I’ll be giving a talk at Purdue University on Saturday, May 3 as part of the 65th Midwest Theory Day. If any readers happen to live in West Lafayette, Indiana and are interested in hearing about some of my recent research, you can register for free by April 28 (one week from today). Lunch and snacks are provided, and the other talks will certainly be interesting too.

Here’s the title and abstract for my talk:

Resilient Coloring and Other Combinatorial Problems

A good property of a problem instance is that it’s easy to solve. And even better property is resilience: that the instance remains easy to solve under arbitrary (but minor) perturbations. We informally define the resilience of an instance of a combinatorial problem, and discuss recent work on resilient promise problems, including resilient satisfiability and resilient graph coloring.

Two years ago, Erik Demaine and three other researchers published a fun paper to the arXiv proving that most incarnations of classic nintendo games are NP-hard. This includes almost every Super Mario Brothers, Donkey Kong, and Pokemon title. Back then I wrote a blog post summarizing the technical aspects of their work, and even gave a talk on it to a room full of curious undergraduate math majors.

But while bad tech-writers tend to interpret NP-hard as “really really hard,” the truth is more complicated. It’s really a statement about computational complexity, which has a precise mathematical formulation. Sparing the reader any technical details, here’s what NP-hard implies for practical purposes:

You should abandon hope of designing an algorithm that can solve any instance of your NP-hard problem, but many NP-hard problems have efficient practical “good-enough” solutions.

The very definition of NP-hard means that NP-hard problems need only be hard in the worst case. For illustration, the fact that Pokemon is NP-hard boils down to whether you can navigate a vastly complicated maze of trainers, some of whom are guaranteed to defeat you. It has little to do with the difficulty of the game Pokemon itself, and everything to do with whether you can stretch some subset of the game’s rules to create a really bad worst-case scenario.

So NP-hardness has very little to do with human playability, and it turns out that in practice there are plenty of good algorithms for winning at Super Mario Brothers. They work really well at beating levels designed for humans to play, but we are highly confident that they would fail to win in the worst-case levels we can cook up. Why don’t we know it for a fact? Well that’s the $ P \ne NP$ conjecture.

Since Demaine’s paper (and for a while before it) a lot of popular games have been inspected under the computational complexity lens. Recently, Candy Crush Saga was proven to be NP-hard, but the list doesn’t stop with bad mobile apps. This paper of Viglietta shows that Pac-man, Tron, Doom, Starcraft, and many other famous games all contain NP-hard rule-sets. Games like Tetris are even known to have strong hardness-of-approximation bounds. Many board games have also been studied under this lens, when you generalize them to an $ n \times n$ sized board. Chess and checkers are both what’s called EXP-complete. A simplified version of Go fits into a category called PSPACE-complete, but with the general ruleset it’s believed to be EXP-complete [1]. Here’s a list of some more classic games and their complexity status.

So we have this weird contrast: lots of NP-hard (and worse!) games have efficient algorithms that play them very well (checkers is “solved,” for example), but in the worst case we believe there is no efficient algorithm that will play these games perfectly. We could ask, “We can still write algorithms to play these games well, so what’s the point of studying their computational complexity?”

I agree with the implication behind the question: it really is just pointless fun. The mathematics involved is the very kind of nuanced manipulations that hackers enjoy: using the rules of a game to craft bizarre gadgets which, if the player is to surpass them, they must implicitly solve some mathematical problem which is already known to be hard.

But we could also turn the question right back around. Since all of these great games have really hard computational hardness properties, could we use theoretical computer science, and to a broader extent mathematics, to design great games? I claim the answer is yes.

[1] EXP is the class of problems solvable in exponential time (where the exponent is the size of the problem instance, say $ n$ for a game played on an $ n \times n$ board), so we’re saying that a perfect Chess or Checkers solver could be used to solve any problem that can be solved in exponential time. PSPACE is strictly smaller (we think; this is open): it’s the class of all problems solvable if you are allowed as much time as you want, but only a polynomial amount of space to write down your computations. ↑

A Case Study: Greedy Spiders

Greedy spiders is a game designed by the game design company Blyts. In it, you’re tasked with protecting a set of helplessly trapped flies from the jaws of a hungry spider.

A screenshot from Greedy Spiders. Click to enlarge.

In the game the spider always moves in discrete amounts (between the intersections of the strands of spiderweb) toward the closest fly. The main tool you have at your disposal is the ability to destroy a strand of the web, thus prohibiting the spider from using it. The game proceeds in rounds: you cut one strand, the spider picks a move, you cut another, the spider moves, and so on until the flies are no longer reachable or the spider devours a victim.

Aside from being totally fun, this game is obviously mathematical. For the reader who is familiar with graph theory, there’s a nice formalization of this problem.

The Greedy Spiders Problem: You are given a graph $ G_0 = (V, E_0)$ and two sets $ S_0, F \subset V$ denoting the locations of the spiders and flies, respectively. There is a fixed algorithm $ A$ that the spiders use to move. An instance of the game proceeds in rounds, and at the beginning of each round we call the current graph $ G_i = (V, E_i)$ and the current location of the spiders $ S_i$. Each round has two steps:

You pick an edge $ e \in E_i$ to delete, forming the new graph $ G_{i+1} = (V, E_i)$.

The spiders jointly compute their next move according to $ A$, and each spider moves to an adjacent vertex. Thus $ S_i$ becomes $ S_{i+1}$.

Your task is to decide whether there is a sequence of edge deletions which keeps $ S_t$ and $ F$ disjoint for all $ t \geq 0$. In other words, we want to find a sequence of edge deletions that disconnects the part of the graph containing the spiders from the part of the graph containing the flies.

