degree $e$ and $Q(x)$ as a generic polynomial of degree $e+k-1$. So their coefficients are unspecified variables. Now we can plug in all the points $a_i, b_i$ to the equations $b_i E(a_i) = Q(a_i)$, and this will form a linear system of $2e + k-1$ unknowns ($e$ unknowns come from $E(x)$ and $e+k-1$ come from $Q(x)$).

Now we know that this system has good solution, because if we take the true error locator polynomial and $Q = E(x)P(x)$ with the true $P(x)$ we win. The worry is that we’ll solve this system and get two different polynomials $Q', E'$ whose quotient will be something crazy and unrelated to $P$. But as it turns out this will never happen, and any solution will give the quotient $P$. Here’s a proof you can skip if you hate proofs.

Proof. Say you have two pairs of solutions to the system, $(Q_1, E_1)$ and $(Q_2, E_2)$, and you want to show that $Q_1/E_1 = Q_2/E_2$. Well, they might not be divisible, but we can multiply the previous equation through to get $Q_1E_2 = Q_2E_1$. Now we show two polynomials are equal in the same way as always: subtract and show there are too many roots. Define $R(x) = Q_1E_2 - Q_2E_1$. The claim is that $R(x)$ has $n$ roots, one for every point $(a_i, b_i)$. Indeed,

$\displaystyle R(a_i) = (b_i E_1(a_i))E_2(a_i) - (b_iE_2(a_i)) E_1(a_i) = 0$

But the degree of $R(x)$ is $2e + k - 1$ which is less than $n$ by the assumption that $e < \frac{n-k+1}{2}$. So $R(x)$ has too many roots and must be the zero polynomial, and the two quotients are equal.

$\square$

So the core python routine is just two steps: solve the linear equation, and then divide two polynomials. However, it turns out that no python module has any decent support for solving linear systems of equations over finite fields.  Luckily, I wrote a linear solver way back when and so we’ll adapt it to our purposes. I’ll leave out the gory details of the solver itself, but you can see them in the source for this post. Here is the code that sets up the system

   def solveSystem(encodedMessage):
for e in range(maxE, 0, -1):
ENumVars = e+1
QNumVars = e+k
def row(i, a, b):
return ([b * a**j for j in range(ENumVars)] +
[-1 * a**j for j in range(QNumVars)] +
[0]) # the "extended" part of the linear system

system = ([row(i, a, b) for (i, (a,b)) in enumerate(encodedMessage)] +
[[0] * (ENumVars-1) + [1] + [0] * (QNumVars) + [1]])
# ensure coefficient of x^e in E(x) is 1

solution = someSolution(system, freeVariableValue=1)
E = Poly([solution[j] for j in range(e + 1)])
Q = Poly([solution[j] for j in range(e + 1, len(solution))])

P, remainder = Q.__divmod__(E)
if remainder == 0:
return Q, E

raise Exception("found no divisors!")

def decode(encodedMessage):
Q,E = solveSystem(encodedMessage)

P, remainder = Q.__divmod__(E)
if remainder != 0:
raise Exception("Q is not divisibly by E!")

return P.coefficients


## A simple example

Now let’s go through an extended example with small numbers. Let’s work modulo 7 and say that our message is

2, 3, 2 (mod 7)


In particular, $k=3$ is the length of the message. We’ll encode it as a polynomial in the way we described:

$\displaystyle m(x) = 2 + 3x + 2x^2 (\mod 7)$

If we pick $n = 5$, then we will encode the message as a sequence of five points on $m(x)$, namely $m(0)$ through $m(4)$.

[[0, 2], [1, 0], [2, 2], [3, 1], [4, 4]] (mod 7)


Now let’s add a single error. First remember that our theoretical guarantee says that we can correct any number of errors up to $\frac{n+k-1}{2} - 1$, which in this case is $[(5+3-1) / 2] - 1 = 2$, so we can definitely correct one error. We’ll add 1 to the third point, giving the received corrupted message as

[[0, 2], [1, 0], [2, 3], [3, 1], [4, 4]] (mod 7)


Now we set up the system of equations $b_i E(a_i) = Q(a_i)$ for all $(a_i, b_i)$ above. Rewriting the equations as $b_iE(a_i) - Q(a_i) = 0$, and adding as the last equation the constraint that $x^e = 1$. The columns represent the variables, with the last column being the right-hand-side of the equality as is the standard for Gaussian elimination.

