The Computational Complexity of Games and Puzzles. Valia Mitsou

Transcription

1 The Computational Complexity of Games and Puzzles Valia Mitsou

2 Abstract The subject of my thesis is studying the algorithmic properties of one and two-player games people enjoy playing, such as chess or Sudoku. This research falls into a wider area known as combinatorial game theory. One of the main questions asked about games in this context is whether they are algorithmically tractable, that is, whether they can be solved with efficient algorithms. So far, more than fifty popular games have been categorized as intractable. This is probably because people enjoy games and puzzles for which designing a winning strategy or finding the solution requires some cleverness.

3 Contents 1 Introduction 2 2 Computational complexity background Complexity classes for games and puzzles The class PSPACE Generalized Geography Parameterized complexity primer Structural Parameters Parameterized Complexity for puzzles and 2-player games 15 3 The Game of UNO Single player version of UNO Many-player uncooperative version The Game of SET One round of SET Multi-round versions A two-player game

4 1 Introduction Combinatorial Game Theory is a very intriguing and widely developed field of mathematics and computer science under the wide umbrella of Recreational Mathematics. A combinatorial game is usually considered to be a 2-player turn-based finite game with perfect information (no player hides any information from her opponent) and no chance, like chess or go. Similarly, finite 1-player puzzles of no chance, like sudoku or mastermind also fall into this category. Other than the practical applications that studying games and puzzles has in the gaming market, combinatorial game theory is a field that many people inside and outside the academic community find insteresting and amusing. Since many games and puzzles are entertaining while having at the same time mathematical and algorithmic depth, they serve as a great teaching tool. For example, Prisner in university of Maryland used a river crossing puzzle in his Discrete Mathematics class. Many games and puzzles can be directly reformulated as well-known important algorithmic problems. To give some examples, solving the river crossing riddle we just mentioned is very close to solving Vertex Cover, playing the game UNO as we present in section 3 can be modeled as solving Hamiltonian Path, different versions of the game SET as we present in section 4 can be reformulated as covering or matching problems, the game lights out can be viewed as some version of parity domination [22], etc. To take this to some extent, studying a game or a puzzle with a direct algorithmic application might give results which are important even outside of the field. Furthermore, all people learn and play games and puzzles and this provides a connecting bond between the academic and the non-academic community: research results in combinatorial game theory are more likely to be read, understood and appreciated by people ouside of academia. Recent the- 2

5 oretical results have appeared in blogs of popular science (see for example Tetris is hard, even to approximate: addictive puzzle even stumps computers, Mathematicians Prove Tetris Is Tough, And Scrabble proved PSPACE- Complete, How Complicated is Scrabble, Lemmings Forum, The Complexity of Tip Over and other puzzles). In addition, studying games and puzzles from an algorithmic point of view can enhance cross-disciplinary research, as people outside of the field but interested in the said game might read research results about it. When we study a 2-player game, one of our main goals is to figure out which player is going to win. It is worth mentioning here that, for the class of games that we study (turn-based, finite, perfect information, and no chance), this is always going to be the same player (or always a draw) if both players play optimally. The strongest way of determining (and enforcing) the outcome is by designing a winning strategy: an algorithm that, for any move of the opponent, can provide us with a choice that if the player follows, she will eventually win the game (or force a draw if the outcome is a draw). For most interesting games though, designing an algorithm that works efficiently is particularly difficult due to the combinatorial explosion of the number of potential next moves. In some cases, we could prove that a player has a winning strategy by giving a non-algorithmic proof. Even in that case it can again be particularly complicated to describe the actual winning strategy. For many well-known games, such as chess, it is still unknown if either player has a winning strategy. The complexity to find a winning strategy to a game is usually what makes the game interesting. In the case of an 1-player puzzle, the objective is to find the solution to the puzzle. Likewise, that can be determined by designing a solving alorithm or giving a non-algorithmic mathematical argument. In some cases it is fruitful to consider the question whether the puzzle is solvable or not, which can be tackled in a similar way. Again, solving a puzzle is usually a slow procedure, 3