This is a slightly generalized version of Greedy Spiders proper, but there are some interesting things to note. Perhaps the most obvious question is about the algorithm $ A$. Depending on your tastes you could make it adversarial, devising the smartest possible move at every step of the way. This is just as hard as asking if there is any algorithm that the spiders can use to win. To make it easier, $ A$ could be an algorithm represented by a small circuit to which the player has access, or, as it truly is in the Greedy Spiders game, it could be the greedy algorithm (and the player can exploit this).

Though I haven’t heard of the Greedy Spiders problem in the literature by any other name, it seems quite likely that it would arise naturally. One can imagine the spiders as enemies traversing a network (a city, or a virus in a computer network), and your job is to hinder their movement toward high-value targets. Perhaps people in the defense industry could use a reasonable approximation algorithm for this problem. I have little doubt that this game is NP-hard [2], but the purpose of this article is not to prove new complexity results. The point is that this natural theoretical problem is a really fun game to play! And the game designer’s job is to do what game designers love to do: add features and design levels that are fun to play.

Indeed the Greedy Spiders folks did just that: their game features some 70-odd levels, many with multiple spiders and additional tools for the player. Some examples of new tools are: the ability to delete a vertex of the graph and the ability to place a ‘decoy-fly’ which is (to the greedy-algorithm-following spiders) indistinguishable from a real fly. They player is usually given only one or two uses of these tools per level, but one can imagine that the puzzles become a lot richer.

[2]: In the adversarial case it smells like it’s PSPACE-complete, being very close to known PSPACE-hard problems like Cops and Robbers and Generalized Geography. ↑

Examples

I can point to a number of interesting problems that I can imagine turning into successful games, and I will in a moment, but before I want to make it clear that I don’t propose game developers study theoretical computer science just to turn our problems into games verbatim. No, I imagine that the wealth of problems in computer science can serve as inspiration, as a spring board into a world of interesting gameplay mechanics and puzzles. The bonus for game designers is that adding features usually makes problems harder and more interesting, and you don’t need to know anything about proofs or the details of the reductions to understand the problems themselves (you just need familiarity with the basic objects of consideration, sets, graphs, etc).

For a tangential motivation, I imagine that students would be much more willing to do math problems if they were based on ideas coming from really fun games. Indeed, people have even turned the stunningly boring chore of drawing an accurate graph of a function into a game that kids seem to enjoy. I could further imagine a game that teaches programming by first having a student play a game (based on a hard computational problem) and then write simple programs that seek to do well. Continuing with the spiders example they could play as the defender, and then switch to the role of the spider by writing the algorithm the spiders follow.

But enough rambling! Here is a short list of theoretical computer science problems for which I see game potential. None of them have, to my knowledge, been turned into games, but the common features among them all are the huge potential for creative extensions and interesting level design.

Graph Coloring

Graph coloring is one of the oldest NP-complete problems known. Given a graph $ G$ and a set of colors $ \{ 1, 2, \dots, k \}$, one seeks to choose colors for the vertices of $ G$ so that no edge connects two vertices of the same color.

Now coloring a given graph would be a lame game, so let’s spice it up. Instead of one player trying to color a graph, have two players. They’re given a $ k$-colorable graph (say, $ k$ is 3), and they take turns coloring the vertices. The first player’s goal is to arrive at a correct coloring, while the second player tries to force the first player to violate the coloring condition (that no adjacent vertices are the same color). No player is allowed to break the coloring if they have an option. Now change the colors to jewels or vegetables or something, and you have yourself an award-winning game! (Or maybe: Epic Cartographer Battles of History)

An additional modification: give the two players a graph that can’t be colored with $ k$ colors, and the first player to color a monochromatic edge is the loser. Add additional move types (contracting edges or deleting vertices, etc) to taste.

Art Gallery Problem

Given a layout of a museum, the art gallery problem is the problem of choosing the minimal number of cameras so as to cover the whole museum.

This is a classic problem in computational geometry, and is well-known to be NP-hard. In some variants (like the one pictured above) the cameras are restricted to being placed at corners. Again, this is the kind of game that would be fun with multiple players. Rather than have perfect 360-degree cameras, you could have an angular slice of vision per camera. Then one player chooses where to place the cameras (getting exponentially more points for using fewer cameras), and the opponent must traverse from one part of the museum to the other avoiding the cameras. Make the thief a chubby pig stealing eggs from birds and you have yourself a franchise.

For more spice, allow the thief some special tactics like breaking through walls and the ability to disable a single camera.

This idea has of course been the basis of many single-player stealth games (where the guards/cameras are fixed by the level designer), but I haven’t seen it done as a multiplayer game. This also brings to mind variants like the recent Nothing to Hide, which counterintuitively pits you as both the camera placer and the hero: you have to place cameras in such a way that you’re always in vision as you move about to solve puzzles. Needless to say, this fruit still has plenty of juice for the squeezing.

Pancake Sorting

Pancake sorting is the problem of sorting a list of integers into ascending order by using only the operation of a “pancake flip.”

Just like it sounds, a pancake flip involves choosing an index in the list and flipping the prefix of the list (or suffix, depending on your orientation) like a spatula flips a stack of pancakes. Now I think sorting integers is boring (and it’s not NP-hard!), but when you forget about numbers and that one special configuration (ascending sorted order), things get more interesting. Instead, have the pancakes be letters and have the goal be to use pancake flips to arrive at a real English word. That is, you don’t know the goal word ahead of time, so it’s the anagram problem plus finding an efficient pancake flip to get there. Have a player’s score be based on the number of flips before a word is found, and make it timed to add extra pressure, and you have yourself a classic!

The level design then becomes finding good word scrambles with multiple reasonable paths one could follow to get valid words. My mother would probably play this game!