# e0 e1 q0 q1 q2 q3
[
[2, 0, 6, 0, 0, 0, 0],
[0, 0, 6, 6, 6, 6, 0],
[3, 6, 6, 5, 3, 6, 0],
[1, 3, 6, 4, 5, 1, 0],
[4, 2, 6, 3, 5, 6, 0],
[0, 1, 0, 0, 0, 0, 1],
]


Then we do row-reduction to get

[
[1, 0, 0, 0, 0, 0, 5],
[0, 1, 0, 0, 0, 0, 1],
[0, 0, 1, 0, 0, 0, 3],
[0, 0, 0, 1, 0, 0, 3],
[0, 0, 0, 0, 1, 0, 6],
[0, 0, 0, 0, 0, 1, 2]
]


And reading off the solution gives $E(x) = 5 + x$ and $Q(x) = 2 + 2x + 6x^2 + 2x^3$. Note in particular that the $E(x)$ given in this solution is not the error locator polynomial! Nevertheless, the quotient of the two polynomials is exactly $m(x) = 2 + 3x + 2x^2$ which gives back the original message.

There is one catch here: how does one determine the value of $e$ to use in setting up the system of linear equations? It turns out that an upper bound on $e$ will work just fine, so long as the upper bound you use agrees with the theoretical maximum number of errors allowed (see the Singleton bound from last time). The effect of doing this is that the linear system ends up with some number of free variables that you can set to arbitrary values, and these will correspond to additional shared roots of $E(x)$ and $Q(x)$ that cancel out upon dividing.

## A larger example

Now it’s time for a sad fact. I tried running Welch-Berlekamp on an encoded version of the following tiny image:

And it didn’t finish after running all night.

Berlekamp-Welch is a slow algorithm for decoding Reed-Solomon codes because it requires one to solve a large system of equations. There’s at least one equation for each pixel in a black and white image! To get around this one typically encodes blocks of pixels together into one message character (since $p$ is larger than $n > k$ there is lots of space), and apparently one can balance it to minimize the number of equations. And finally, a nontrivial inefficiency comes from our implementation of everything in Python without optimizations. If we rewrote everything in C++ or Go and fixed the prime modulus, we would likely see reasonable running times. There are also asymptotically much faster methods based on the fast Fourier transform, and in the future we’ll try implementing some of these. For the dedicated reader, these are all good follow-up projects.

For now we’ll just demonstrate that it works by running it on a larger sample of text, the introductory paragraphs of To Kill a Mockingbird:

def tkamTest():
message = '''When he was nearly thirteen, my brother Jem got his arm badly broken at the elbow.  When it healed, and Jem's fears of never being able to play football were assuaged, he was seldom   self-conscious about his injury. His left arm was somewhat shorter than his right; when he stood or walked, the back of his hand was at right angles to his body, his thumb parallel to his thigh. He   couldn't have cared less, so long as he could pass and punt.'''

k = len(message)
n = len(message) * 2
p = 2087
integerMessage = [ord(x) for x in message]

enc, dec, solveSystem = makeEncoderDecoder(n, k, p)
print("encoding...")
encoded = enc(integerMessage)

e = int(k/2)
print("corrupting...")
corrupted = corrupt(encoded[:], e, 0, p)

print("decoding...")
Q,E = solveSystem(corrupted)
P, remainder = (Q.__divmod__(E))

recovered = ''.join([chr(x) for x in P.coefficients])
print(recovered)


Running this with unix time produces the following:

encoding...
corrupting...
decoding...
When he was nearly thirteen, my brother Jem got his arm badly broken at the elbow. When it healed, and Jem's fears of never being able to play football were assuaged, he was seldom self-conscious about his injury. His left arm was somewhat shorter than his right; when he stood or walked, the back of his hand was at right angles to his body, his thumb parallel to his thigh. He couldn't have cared less, so long as he could pass and punt.

real	82m9.813s
user	81m18.891s
sys	0m27.404s


So it finishes in “only” an hour or so.

In any case, the decoding algorithm is an interesting one. In future posts we’ll explore more efficient algorithms and faster implementations.

Until then!