6 because most of the times it requires an exaustive search in the set of potential plays. For example, though Sudoku is easy for modern computers for 9 9 boards, it can quickly become impossible to decide if a larger Sudoku instance has a solution, because we know of no (significantly) better algorithm than trying out all solutions. Again, the complexity of a puzzle is what makes it interesting to play, unless the amusement comes from figuring out a clever trick to solve it (which renders the puzzle useless once you know the trick). It thus becomes natural to ask this question: for which games or puzzles is it always possible to decide on an optimal next move efficiently, even for large instances? Such questions are usually answered with the tools of computational complexity theory, a theory which categorizes algorithmic problems as tractable or intractable depending on the resources needed to solve them. 2 Computational complexity background 2.1 Complexity classes for games and puzzles In computational complexity theory resources are measured as functions of the size of the input. Thus any finite-sized puzzle or game is considered trivially solvable. So, when we study a puzzle or a 2-player game from the computational complexity point of view, we first need to create a natural unbounded generalization of the puzzle or game. For example, in a board game we consider a variation where the board is of size n n. Usually, algorithms with a polynomial (in n) complexity are considered efficient and algorithms with an expenential complexity are considered inefficient or impractical. To characterize the computational complexity of a game we can do two things. If the game is tractable we must demonstrate an efficient algorithm that solves it. If it is not, we must show that such an algorithm cannot exist. The second task is generally more complicated (since we must rule out all 4

7 conceivable algorithms). Usually, evidence of intractability takes the form of a hardness proof that shows that if an efficient algorithm for the game exists some widely believed complexity conjectures are proved false. The most celebrated theory for proving hardness results is the theory of NP-Completeness, invented by Cook in 1971 ([6]) and further developed by Karp in 1972 ([26]). From the time that it was introduced, more than 1000 problems have been characterized as NP-complete (i.e. problems that it is efficient to verify a suggested solution but probably inefficient to compute it). Furthermore, one of the most important open problems in complexity theory (and computer science in gereral) is the famous P=NP question (i.e. whether NP-complete problems admit efficient algorithms). Similar to the example of NP-completeness, theorists have introduced several other hardness complexity classes to characterize possibly more difficult problems. Some of them are for example the class of PSPACE-complete problems (problems not requiring a lot of additional memory space to be worked out but probably requiring a lot of time) and the class of EXPTIME-complete problems (problems known to be inefficient from a time perspective). Intractability for games and puzzles takes the form of a hardness proof. When we say that a game or puzzle is NP-complete, we mean that it is hard to find its solution efficiently, but easy to verify the correctness of a given solution. For a PSPACE-complete problem, verification is also hard but at least polynomial termination of the game is guaranteed. Harder games without such a termination guarantee usually turn out to be EXP-complete. Many of the famous 1-player games and puzzles are usually shown to be NP-Complete. Examples include (among others) sudoku [32], mastermind [31] and tetris [11]. This is definitely not a coincidence: people enjoy spending time on an intractable puzzle, such as Sudoku, but also appreciate the ability to easily check the solution for correctness, once it s done. On the other hand, hard 2-player games are usually at least PSPACE- 5

8 hard. That is so because in order to be sure that some player has a winning strategy, for each move we have to check all the possible moves that the opponent can make. Examples include (among others) chess [19], go [28] and othello [25]. For an extensive list of NP-Complete and PSPACE-Complete games and puzzles see problems#games_and_puzzles and of_pspace-complete_problems#games_and_puzzles. 2.2 The class PSPACE The class PSPACE is considered the natural home of 2 player games. The reason is that Quantified Boolean Formula, the variation of satisfiability which is complete for PSPACE, can be viewed as a two player game where players set truth values of the variables used in a formula φ interchangeably. If φ is satisfied then player A wins, else player B wins. Definition 2.1. Problem: Quantified Boolean Formula. Input: A first order formula x 1 x 2 x 3... x n φ(x 1, x 2, x 3,... x n ) with φ being in CNF. Question: Is φ satisfiable? Theorem 2.1. (from [29]) Quantified Boolean Formula is PSPACE- Complete. Quantified Boolean Formula can be seen as a two player game: If player A can pick a truth value for x 1 such that for any truth value player B might pick for x 2 then player A can pick a truth value for x 3, and so on, such that in the end the formula is satisfied, player A wins. On the other hand, if for any x 1 that player A picks there is a truth value for x 2 that player B picks, and so on, such that in the end the formula is false, then player B wins. Quantified Boolean Formula is a very useful problem 6