Bin Packing

Young Mikio is making sushi for his family! He’s got a table full of ingredients of various sizes, but there is a limit to how much he can fit into each roll. His family members have different tastes, and so his goal is to make everyone as happy as possible with his culinary skills and the options available to him.

Another name for this problem is bin packing. There are a collection of indivisible objects of various sizes and values, and a set of bins to pack them in. Your goal is to find the packing that doesn’t exceed the maximum in any bin and maximizes the total value of the packed goods.

I thought of sushi because I recently played a ridiculously cute game about sushi (thanks to my awesome friend Yen over at Baking And Math), but I can imagine other themes that suggest natural modifications of the problem. The objects being packed could be two-dimensional, there could be bonuses for satisfying certain family members (or penalties for not doing so!), or there could be a super knife that is able to divide one object in half.

I could continue this list for quite a while, but perhaps I should keep my best ideas to myself in case any game companies want to hire me as a consultant. 🙂

Do you know of games that are based on any of these ideas? Do you have ideas for features or variations of the game ideas listed above? Do you have other ideas for how to turn computational problems into games? I’d love to hear about it in the comments.

I’m pleased to announce that another paper of mine is finished. This one just got accepted to MFCS 2014, which is being held in Budapest this year (this whole research thing is exciting!). This is joint work with my advisor, Lev Reyzin. As with my first paper, I’d like to explain things here on my blog a bit more informally than a scholarly article allows.

A Recent History of Graph Coloring

One of the first important things you learn when you study graphs is that coloring graphs is hard. Remember that coloring a graph with $ k$ colors means that you assign each vertex a color (a number in $ \left \{ 1, 2, \dots, k \right \}$) so that no vertex is adjacent to a vertex of the same color (no edge is monochromatic). In fact, even deciding whether a graph can be colored with just $ 3$ colors (not to mention finding such a coloring) has no known polynomial time algorithm. It’s what’s called NP-hard, which means that almost everyone believes it’s hopeless to solve efficiently in the worst case.

One might think that there’s some sort of gradient to this problem, that as the graphs get more “complicated” it becomes algorithmically harder to figure out how colorable they are. There are some notions of “simplicity” and “complexity” for graphs, but they hardly fall on a gradient. Just to give the reader an idea, here are some ways to make graph coloring easy:

Make sure your graph is planar. Then deciding 4-colorability is easy because the answer is always yes.

Make sure your graph is triangle-free and planar. Then finding a 3-coloring is easy.

Make sure your graph is perfect (which again requires knowledge about how colorable it is).

Make sure your graph doesn’t have a certain kind of induced subgraph (such as having no induced paths of length 4 or 5).

Let me emphasize that these results are very difficult and tricky to compare. The properties are inherently discrete (either perfect or imperfect, planar or not planar). The fact that the world has not yet agreed upon a universal measure of complexity for graphs (or at least one that makes graph coloring easy to understand) is not a criticism of the chef but a testament to the challenge and intrigue of the dish.

Coloring general graphs is much bleaker, where the focus has turned to approximations. You can’t “approximate” the answer to whether a graph is colorable, so now the key here is that we are actually trying to find an approximate coloring. In particular, if you’re given some graph $ G$ and you don’t know the minimum number of colors needed to color it (say it’s $ \chi(G)$, this is called the chromatic number), can you easily color it with what turns out to be, say, $ 2 \chi(G)$ colors?

Garey and Johnson (the gods of NP-hardness) proved this problem is again hard. In fact, they proved that you can’t do better than twice the number of colors. This might not seem so bad in practice, but the story gets worse. This lower bound was improved by Zuckerman, building on the work of Håstad, to depend on the size of the graph! That is, unless $ P=NP$, all efficient algorithms will use asymptotically more than $ \chi(G) n^{1 – \varepsilon}$ colors for any $ \varepsilon > 0$ in the worst case, where $ n$ is the number of vertices of $ G$. So the best you can hope for is being off by something like a multiplicative factor of $ n / \log n$. You can actually achieve this (it’s nontrivial and takes a lot of work), but it carries that aura of pity for the hopeful graph colorer.

The next avenue is to assume you know the chromatic number of your graph, and see how well you can do then. For example: if you are given the promise that a graph $ G$ is 3-colorable, can you efficiently find a coloring with 8 colors? The best would be if you could find a coloring with 4 colors, but this is already known to be NP-hard.

The best upper bounds, algorithms to find approximate colorings of 3-colorable graphs, also pitifully depend on the size of the graph. Remember I say pitiful not to insult the researchers! This decades-long line of work was extremely difficult and deserves the highest praise. It’s just frustrating that the best known algorithm to color a 3-colorable graph requires as many as $ n^{0.2}$ colors. At least it bypasses the barrier of $ n^{1 – \varepsilon}$ mentioned above, so we know that knowing the chromatic number actually does help.

The lower bounds are a bit more hopeful; it’s known to be NP-hard to color a $ k$-colorable graph using $ 2^{\sqrt[3]{k}}$ colors if $ k$ is sufficiently large. There are a handful of other linear lower bounds that work for all $ k \geq 3$, but to my knowledge this is the best asymptotic result. The big open problem (which I doubt many people have their eye on considering how hard it seems) is to find an upper bound depending only on $ k$. I wonder offhand whether a ridiculous bound like $ k^{k^k}$ colors would be considered progress, and I bet it would.

Our Idea: Resilience

So without big breakthroughs on the front of approximate graph coloring, we propose a new front for investigation. The idea is that we consider graphs which are not only colorable, but remain colorable under the adversarial operation of adding a few new edges. More formally,

Definition: A graph $ G = (V,E)$ is called $ r$-resiliently $ k$-colorable if two properties hold

$ G$ is $ k$-colorable.