# The Many Faces of Set Cover

A while back Peter Norvig posted a wonderful pair of articles about regex golf. The idea behind regex golf is to come up with the shortest possible regular expression that matches one given list of strings, but not the other.

“Regex Golf,” by Randall Munroe.

In the first article, Norvig runs a basic algorithm to recreate and improve the results from the comic, and in the second he beefs it up with some improved search heuristics. My favorite part about this topic is that regex golf can be phrased in terms of a problem called set cover. I noticed this when reading the comic, and was delighted to see Norvig use that as the basis of his algorithm.

The set cover problem shows up in other places, too. If you have a database of items labeled by users, and you want to find the smallest set of labels to display that covers every item in the database, you’re doing set cover. I hear there are applications in biochemistry and biology but haven’t seen them myself.

If you know what a set is (just think of the “set” or “hash set” type from your favorite programming language), then set cover has a simple definition.

Definition (The Set Cover Problem): You are given a finite set $U$ called a “universe” and sets $S_1, \dots, S_n$ each of which is a subset of $U$. You choose some of the $S_i$ to ensure that every $x \in U$ is in one of your chosen sets, and you want to minimize the number of $S_i$ you picked.

It’s called a “cover” because the sets you pick “cover” every element of $U$. Let’s do a simple. Let $U = \{ 1,2,3,4,5 \}$ and

$\displaystyle S_1 = \{ 1,3,4 \}, S_2 = \{ 2,3,5 \}, S_3 = \{ 1,4,5 \}, S_4 = \{ 2,4 \}$

Then the smallest possible number of sets you can pick is 2, and you can achieve this by picking both $S_1, S_2$ or both $S_2, S_3$. The connection to regex golf is that you pick $U$ to be the set of strings you want to match, and you pick a set of regexes that match some of the strings in $U$ but none of the strings you want to avoid matching (I’ll call them $V$). If $w$ is such a regex, then you can form the set $S_w$ of strings that $w$ matches. Then if you find a small set cover with the strings $w_1, \dots, w_t$, then you can “or” them together to get a single regex $w_1 \mid w_2 \mid \dots \mid w_t$ that matches all of $U$ but none of $V$.

Set cover is what’s called NP-hard, and one implication is that we shouldn’t hope to find an efficient algorithm that will always give you the shortest regex for every regex golf problem. But despite this, there are approximation algorithms for set cover. What I mean by this is that there is a regex-golf algorithm $A$ that outputs a subset of the regexes matching all of $U$, and the number of regexes it outputs is such-and-such close to the minimum possible number. We’ll make “such-and-such” more formal later in the post.

What made me sad was that Norvig didn’t go any deeper than saying, “We can try to approximate set cover, and the greedy algorithm is pretty good.” It’s true, but the ideas are richer than that! Set cover is a simple example to showcase interesting techniques from theoretical computer science. And perhaps ironically, in Norvig’s second post a header promised the article would discuss the theory of set cover, but I didn’t see any of what I think of as theory. Instead he partially analyzes the structure of the regex golf instances he cares about. This is useful, but not really theoretical in any way unless he can say something universal about those instances.

I don’t mean to bash Norvig. His articles were great! And in-depth theory was way beyond scope. So this post is just my opportunity to fill in some theory gaps. We’ll do three things:

1. Show formally that set cover is NP-hard.
2. Prove the approximation guarantee of the greedy algorithm.
3. Show another (very different) approximation algorithm based on linear programming.

Along the way I’ll argue that by knowing (or at least seeing) the details of these proofs, one can get a better sense of what features to look for in the set cover instance you’re trying to solve. We’ll also see how set cover depicts the broader themes of theoretical computer science.

## NP-hardness

The first thing we should do is show that set cover is NP-hard. Intuitively what this means is that we can take some hard problem $P$ and encode instances of $P$ inside set cover problems. This idea is called a reduction, because solving problem $P$ will “reduce” to solving set cover, and the method we use to encode instance of $P$ as set cover problems will have a small amount of overhead. This is one way to say that set cover is “at least as hard as” $P$.