9 for proving hardness of 2-player games. Below we present a famous 2-player game called Geography Generalized Geography The problem Generalized Geography is inspired by the school game of Geography usually played by two players: Player A starts the game with introducing the name of a city, for example New York. Then player B should find a city the name of which starts with the last letter of the previous city played, for example Kyoto. Player A continues in the same way, playing for example Osaka, player B playing Athens and so on and so forth. A city that was already played cannot be played again. The first player unable to find a city to play loses. Variations of this game can be found in many countries. For example in the US the game is called States (the theme is US states), in Russia it s called Goroda (trans. cities ), and in Japan it is called Shiritori (trans. taking the end ). In a similar game played in Greece, the theme is songs. Definition 2.2. Problem: Generalized Geography. Input: A directed graph G(V, E) and a starting vertex v V. Description: Two players take turns moving a token (initially placed in v) from a current vertex to a neighboring vertex following the direction of the arcs. When a vertex is visited, it is removed from the graph. First player unable to move loses. Question: Is there a strategy so that player 1 wins? Generalized Geography is one of the most common games used in proving PSPACE-hardness for other games. Here we prove that it is PSPACE-hard. Theorem 2.2. (from [29]) Directed Generalized Geography is PSPACEhard 7

10 Figure 2.1: Geography game: the variable gadget. Proof. We reduce Quantified Boolean Formula to Generalized Geography. The construction is as follows. Suppose we are given a first order formula x 1 x 2... x n φ. We construct a graph as follows: For each variable x i we construct a diamond-like variable gadget as shown in figure 2.1. The player who gets to choose whether some variable x i will be set to true or false is the one responsible to move the token in the corresponding variable gadget to up or down. We put many variable gadgets one after the other to represent the alternating choice in truth values of variables by the players. In the end, player B should choose which clause player A should satisfy. So we create one vertex c j for each clause and connect the last gadget vertex with each clause vertex with arcs. We also connect a clause vertex with the literals that it contains in the following manner: if variable x i appears in the clause c j as positive, then we connect vertex c j with the vertex that corresponds to x i = F, otherwise if it appears as negative, we connect c j with vertex from the variable gadget x i that corresponds to x i = T. We need to make sure that the choice of clause to be satisfied is made by player B. So we might need to add an additional vertex after the variable gadgets depending on the parity of the number of the alternating quantifiers. For an example see figure 2.2. There are many variations of Generalized Geography. One is Edge 8

11 Figure 2.2: A complete example transforming the first order formula x y z(( x y z) (x y z)) into a geography game. Game starts on the purple node with player A setting the truth value of variable x. Player B on the yellow node picks which clause is to be satisfied. Generalized Geography, where instead of vertices we remove visited edges from the graph. Generalized Geography and Edge Generalized Geography are both proven to be PSPACE-complete even on bipartite or on planar graphs with in/out degrees at most 2 and total degree at most 3 (see [21] and [28]). Furthermore, both vertex and edge versions can be played in an undirected graph instead of a directed. In [20], Undirected Vertex Generalized Geography is proven to be in P, whereas Undirected Edge Generalized Geography is proven PSPACE-complete (though its restriction to bipartite graphs is shown to be in P). Theorem 2.3. (from [20]) Undirected Generalized Geography is solvable in polynomial time. Proof. We reduce Undirected Generalized Geography to Maximum Matching. We prove that player 1 has a winning strategy iff v is matched in every maximum matching of G. If v is matched in every maximum matching, then consider one such 9