For any set $ E’$ of $ r$ edges disjoint from $ E$, the graph $ G’ = (V, E \cup E’)$ is $ k$-colorable.

The simplest nontrivial example of this is 1-resiliently 3-colorable graphs. That is a graph that is 3-colorable and remains 3-colorable no matter which new edge you add. And the question we ask of this example: is there a polynomial time algorithm to 3-color a 1-resiliently 3-colorable graph? We prove in our paper that this is actually NP-hard, but it’s not a trivial thing to see.

The chief benefit of thinking about resiliently colorable graphs is that it provides a clear gradient of complexity from general graphs (zero-resilient) to the empty graph (which is $ (\binom{k+1}{2} – 1)$-resiliently $ k$-colorable). We know that the most complex case is NP-hard, and maximally resilient graphs are trivially colorable. So finding the boundary where resilience makes things easy can shed new light on graph coloring.

Indeed, we argue in the paper that lots of important graphs have stronger resilience properties than one might expect. For example, here are the resilience properties of some famous graphs.

From left to right: the Petersen graph, 2-resiliently 3-colorable; the Dürer graph, 4-resiliently 4-colorable; the Grötzsch graph, 4-resiliently 4-colorable; and the Chvátal graph, 3-resiliently 4-colorable. These are all maximally resilient (no graph is more resilient than stated) and chromatic (no graph is colorable with fewer colors)

If I were of a mind to do applied graph theory, I would love to know about the resilience properties of graphs that occur in the wild. For example, the reader probably knows the problem of register allocation is a natural graph coloring problem. I would love to know the resilience properties of such graphs, with the dream that they might be resilient enough on average to admit efficient coloring algorithms.

Unfortunately the only way that I know how to compute resilience properties is via brute-force search, and of course this only works for small graphs and small $ k$. If readers are interested I could post such a program (I wrote it in vanilla python), but for now I’ll just post a table I computed on the proportion of small graphs that have various levels of resilience (note this includes graphs that vacuously satisfy the definition).

Percentage of k-colorable graphs on 6 vertices which are n-resilient
k\n 1 2 3 4
----------------------------------------
3 58.0 22.7 5.9 1.7
4 93.3 79.3 58.0 35.3
5 99.4 98.1 94.8 89.0
6 100.0 100.0 100.0 100.0
Percentage of k-colorable graphs on 7 vertices which are n-resilient
k\n 1 2 3 4
----------------------------------------
3 38.1 8.2 1.2 0.3
4 86.7 62.6 35.0 14.9
5 98.7 95.6 88.5 76.2
6 99.9 99.7 99.2 98.3
Percentage of k-colorable graphs on 8 vertices which are n-resilient
k\n 1 2 3 4
----------------------------------------
3 21.3 2.1 0.2 0.0
4 77.6 44.2 17.0 4.5

The idea is this: if this trend continues, that only some small fraction of all 3-colorable graphs are, say, 2-resiliently 3-colorable graphs, then it should be easy to color them. Why? Because resilience imposes structure on the graphs, and that structure can hopefully be realized in a way that allows us to color easily. We don’t know how to characterize that structure yet, but we can give some structural implications for sufficiently resilient graphs.

For example, a 7-resiliently 5-colorable graph can’t have any subgraphs on 6 vertices with $ \binom{6}{2} – 7$ edges, or else we can add enough edges to get a 6-clique which isn’t 5-colorable. This gives an obvious general property about the sizes of subgraphs in resilient graphs, but as a more concrete instance let’s think about 2-resilient 3-colorable graphs $ G$. This property says that no set of 4 vertices may have more than $ 4 = \binom{4}{2} – 2$ edges in $ G$. This rules out 4-cycles and non-isolated triangles, but is it enough to make 3-coloring easy? We can say that $ G$ is a triangle-free graph and a bunch of disjoint triangles, but it’s known 3-colorable non-planar triangle-free graphs can have arbitrarily large chromatic number, and so the coloring problem is hard. Moreover, 2-resilience isn’t enough to make $ G$ planar. It’s not hard to construct a non-planar counterexample, but proving it’s 2-resilient is a tedious task I relegated to my computer.

Speaking of which, the problem of how to determine whether a $ k$-colorable graph is $ r$-resiliently $ k$-colorable is open. Is this problem even in NP? It certainly seems not to be, but if it had a nice characterization or even stronger necessary conditions than above, we might be able to use them to find efficient coloring algorithms.

In our paper we begin to fill in a table whose completion would characterize the NP-hardness of coloring resilient graphs

The known complexity of k-coloring r-resiliently k-colorable graphs

Ignoring the technical notion of 2-to-1 hardness, the paper accomplishes this as follows. First, we prove some relationships between cells. In particular, if a cell is NP-hard then so are all the cells to the left and below it. So our Theorem 1, that 3-coloring 1-resiliently 3-colorable graphs is NP-hard, gives us the entire black region, though more trivial arguments give all except the (3,1) cell. Also, if a cell is in P (it’s easy to $ k$-color graphs with that resilience), then so are all cells above and to its right. We prove that $ k$-coloring $ \binom{k}{2}$-resiliently $ k$-colorable graphs is easy. This is trivial: no vertex may have degree greater than $ k-1$, and the greedy algorithm can color such graphs with $ k$ colors. So that gives us the entire light gray region.

There is one additional lower bound comes from the fact that it’s NP-hard to $ 2^{\sqrt[3]{k}}$-color a $ k$-colorable graph. In particular, we prove that if you have any function $ f(k)$ that makes it NP-hard to $ f(k)$-color a $ k$-colorable graph, then it is NP-hard to $ f(k)$-color an $ (f(k) – k)$-resiliently $ f(k)$-colorable graph. The exponential lower bound hence gives us a nice linear lower bound, and so we have the following “sufficiently zoomed out” picture of the table

The zoomed out version of the classification table above.