The hard problem we’ll reduce to set cover is called 3-satisfiability (3-SAT). In 3-SAT, the input is a formula whose variables are either true or false, and the formula is expressed as an OR of a bunch of clauses, each of which is an AND of three variables (or their negations). This is called 3-CNF form. A simple example:

$\displaystyle (x \vee y \vee \neg z) \wedge (\neg x \vee w \vee y) \wedge (z \vee x \vee \neg w)$

The goal of the algorithm is to decide whether there is an assignment to the variables which makes the formula true. 3-SAT is one of the most fundamental problems we believe to be hard and, roughly speaking, by reducing it to set cover we include set cover in a class called NP-complete, and if any one of these problems can be solved efficiently, then they all can (this is the famous P versus NP problem, and an efficient algorithm would imply P equals NP).

So a reduction would consist of the following: you give me a formula $\varphi$ in 3-CNF form, and I have to produce (in a way that depends on $\varphi$!) a universe $U$ and a choice of subsets $S_i \subset U$ in such a way that

$\varphi$ has a true assignment of variables if and only if the corresponding set cover problem has a cover using $k$ sets.

In other words, I’m going to design a function $f$ from 3-SAT instances to set cover instances, such that $x$ is satisfiable if and only if $f(x)$ has a set cover with $k$ sets.

Why do I say it only for $k$ sets? Well, if you can always answer this question then I claim you can find the minimum size of a set cover needed by doing a binary search for the smallest value of $k$. So finding the minimum size of a set cover reduces to the problem of telling if theres a set cover of size $k$.

Now let’s do the reduction from 3-SAT to set cover.

If you give me $\varphi = C_1 \wedge C_2 \wedge \dots \wedge C_m$ where each $C_i$ is a clause and the variables are denoted $x_1, \dots, x_n$, then I will choose as my universe $U$ to be the set of all the clauses and indices of the variables (these are all just formal symbols). i.e.

$\displaystyle U = \{ C_1, C_2, \dots, C_m, 1, 2, \dots, n \}$

The first part of $U$ will ensure I make all the clauses true, and the last part will ensure I don’t pick a variable to be both true and false at the same time.

To show how this works I have to pick my subsets. For each variable $x_i$, I’ll make two sets, one called $S_{x_i}$ and one called $S_{\neg x_i}$. They will both contain $i$ in addition to the clauses which they make true when the corresponding literal is true (by literal I just mean the variable or its negation). For example, if $C_j$ uses the literal $\neg x_7$, then $S_{\neg x_7}$ will contain $C_j$ but $S_{x_7}$ will not. Finally, I’ll set $k = n$, the number of variables.

Now to prove this reduction works I have to prove two things: if my starting formula has a satisfying assignment I have to show the set cover problem has a cover of size $k$. Indeed, take the sets $S_{y}$ for all literals $y$ that are set to true in a satisfying assignment. There can be at most $n$ true literals since half are true and half are false, so there will be at most $n$ sets, and these sets clearly cover all of $U$ because every literal has to be satisfied by some literal or else the formula isn’t true.

The reverse direction is similar: if I have a set cover of size $n$, I need to use it to come up with a satisfying truth assignment for the original formula. But indeed, the sets that get chosen can’t include both a $S_{x_i}$ and its negation set $S_{\neg x_i}$, because there are $n$ of the elements $\{1, 2, \dots, n \} \subset U$, and each $i$ is only in the two $S_{x_i}, S_{\neg x_i}$. Just by counting if I cover all the indices $i$, I already account for $n$ sets! And finally, since I have covered all the clauses, the literals corresponding to the sets I chose give exactly a satisfying assignment.

Whew! So set cover is NP-hard because I encoded this logic problem 3-SAT within its rules. If we think 3-SAT is hard (and we do) then set cover must also be hard. So if we can’t hope to solve it exactly we should try to approximate the best solution.

## The greedy approach

The method that Norvig uses in attacking the meta-regex golf problem is the greedy algorithm. The greedy algorithm is exactly what you’d expect: you maintain a list $L$ of the subsets you’ve picked so far, and at each step you pick the set $S_i$ that maximizes the number of new elements of $U$ that aren’t already covered by the sets in $L$. In python pseudocode:

def greedySetCover(universe, sets):
chosenSets = set()
leftToCover = universe.copy()
unchosenSets = sets

covered = lambda s: leftToCover & s

while universe != 0:
if len(chosenSets) == len(sets):
raise Exception("No set cover possible")

nextSet = max(unchosenSets, key=lambda s: len(covered(s)))
unchosenSets.remove(nextSet)
leftToCover -= nextSet

return chosenSets


This is what theory has to say about the greedy algorithm:

Theorem: If it is possible to cover $U$ by the sets in $F = \{ S_1, \dots, S_n \}$, then the greedy algorithm always produces a cover that at worst has size $O(\log(n)) \textup{OPT}$, where $\textup{OPT}$ is the size of the smallest cover. Moreover, this is asymptotically the best any algorithm can do.