12 matching M in G. Player 1 will play over edges of M. Observe that the edges that player 2 plays over also constitute a matching M in G. However, M can t be as large as M because then it would be a maximum matching that doesn t cover v. So M M and player 1 wins. On the other hand, if M is a maximum matching that doesn t match v, it should match all of its neighbors (otherwise if there was a u neighbor of v that was also unmatched, we could include (u, v) in M). So now player 2 can play on the edges of the matching and guarantee a winning strategy. Deciding whether all maximum matchings match v can be done by computing the maximum matching on G and then on G \ v and comparing the sizes. If they are equal, there exists a maximum matching in G that doesn t match v, otherwise v is always matched by a maximum matching in G. 2.3 Parameterized complexity primer Classical complexity theory mostly focuses on the distinction between polynomially solvable vs exponentially solvable problems in terms of the size of the input (n). Parameterized Complexity refines this distinction by introducing yet another parameter of the problem (k) and classifying problems as tractable or intractable when this parameter is small or moderately large. For example, consider the following problems: Vertex Cover, Independent Set, Graph Coloring. All 3 are well-known NP-Hard problems. However, the first two become polynomially solvable if we know that the size of the solution is a constant, whereas the third is NP-hard even for 3 colors. The class of problems that admit polynomial time algorithms when the parameter is a constant is called XP. Parameterized complexity theory comes with its own notions of tractability and intractability. A problem is considered Fixed Parameter Tractable (abbrev. FPT) when exponential growth in the complexity of the problem 10

13 is confined in a function of just the parameter, in other words if the problem admits an algorithm with complexity O(f(k) n c ). FPT algorithms are considered more efficient than their XP counterpart with running time O(n f(k) ) for moderately large values of k. Vertex Cover parameterized by the size of the vertex cover admits an easy FPT algorithm: Pick an uncovered edge (u, v) and consider covering it first by u and then by v. Remove (u, v) together with all edges having either u or v as an endpoint. Repeat for up to k steps. If all edges are covered reply yes. If not, reply no. On the other hand, Independent Set parameterized by the size of the independent set is not known to admit an FPT algorithm. It is also unlikely 1 to admit one, as this problem is proven to be W-hard. The W-hierarchy is an analogue of the polynomial hierarchy of classical complexity theory to parameterized complexity. Classes W[t] are defined as the classes for which the Weighted t Normalized Satisfiability problem is complete. Definition 2.3. Problem: Weighted t Normalized Satisfiability Input: A t-normalized boolean expression F, a positive integer k Parameter : k Question: Does F have a satisfying truth assignment of weight k? That is, a truth assignment where precisely k of the variables which appear in F are true? In the definition, t stands for the number of alternations between conjunctions and disjunctions in the formula F. A CNF formula can be viewed as the conjunction of bracketed disjunctions, so it has only one alternation level and is 1-normalized. If we were to substitute some literals in a CNF formula with bracketed conjunctions, the result would be a 2-normalized formula. As with classical complexity theory, in parameterized complexity theory we have developed a reduction program. If a problem A is FPT-reducible 1 unless the exponential time hypothesis (abbrev. ETH) is proven false, that is there exists a 2 o(n) algorithm to solve 3 SAT. 11

14 to another problem B and we can solve B in FPT time, then we can also solve A in FPT time. On the other hand, a problem B can be proven to be W-hard (or AW-hard) when, starting by a known hard problem A we can reduce A to B via an FPT time reduction, as long as the parameter in B is a function of just the parameter of A (independent of the input size). Classes W[1] and W[2] are the most commonly seen, and many parameterized problems have been categorized as complete for these classes. Independent Set is an example of a W[1]-complete problem, together with Clique. On the other hand, Dominating Set and Hitting Set are W[2]- complete. Yet another complete problem for W[1] is the k-multicolored Clique problem, one of the most famous problems that is used to prove parameterized hardness (see [16]). A quantified variation of t normalized SAT Parameterized QBFSATt, is complete for the class AW[t] [1]. AW[t] for t = 1, 2,..., is also a hierarchy of classes which has been collapsed (see [1]) and is now known as AW[*]. In this parameterized geography of classes, the class XP can be seen as the parameterized equivalent of the class EXP of classical complexity. A problem which is hard for XP cannot be in FPT [14] Structural Parameters In addition to natural parameters (i.e. the parameter being the size of the solution), one can consider other parameters - quantities or measures which when they are small constants the problem can be solved in polynomial time. One of the most celebrated such examples is the notion of treewidth. In simple terms, treewidth measures how much a graph resembles a tree. For a graph, having small treewidth essentially means that it has small cuts. A graph with small treewidth can be re-written in a tree-like structure called a tree decomposition where every node is a small cut of the original graph. Then, one can explore this tree-like structure to develop dynamic programs 12