The paper contains the details of how these observations are proved, in addition to the NP-hardness proof for 1-resiliently 3-colorable graphs. This leaves the following open problems:

Get an unconditional, concrete linear resilience lower bound for hardness.

Find an algorithm that colors graphs that are less resilient than $ O(k^2)$. Even determining specific cells like (4,5) or (5,9) would likely give enough insight for this.

Classify the tantalizing (3,2) cell (determine if it’s hard or easy to 3-color a 2-resiliently 3-colorable graph) or even better the (4,2) cell.

Find a way to relate resilient coloring back to general coloring. For example, if such and such cell is hard, then you can’t approximate k-coloring to within so many colors.

But Wait, There’s More!

Though this paper focuses on graph coloring, our idea of resilience doesn’t stop there (and this is one reason I like it so much!). One can imagine a notion of resilience for almost any combinatorial problem. If you’re trying to satisfy boolean formulas, you can define resilience to mean that you fix the truth value of some variable (we do this in the paper to build up to our main NP-hardness result of 3-coloring 1-resiliently 3-colorable graphs). You can define resilient set cover to allow the removal of some sets. And any other sort of graph-based problem (Traveling salesman, max cut, etc) can be resiliencified by adding or removing edges, whichever makes the problem more constrained.

So this resilience notion is quite general, though it’s hard to define precisely in a general fashion. There is a general framework called Constraint Satisfaction Problems (CSPs), but resilience here seem too general. A CSP is literally just a bunch of objects which can be assigned some set of values, and a set of constraints (k-ary 0-1-valued functions) that need to all be true for the problem to succeed. If we were to define resilience by “adding any constraint” to a given CSP, then there’s nothing to stop us from adding the negation of an existing constraint (or even the tautologically unsatisfiable constraint!). This kind of resilience would be a vacuous definition, and even if we try to rule out these edge cases, I can imagine plenty of weird things that might happen in their stead. That doesn’t mean there isn’t a nice way to generalize resilience to CSPs, but it would probably involve some sort of “constraint class” of acceptable constraints, and I don’t know a reasonable property to impose on the constraint class to make things work.

So there’s lots of room for future work here. It’s exciting to think where it will take me.

I’m pleased to announce that my first paper, titled “Anti-Coordination Games and Stable Colorings,” has been accepted for publication! The venue is the Symposium on Algorithmic Game Theory, which will take place in Aachen, Germany this October. A professor of mine once told me that everyone puts their first few publications on a pedestal, so I’ll do my best to keep things down to earth by focusing on the contents of the paper and not my swirling cocktail of pride. The point of this post is to explain the results of my work to a non-research-level audience; the research level folks will likely feel more comfortable reading the paper itself. So here we’ll spend significantly longer explaining the proofs and the concepts, and significantly less time on previous work.

The central question in the paper is how to find stable strategy profiles for anti-coordination games played on graphs. This section will flush out exactly what all of that means.

The easiest way to understand the game is in terms of fashion. Imagine there is a group of people. Every day they choose their outfits individually and interact with their friends. If any pair of friends happens to choose the same clothing, then they both suffer some embarrassment. We can alternatively say that whenever two friends anti-coordinate their outfits, they each get some kind of reward. If not being embarrassed is your kind of reward, then these really are equivalent. Not every pair of people are friends, so perhaps the most important aspect of this problem is how the particular friendship network considered affects their interactions. This kind of game is called an anti-coordination game, and the network of friends makes it a “game on a graph.” We’ll make this more rigorous shortly.

We can ask questions like, if everyone is acting independently and greedily will their choices converge over time to a single choice of outfit? If so how quickly? How much better could a centralized fashion-planner who knows the entire friendship network fare in choosing outfits? Is the problem of finding a best strategy for picking outfits computationally hard? What if some pairs of people want to coordinate their outfits and others don’t? What if caring about another’s fashion is only one-sided in some cases?

Already this problem is rooted in the theory of social networks, but the concept of an anti-coordination game played on a graph is quite broad, and the relevance of this model to the real world comes from the generality of a graph. For example, one may consider the trading networks of various countries; in this case not all countries are trading partners, and it is beneficial to produce different commodities than your trading partners so that you actually benefit from the interaction. Likewise, neighboring radio towers want to emit signals on differing wavelengths to minimize interference, and commuters want to pick different roadways to minimize traffic. These are all examples of this model which we’re about to formalize.

In place of our “network of friends,” the game entails a graph $ G = (V,E)$ in which each player is represented by a vertex, and there is an edge between two vertices whenever the corresponding players are trying to anti-coordinate. We will use the terms player and vertex interchangeably. For now the graph is undirected, but later we will relax this assumption and work with directed graphs. In place of “outfits” we’ll have a generic set of strategies denoted by the numbers $ 1, \dots, k$, and each vertex will choose a strategy from this set. In one round of the game, each vertex $ v$ chooses a strategy, and this defines a function $ f : V \to \left \{ 1, \dots, k \right \}$ from the set of vertices to the set of strategies. Then the payoff of a vertex $ v$ in a round, which we denote $ \mu_f(v)$, is the number of neighbors of $ v$ which have chosen a different strategy than $ v$. That is, it is

Where $ \mathbf{1}_{A}$ denotes the indicator function for the event $ A$, which assumes a value of 1 when the event occurs and 0 otherwise. Here is an example of an instance of the game. We have three strategies, denoted by colors, and the payoff for the vertex labeled $ v$ is three.