One simple fact we need from calculus is that the following sum is asymptotically the same as $\log(n)$:

$\displaystyle H(n) = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} = \log(n) + O(1)$

Proof. [adapted from Wan] Let’s say the greedy algorithm picks sets $T_1, T_2, \dots, T_k$ in that order. We’ll set up a little value system for the elements of $U$. Specifically, the value of each $T_i$ is 1, and in step $i$ we evenly distribute this unit value across all newly covered elements of $T_i$. So for $T_1$ each covered element gets value $1/|T_1|$, and if $T_2$ covers four new elements, each gets a value of 1/4. One can think of this “value” as a price, or energy, or unit mass, or whatever. It’s just an accounting system (albeit a clever one) we use to make some inequalities clear later.

In general call the value $v_x$ of element $x \in U$ the value assigned to $x$ at the step where it’s first covered. In particular, the number of sets chosen by the greedy algorithm $k$ is just $\sum_{x \in U} v_x$. We’re just bunching back together the unit value we distributed for each step of the algorithm.

Now we want to compare the sets chosen by greedy to the optimal choice. Call a smallest set cover $C_{\textup{OPT}}$. Let’s stare at the following inequality.

$\displaystyle \sum_{x \in U} v_x \leq \sum_{S \in C_{\textup{OPT}}} \sum_{x \in S} v_x$

It’s true because each $x$ counts for a $v_x$ at most once in the left hand side, and in the right hand side the sets in $C_{\textup{OPT}}$ must hit each $x$ at least once but may hit some $x$ more than once. Also remember the left hand side is equal to $k$.

Now we want to show that the inner sum on the right hand side, $\sum_{x \in S} v_x$, is at most $H(|S|)$. This will in fact prove the entire theorem: because each set $S_i$ has size at most $n$, the inequality above will turn into

$\displaystyle k \leq |C_{\textup{OPT}}| H(|S|) \leq |C_{\textup{OPT}}| H(n)$

And so $k \leq \textup{OPT} \cdot O(\log(n))$, which is the statement of the theorem.

So we want to show that $\sum_{x \in S} v_x \leq H(|S|)$. For each $j$ define $\delta_j(S)$ to be the number of elements in $S$ not covered in $T_1, \cup \dots \cup T_j$. Notice that $\delta_{j-1}(S) - \delta_{j}(S)$ is the number of elements of $S$ that are covered for the first time in step $j$. If we call $t_S$ the smallest integer $j$ for which $\delta_j(S) = 0$, we can count up the differences up to step $t_S$, we get

$\sum_{x \in S} v_x = \sum_{i=1}^{t_S} (\delta_{i-1}(S) - \delta_i(S)) \cdot \frac{1}{T_i - (T_1 \cup \dots \cup T_{i-1})}$

The rightmost term is just the cost assigned to the relevant elements at step $i$. Moreover, because $T_i$ covers more new elements than $S$ (by definition of the greedy algorithm), the fraction above is at most $1/\delta_{i-1}(S)$. The end is near. For brevity I’ll drop the $(S)$ from $\delta_j(S)$.

\displaystyle \begin{aligned} \sum_{x \in S} v_x & \leq \sum_{i=1}^{t_S} (\delta_{i-1} - \delta_i) \frac{1}{\delta_{i-1}} \\ & \leq \sum_{i=1}^{t_S} (\frac{1}{1 + \delta_i} + \frac{1}{2+\delta_i} \dots + \frac{1}{\delta_{i-1}}) \\ & = \sum_{i=1}^{t_S} H(\delta_{i-1}) - H(\delta_i) \\ &= H(\delta_0) - H(\delta_{t_S}) = H(|S|) \end{aligned}

And that proves the claim.