15 to solve a number of different problems. Treewidth is one of the most celebrated structural parameters. First of all, the problem of finding whether a graph has treewidth at most k is FPT. Second, by Courcelle s theorem [8], every problem expressible in monadic second order logic is FPT parameterized by treewidth and solvable in linear time when treewidth is a constant. Researcher have tried to reproduce treewidth s success and have introduced many different structural parameters. We will analyse clique-width, since this will prove useful later. In order to describe clique-width we need to define the following 4 operations: 1. Construct a new vertex with any color among {1, 2,... k}; 2. Take the union of two colored graphs of up to k colors each; 3. Take the join between two colors i, j, in other words connect all vertices of color i to vertices of color j; 4. Rename color i to color j. A graph G(V, E) is said to have click-width k if it can be constructed using at most k colors and the above operations. Graphs of bounded treewidth also have bounded clique-width but not vice versa, which makes the class of graphs of bounded clique-width more general [7]. Clique-width was introduced in lack of a measure to describe classes of dense graphs which might have a simple enough structure. For example, cliques have unbounded treewidth, but have clique-width 2: First, construct a vertex of color 1, then repeatedly: construct a vertex of color 2 ; join all vertices of color 1 to the vertex of color 2; 13

16 recolor vertex of color 2 to color 1. Unfortunately, it is shown that computing the clique-width of a graph is NP-hard [17], though it is still unknown if it is W-hard. However, the class of graphs of bounded clique-width is interesting, because many NP-Complete problems become polynomially solvable for this class. For example, all graph properties expressible in MSO 1 logic, in other words if we allow quantification over sets of vertices (but not sets of edges) are decidable in linear time in graphs of constant clique-width [9]. Furthermore, many other problems have been shown to be tractable for constant clique-width, see [15]. Here we present an algorithm to solve Hamiltonian Path. We design a dynamic program that works as follows: we store information about all different path covers over all pairs of colors of the graph in a multiset M, then we extend the multiset for each operation that we perform. A path cover of the graph is a union of vertex disjoint paths that cover the graph. A Hamilton path is a path cover that is a unique path. Thus, if after computing M for graph G, if M contains some path cover which is a unique path we reply yes, else we reply no. Here we show how to update the multiset after performing one of the four operations described above. Suppose that we have a multiset M that contains information about all different path covers of the graph so far. The operations 1,2, and 4 (add vertex, union, rename) are easy. For the first, if the added vertex has color i, add the path < i, i > in all path covers of M. For the second, combine all the path covers of the first with the path covers of the second. For the last, if we are to rename all vertices of color i to color j, we construct M by substituting i with j everywhere in M. The join operation is the most difficult to explain. If we are to connect all vertices of color i to vertices of color j then potentially some new paths will be constructed. Thus, for every path cover in M that contains < i, k > and < l, j >, we add in the multiset this path cover and also a path cover where 14

17 < k, i >, < j, l > is replaced by < k, l >. The size of the multiset in any given time is at most O( V k2 ). Furthermore, it is polynomial in the size of the multiset to compute M for each operation, and the number of operations for any reasonable expression of clique-width is polynomial in the size of the input. Thus: Theorem 2.4. Hamiltonian Path can be solved in polynomial time in graphs of bounded clique-width Parameterized Complexity for puzzles and 2-player games In puzzles and games as well as in regular problems, one can identify important parameters that are known to be small for the particular game or puzzle and examine its complexity from the parameterized point of view. It is quite usual for games and puzzles to have natural parameters apart from the input size. For example, in a card game like UNO, one can consider the number of colors as a parameter (in the real game this number is 4: red, blue, green, yellow). In [1], Abrahamson, Downey, and Fellows introduced the concept of a short or k-move game. This concept can be viewed as a generalization of the mate-in-k-moves problem from chess, and creates a parameter which can be applied to every combinatorial game which can be played in rounds. Short games are also very much in line with the standard heuristic approach to playing combinatorial games, which involves the evaluation of future moves to some upper limit. In [1], AW[*] is conjectured to be the natural home of short 2 player games. 15