If this game is played over many many rounds, we can ask if there is a so-called Nash equilibrium. That is, is there a choice of strategies for the players so that no single player will have an incentive to change in future rounds, assuming nobody else changes? We restrict even further to thinking about pure strategy Nash equilibria, which means there are no probabilistic choices made in choosing a strategy. Equivalently, a pure strategy equilibrium is just a choice of a strategy for each vertex which doesn’t change across rounds. In terms of the graph, we call a strategy function $ f$ which is a Nash equilibrium a stable equilibrium (or, as will be made clear in the next paragraph, a stable coloring). It must satisfy the property that no vertex can increase its payoff by switching to a different strategy. So our question now becomes: how can we find a stable coloring which as good as possible for all players involved? Slightly more generally, we call a Nash equilibrium a strictly stable equilibrium (or a strictly stable coloring) if every vertex would strictly decrease its payoff by switching to another strategy. As opposed to a plain old stable coloring where one could have the same payoff by switching strategies, if any player tries to switch strategy then it will get a necessarily worse payoff. Though it’s not at all clear now, we will see that this distinction is the difference between computational tractability and infeasibility.

We can see a very clear connection between this game and graph coloring. Here an edge produces a payoff of 1 for each of its two vertices if and only if it’s properly colored. And so if the strategy choice function $ f$ is also a proper coloring, this will produce the largest possible payoff for all vertices in the graph. But it may not be the case that (for a fixed set of strategies) the graph is properly colorable, and we already know that finding a proper coloring with more than two colors is a computationally hard problem. So this isn’t a viable avenue for solving our fashion game. In any case, the connection confuses us enough to interchangeably call the strategy choice function $ f$ a coloring of $ G$.

As an interesting side note, a slight variation of this game was actually tested on humans (with money as payoff!) to see how well they could do. Each human player was only shown the strategies of their neighbors, and received $5 for every round in which they collectively arrived at a proper coloring of the graph. See this article in Science for more details.

Since our game allows for the presence of improperly colored edges, we could instead propose to find an assignment of colors to vertices which maximizes the sum of the payoffs of all players. In this vein, we define the social welfare of a graph and a coloring, denoted $ W(G,f)$, to be the sum of the payoffs for all vertices $ \sum_v \mu_f(v)$. This is a natural quantity one wants to analyze. Unfortunately, even in the case of two strategies, this quantity is computationally difficult (NP-hard) to maximize. It’s a version of the MAX-CUT problem, in which we try to separate the graph into two sets $ X, Y$ such that the largest number of edges crosses from $ X$ to $ Y$. The correspondence between the two problems is seen by having $ X$ represent those vertices which get strategy 1 and $ Y$ represent strategy 2.

So we can’t hope to find an efficient algorithm maximizing social welfare. The next natural question is: can we find stable or strictly stable colorings at all? Will they even necessarily exist? The answers to these questions form the main results of our paper.

An Algorithm for Stable Colorings, and the Price of Anarchy

It turns out that there is a very simple greedy algorithm for finding stable colorings of a graph. We state it in the form of a proposition. By stable $ k$-coloring we mean a stable coloring of the graph with $ k$ colors (strategies).

Proposition: For every graph $ G$ and every $ k \geq 1$, $ G$ admits a stable $ k$-coloring, and such a coloring can be found in polynomial time.

Proof. The proof operates by using the social welfare function as a so-called potential function. That is, a change in a player’s strategy which results in a higher payoff results in a higher value of the social welfare function. It is easy to see why: if a player $ v$ changes to a color that helps him, then it will result in more properly colored edges (adjacent to $ v$) than there were before. This means that more of $ v$’s neighbors receive an additional 1 unit of payoff than those that lost 1 as a result of $ v$’s switch. We call a vertex which has the potential to improve its payoff unhappy, and one which cannot improve its payoff happy.

And so our algorithm to find a stable coloring simply finds some unhappy vertex, switches its color to the most uncommon color among its neighbors, and repeats the process until all vertices are happy. Indeed, this is a local maximum of the social welfare function, and the very definition of a stable coloring.

$ \square$

So that was nice, but we might ask: how much worse is the social welfare arrived at by this algorithm than the optimal social welfare? How much do we stand to lose by accepting the condemnation of NP-hardness and settling for the greedy solution we found? More precisely, if we call $ Q$ the set of stable colorings and $ C$ the set of all possible colorings, what is the value of

This is a well-studied quantity in many games, called the price of anarchy. The name comes from the thought: what do we stand to gain by having a central authority, who can see the entire network topology and decide what is best for us, manage our strategies? The alternative (anarchy) is to have each player in the game act as selfishly and rationally as possible without complete information. It turns out that as the number of strategies grows large in our anti-coordination game, there is no price of anarchy. For our game this obviously depends on the choice of graph, but we know what it is and we formally state the result as a proposition:

Proposition: For any graph, the price of anarchy for the $ k$ strategy anti-coordination game is at most $ k/(k-1)$ and this value is actually achieved by some instances of the game.

Proof. The pigeonhole principle says that every vertex can always achieve at least a $ (k-1)/k$ fraction of its maximum possible payoff. Specifically, if a vertex $ v_i$ can’t achieve a proper coloring, then every color must be accounted for among $ v_i$’s neighbors. Choosing the minimally occurring color will give $ v_i$ at least a payoff of $ d_i(k-1)/k$ where $ d_i$ is the number of neighbors of $ v_i$. Since every stable coloring has to satisfy the condition that no vertex can do any better than the strategy it already has, even in the worst stable coloring every vertex already has chosen such a minority color. Since the maximum payoff is twice the number of edges $ 2 |E|$, and the sum of the degrees $ \sum_i d_i = 2 |E|$, we have that the price of anarchy is at most

Indeed, we can’t do any better than this in general, because the following graph gives an example where the price of anarchy exactly meets this bound.

An instance of the anti-coordination game with 5 strategies which meets the upper bound on price of anarchy.