$\square$

I have three postscripts to this proof:

1. This is basically the exact worst-case approximation that the greedy algorithm achieves. In fact, Petr Slavik proved in 1996 that the greedy gives you a set of size exactly $(\log n - \log \log n + O(1)) \textup{OPT}$ in the worst case.
2. This is also the best approximation that any set cover algorithm can achieve, provided that P is not NP. This result was basically known in 1994, but it wasn’t until 2013 and the use of some very sophisticated tools that the best possible bound was found with the smallest assumptions.
3. In the proof we used that $|S| \leq n$ to bound things, but if we knew that our sets $S_i$ (i.e. subsets matched by a regex) had sizes bounded by, say, $B$, the same proof would show that the approximation factor is $\log(B)$ instead of $\log n$. However, in order for that to be useful you need $B$ to be a constant, or at least to grow more slowly than any polynomial in $n$, since e.g. $\log(n^{0.1}) = 0.1 \log n$. In fact, taking a second look at Norvig’s meta regex golf problem, some of his instances had this property! Which means the greedy algorithm gives a much better approximation ratio for certain meta regex golf problems than it does for the worst case general problem. This is one instance where knowing the proof of a theorem helps us understand how to specialize it to our interests.

Norvig’s frequency table for president meta-regex golf. The left side counts the size of each set (defined by a regex)

## The linear programming approach

So we just said that you can’t possibly do better than the greedy algorithm for approximating set cover. There must be nothing left to say, job well done, right? Wrong! Our second analysis, based on linear programming, shows that instances with special features can have better approximation results.

In particular, if we’re guaranteed that each element $x \in U$ occurs in at most $B$ of the sets $S_i$, then the linear programming approach will give a $B$-approximation, i.e. a cover whose size is at worst larger than OPT by a multiplicative factor of $B$. In the case that $B$ is constant, we can beat our earlier greedy algorithm.

The technique is now a classic one in optimization, called LP-relaxation (LP stands for linear programming). The idea is simple. Most optimization problems can be written as integer linear programs, that is there you have $n$ variables $x_1, \dots, x_n \in \{ 0, 1 \}$ and you want to maximize (or minimize) a linear function of the $x_i$ subject to some linear constraints. The thing you’re trying to optimize is called the objective. While in general solving integer linear programs is NP-hard, we can relax the “integer” requirement to $0 \leq x_i \leq 1$, or something similar. The resulting linear program, called the relaxed program, can be solved efficiently using the simplex algorithm or another more complicated method.

The output of solving the relaxed program is an assignment of real numbers for the $x_i$ that optimizes the objective function. A key fact is that the solution to the relaxed linear program will be at least as good as the solution to the original integer program, because the optimal solution to the integer program is a valid candidate for the optimal solution to the linear program. Then the idea is that if we use some clever scheme to round the $x_i$ to integers, we can measure how much this degrades the objective and prove that it doesn’t degrade too much when compared to the optimum of the relaxed program, which means it doesn’t degrade too much when compared to the optimum of the integer program as well.

If this sounds wishy washy and vague don’t worry, we’re about to make it super concrete for set cover.

We’ll make a binary variable $x_i$ for each set $S_i$ in the input, and $x_i = 1$ if and only if we include it in our proposed cover. Then the objective function we want to minimize is $\sum_{i=1}^n x_i$. If we call our elements $X = \{ e_1, \dots, e_m \}$, then we need to write down a linear constraint that says each element $e_j$ is hit by at least one set in the proposed cover. These constraints have to depend on the sets $S_i$, but that’s not a problem. One good constraint for element $e_j$ is

$\displaystyle \sum_{i : e_j \in S_i} x_i \geq 1$

In words, the only way that an $e_j$ will not be covered is if all the sets containing it have their $x_i = 0$. And we need one of these constraints for each $j$. Putting it together, the integer linear program is

The integer program for set cover.

Once we understand this formulation of set cover, the relaxation is trivial. We just replace the last constraint with inequalities.