This example can easily be generalized to work with arbitrary $ k$. We leave the details as an exercise to the reader.

$ \square$

Strictly Stable Colorings are Hard to Find

Perhaps surprisingly, the relatively minor change of adding strictness is enough to make computability intractable. We’ll give an explicit proof of this, but first let’s recall what it means to be intractable.

Recall that a problem is in NP if there is an efficient (read, polynomial-time) algorithm which can verify a solution to the problem is actually a solution. For example, the problem of proper graph $ k$-coloring is in NP, because if someone gives you a purported coloring all you have to do is verify that each of the $ O(n^2)$ edges are properly colored. Similarly, the problem of strictly stable coloring is in NP; all one need do is verify that no choice of a different color for any vertex improves its payoff, and it is trivial to come up with an algorithm which checks this.

Next, call a problem $ A$ NP-hard if a solution to $ A$ allows you to solve any problem in NP. More formally, $ A$ being NP-hard means that there is a polynomial-time reduction from any problem in NP $ B$ to $ A$ in the following (rough) sense: there is a polynomial-time computable function (i.e. deterministic program) $ f$ which takes inputs for $ B$ and transforms them into inputs for $ A$ such that:

$ w$ is a solvable instance of $ B$ is if and only if $ f(w)$ is solvable for $ A$.

This is not a completely formal definition (see this primer on NP-completeness for a more serious treatment), but it’s good enough for this post. In order to prove a problem $ C$ is NP-hard, all you need to do is come up with a polynomial-time reduction from a known NP-hard problem $ A$ to your new problem $ C$. The composition of the reduction used for $ A$ can be composed with the reduction for $ C$ to get a new reduction proving $ C$ is NP-hard.

Finally, we call a problem NP-complete if it is both in NP and NP-hard. One natural question to ask is: if we don’t already know of any NP-hard problems, how can we prove anything is NP-hard? The answer is: it’s very hard, but it was done once and we don’t need to do it again (but if you really want to, see these notes). As a result, we have generated a huge list of problems that are NP-complete, and unless P = NP none of these algorithms have polynomial-time algorithms to solve them. We need two examples of NP-hard problems for this paper: graph coloring, and boolean satisfiability. Since we assume the reader is familiar with the former, we recall the latter.

Given a set of variables $ x_i$, we can form a boolean formula over those variables of the form $ \varphi = C_1 \wedge C_2 \wedge \dots \wedge C_m$ where each clause $ C_i$ is a disjunction of three literals (negated or unnegated variables). For example, $ C_i = (x_2 \vee \bar{x_5} \vee \bar{x_9})$ might be one clause. Here interpret a formula as the $ x_i$ having the value true or false, the horizontal bars denoting negation, the wedges $ \wedge$ meaning “and” and the vees $ \vee$ meaning “or.” We call this particular form conjunctive normal form. A formula $ \varphi$ is called satisfiable if there is a choice of true and false assignments to the variables which makes the entire logical formula true. The problem of determining whether there is any satisfying assignment of such a formula, called 3-SAT, is NP-hard.

Going back to strictly stable equilibria and anti-coordination games, we will prove that the problem of determining whether a graph has a strictly stable coloring with $ k$ colors is NP-hard. As a consequence, finding such an equilibrium is NP-hard. Since the problem is also in NP, it is in fact NP-complete.

Theorem: For all $ k \geq 2$, the problem of determining whether a graph $ G$ has a strictly stable coloring with $ k$ colors is NP-complete.

Proof. The hardest case is $ k =2$, but $ k \geq 3$ is a nice warmup to understand how a reduction works, so we start there.

The $ k \geq 3$ part works by reducing from graph coloring. That is, our reduction will take an input to the graph $ k$-coloring problem (a graph $ G$ whose $ k$-colorability is in question) and we produce a graph $ G’$ such that $ G$ is $ k$-colorable if and only if $ G’$ has a strictly stable coloring with $ k$ colors. Since graph coloring is hard for $ k \geq 3$, this will prove our problem is NP-hard. More specifically, we will construct $ G’$ in such a way that the strictly stable colorings also happen to be proper colorings! So finding a strictly stable coloring of $ G’$ will immediately give us a proper coloring of $ G$.

The construction of $ G’$ is quite straightforward. We start with $ G’ = G$, and then for each edge $ e = (u,v)$ we add a new subgraph which we call $ H_e$ that looks like:

By $ K_{k-2}$ we mean the complete graph on $ k-2$ vertices (all possible edges are present), and the vertices $ u,v$ are adjacent to all vertices of the $ H_e = K_{k-2}$ part. That is, the graph $ H_e \cup \left \{ u,v \right \}$ is the complete graph on $ k$ vertices. Now if the original graph $ G$ was $ k$-colorable, then we can use the same colors for the corresponding vertices in $ G’$, and extend to a proper coloring (and hence a strictly stable equilibrium) of all of $ G’$. Indeed, for any $ H_e$ we can use one different color for each vertex of the $ K_{k-2}$ part if we don’t use either of the colors used for $ u,v$, then we’ll have a proper coloring.

On the other hand, if $ G’$ has a strictly stable equilibrium, then no edge $ e$ which originally came from $ G$ can be improperly colored. If some edge $ e = (u,v)$ were improperly colored, then there would be some vertex in the corresponding $ H_e$ which is not strictly stable. To see this, notice that in the $ k$ vertices among $ H_e \cup \left \{ u,v \right \}$ there can be at most $ k-1$ colors used, and so any vertex will always be able to switch to that color without hurting his payoff. That is, the coloring might be stable, but it won’t be strictly so. So strictly stable colorings are the same as proper colorings, and we already see that the subgraph $ G \subset G’$ is $ k$-colorable, completing the reduction.

Well that was a bit of a cheap trick, but it shows the difficulty of working with strictly stable equilibria: preventing vertices from changing with no penalty is hard! What’s worse is that it’s still hard even if there are only two colors. The reduction here is a lot more complicated, so we’ll give a sketch of how it works.

The reduction is from 3-SAT. So given a boolean formula $ \varphi = C_1 \wedge \dots \wedge C_m$ we produce a graph $ G$ so that $ \varphi$ has a satisfying assignment if and only if $ G$ has a strictly stable coloring with two colors. The principle part of the reduction is the following gadget which represents the logic inherent in a clause. We pulled the figure directly from our paper, since the caption gives a good explanation of how it works.

To reiterate, the two “appendages” labeled by $ x$ correspond to the literal $ x$, and the choice of colors for these vertices correspond to truth assignments in $ \varphi$. In particular, if the two vertices have the same color, then the literal is assigned true. Of course, we have to ensure that any $ x$’s showing up in other clause gadgets agree, and any $ \bar{x}$’s will have opposite truth values. That’s what the following two gadgets do:

The gadget on the left enforces x’s to have the same truth assignment across gadgets (otherwise the center vertex won’t be in strict equilibrium). The gadget on the right enforces two literals to be opposites.

And if we stitch the clauses together in a particular way (using the two gadgets above) then we will guarantee consistency across all of the literals. All that’s left to check is that the clauses do what they’re supposed to. That is, we need it to be the case that if all of the literals in a clause gadget are “false,” then we can’t complete the coloring to be strictly stable, and otherwise we can. Indeed, the following diagram gives all possible cases of this up to symmetry:

The last figure deserves an explanation: if the three literals are all false, then we can pick any color we wish for $ v_1$, and its two remaining neighbors must both have the same color (or else $ v_1$ is not in strict equilibrium). Call this color $ a$, and using the same reasoning call the neighbors of $ v_2$ and $ v_3$ $ b$ and $ c$, respectively. Now by the pigeonhole principle, either $ a=b, b=c,$ or $ b=c$. Suppose without loss of generality that $ a=b$, then the edge labeled $ (a,b)$ will have the $ a$ part not in strict equilibrium (it will have two neighbors of its same color and only one of the other color). This shows that no strict equilibrium can exist.

The reduction then works by taking a satisfying assignment for the variables, coloring the literals in $ G$ appropriately, and extending to a strictly stable equilibrium of all of $ G$. Conversely, if $ G$ has a strictly stable coloring, then the literals must be consistent and each clause must be fully colorable, which the above diagram shows is the same as the clauses being satisfiable. So all of $ \varphi$ is satisfiable and we’re done (excluding a few additional details we describe in the paper).

$ \square$

Directed Graphs and Cooperation

That was the main result of our paper, but we go on to describe some interesting generalizations. Since this post is getting quite long, we’ll just give a quick overview of the interesting parts.

The rest of the paper is dedicated to directed graphs, where we define the payoff of a directed edge $ (u,v)$ to go to the $ u$ player if $ u$ and $ v$ anti-coordinate, but $ v$ gets nothing. Here the computational feasibility is even worse than it was in the undirected case, but the structure is noticeably more interesting. For the former, not only is in NP-hard to compute strictly stable colorings, it’s even NP-hard to do so in the non-strict case! One large part of the reason for this is that stable colorings might not even exist: a directed 3-cycle has no stable equilibrium. We use this fact as a tool in our reductions to prove the following theorem.

Theorem: For all $ k \geq 2$, determining whether a directed graph has a stable $ k$-coloring is NP-complete.

To address the interesting structure that arises in the directed case, we observe that we can use a directed graph to simulate the desire of one vertex to actually cooperate with another. To see this for two colors, instead of adding an edge $ (u,v)$ we add a proxy edge $ u’$ and directed edges $ (u,u’), (u’,v)$. To be in equilibrium, the proxy has no choice but to anti-coordinate with $ v$, and this will give $ u$ more incentive to cooperate with $ v$ by anti-cooperating with its proxy. This can be extended to $ k$ colors by using an appropriately (acyclically) directed copy of $ K_{k-1}$.

Thoughts, and Future Work

While the results in this paper are nice, and I’m particularly proud that I came up with a novel NP-hardness reduction, it is unfortunate that the only results in this paper were hardness results. Because of the ubiquity of NP-hard problems, it’s far more impressive to have an algorithm which actually does something (approximate a good solution, do well under some relaxed assumption, do well in expectation with some randomness) than to prove something is NP-hard. To get an idea of the tone set by researchers, NP-hardness results are often called “negative” results (in the sense that they give a “no” answer to the question of whether there is an efficient algorithm) and finding an algorithm that does something is called a positive result. That being said the technique of using two separate vertices to represent a single literal in a reduction proof is a nice trick, and I have since used it in another research paper, so I’m happy with my work.

On the positive side, though, there is some interesting work to be done. We could look at varying types of payoff structures, where instead of a binary payoff it is a function of the colors involved (say, $ |i – j|$. Another interesting direction is to consider distributed algorithms (where each player operates independently and in parallel) and see what kinds of approximations of the optimal payoff can be achieved in that setting. Yet another direction favored by a combinatorialist is to generalize the game to hypergraphs, which makes me wonder what type of payoff structure is appropriate (payoff of 1 for a rainbow edge? a non-monochromatic edge?). There is also some more work that can be done in inspecting the relationship between cooperation and anti-cooperation in the directed version. Though I don’t have any immediate open questions about it, it’s a very interesting phenomenon.

In any event, I’m currently scheduled to give three talks about the results in this paper (one at the conference venue in Germany, and two at my department’s seminars). Here’s to starting off my research career!