For a given candidate assignment $x$ to the $x_i$, call $Z(x)$ the objective value (in this case $\sum_i x_i$). Now we can be more concrete about the guarantees of this relaxation method. Let $\textup{OPT}_{\textup{IP}}$ be the optimal value of the integer program and $x_{\textup{IP}}$ a corresponding assignment to $x_i$ achieving the optimum. Likewise let $\textup{OPT}_{\textup{LP}}, x_{\textup{LP}}$ be the optimal things for the linear relaxation. We will prove:

Theorem: There is a deterministic algorithm that rounds $x_{\textup{LP}}$ to integer values $x$ so that the objective value $Z(x) \leq B \textup{OPT}_{\textup{IP}}$, where $B$ is the maximum number of sets that any element $e_j$ occurs in. So this gives a $B$-approximation of set cover.

Proof. Let $B$ be as described in the theorem, and call $y = x_{\textup{LP}}$ to make the indexing notation easier. The rounding algorithm is to set $x_i = 1$ if $y_i \geq 1/B$ and zero otherwise.

To prove the theorem we need to show two things hold about this new candidate solution $x$:

1. The choice of all $S_i$ for which $x_i = 1$ covers every element.
2. The number of sets chosen (i.e. $Z(x)$) is at most $B$ times more than $\textup{OPT}_{\textup{LP}}$.

Since $\textup{OPT}_{\textup{LP}} \leq \textup{OPT}_{\textup{IP}}$, so if we can prove number 2 we get $Z(x) \leq B \textup{OPT}_{\textup{LP}} \leq B \textup{OPT}_{\textup{IP}}$, which is the theorem.

So let’s prove 1. Fix any $j$ and we’ll show that element $e_j$ is covered by some set in the rounded solution. Call $B_j$ the number of times element $e_j$ occurs in the input sets. By definition $B_j \leq B$, so $1/B_j \geq 1/B$. Recall $y$ was the optimal solution to the relaxed linear program, and so it must be the case that the linear constraint for each $e_j$ is satisfied: $\sum_{i : e_j \in S_i} x_i \geq 1$. We know that there are $B_j$ terms and they sums to at least 1, so not all terms can be smaller than $1/B_j$ (otherwise they’d sum to something less than 1). In other words, some variable $x_i$ in the sum is at least $1/B_j \geq 1/B$, and so $x_i$ is set to 1 in the rounded solution, corresponding to a set $S_i$ that contains $e_j$. This finishes the proof of 1.

Now let’s prove 2. For each $j$, we know that for each $x_i = 1$, the corresponding variable $y_i \geq 1/B$. In particular $1 \leq y_i B$. Now we can simply bound the sum.

\displaystyle \begin{aligned} Z(x) = \sum_i x_i &\leq \sum_i x_i (B y_i) \\ &\leq B \sum_{i} y_i \\ &= B \cdot \textup{OPT}_{\textup{LP}} \end{aligned}

The second inequality is true because some of the $x_i$ are zero, but we can ignore them when we upper bound and just include all the $y_i$. This proves part 2 and the theorem.

$\square$

1. The proof works equally well when the sets are weighted, i.e. your cost for picking a set is not 1 for every set but depends on some arbitrarily given constants $w_i \geq 0$.
2. We gave a deterministic algorithm rounding $y$ to $x$, but one can get the same result (with high probability) using a randomized algorithm. The idea is to flip a coin with bias $y_i$ roughly $\log(n)$ times and set $x_i = 1$ if and only if the coin lands heads at least once. The guarantee is no better than what we proved, but for some other problems randomness can help you get approximations where we don’t know of any deterministic algorithms to get the same guarantees. I can’t think of any off the top of my head, but I’m pretty sure they’re out there.
3. For step 1 we showed that at least one term in the inequality for $e_j$ would be rounded up to 1, and this guaranteed we covered all the elements. A natural question is: why not also round up at most one term of each of these inequalities? It might be that in the worst case you don’t get a better guarantee, but it would be a quick extra heuristic you could use to post-process a rounded solution.
4. Solving linear programs is slow. There are faster methods based on so-called “primal-dual” methods that use information about the dual of the linear program to construct a solution to the problem. Goemans and Williamson have a nice self-contained chapter on their website about this with a ton of applications.

I wrote a post a while back giving conditions which, if a problem satisfies those conditions, the greedy algorithm will give a constant-factor approximation. This is much better than the worst case $\log(n)$-approximation we saw in this post. Moreover, I also wrote a post about matroids, which is a characterization of problems where the greedy algorithm is actually optimal.