18 3 The Game of UNO In this section we will analyze known results regarding the game of UNO. UNO is a popular game of cards. Each card consists of two attributes, number and color. There are also some cards with special effects. In the beginning of the game, an equal number of cards is dealt to each of the players. Players take turns discarding one card at a time from their hand to a middle pile of used cards. The rule is that the discarded card should match the top card on the pile either in color or in number. If the player who is about to play doesn t have a matching card to play, then she is required to draw one card from a stack of unused cards. If the new card does not match either, then the player loses her turn. The first player to get rid of all her cards wins. We present results from the work by E. Demaine, M. Demaine, N. Harvey, R. Uehara, T. Uno and Y. Uno [13]. The authors study a simplified version of the game with no special effect cards and no stack of unused cards (if a player doesn t have a card to play she loses the game). However, this inconvenience is somewhat remedied by the fact that the simplified version is a perfect information game, where the opponents cards are open. Below, we present the game definitions more formally. An UNO card is defined by a color-number pair (x, y) X Y, where X = {1, 2,... c} is a set of colors and Y = {1, 2,... b} is a set of numbers. In an UNO game there are p players participating (where p 1). At the beginning, each player i is given a set of C i cards. We assume that the set C = i C i is a multi-set (i.e. two or more cards can have the same number and color). An interesting observation is that the problem can be expressed in a graph-theoretic form, where the dealt cards are the vertices and two vertices are connected with an edge if the cards that they represent can be discarded one after the other in a game play, i.e if they share either the same color 16

19 or the same number, and, in the many-player version, one belongs in the hand of the i-th player and the other in the hand of the i + 1( mod p)-th player. In that sense, the one-player version where a single player is trying to discard all of her cards, is equivalent to finding a Hamiltonian path in the card graph, whereas the p-player un-cooperative version is like playing Generalized Geography on the card graph. We present results regarding the one-player version of the game. This problem is NP-Complete when c and b are unbounded and in XP when c (or equivalently b) is a parameter. We also present some interesting open problems. 3.1 Single player version of UNO As it was mentioned before, each game instance can be expressed as an (undirected) graph, where vertices represent the cards and two vertices are connected if the cards that they represent share the same color or the same number. In that sense, solving the solitaire UNO becomes equivalent to finding a Hamiltonian Path in the graph. Let us now examine the properties of an UNO graph. The vertices of a single-player UNO graph represent cards that can be determined by a pair of attributes (color,number). Thus, UNO graphs can be redrawn in a grid-like form, where cards of the same number are on the same line and cards of the same color are on the same column. It is easy to see that the graph will have edges only within the same lines or the same columns. In fact, the graph induced by a line or a column of this grid-like UNO graph will be a clique. An UNO graph can be seen in figure 3.1 (boxes represent cliques). Observe that UNO graphs are claw-free graphs: there is no induced K 1,3. Furthermore, it is not difficult to see that an UNO graph is the line graph of 17

20 Figure 3.1: A set of UNO cards represented as a grid-like graph. Vertices enclosed in boxes are fully joint with edges. Figure 3.2: A Hamiltonian path in the UNO graph, which can be viewed as a discarding sequence of the cards. Figure 3.3: The bipartite graph whose line graph is shown in figure 3.1. a bipartite graph, with one part having c vertices labeled {1,2,... c} and the other b vertices labeled {1,2,... b}, and each card is represented as an edge connecting a color to a number. The graph of figure 3.1 is the line graph of the graph shown in figure 3.3. This observation, together with the fact that Hamiltonian Path is NP- Hard in line graphs of bipartite graphs leads to the following theorem. Theorem 3.1 (From [2]). The single player version of UNO where the number of colors c and number of numbers b is unbounded is NP-Hard. The parameterized version of the problem where the number of colors c 18

21 is the parameter, can be shown to be in XP, by showing that the underlying card graph has small clique-width and solving Hamiltonian Path with dynamic programming, see section 2. Lemma 3.1. Single player UNO graphs have clique-width at most 2c. Proof. An arbitrary UNO graph can be constructed line by line as follows: Construct the vertices of the first line and give them all different colors among {1, 2,...,c}. Assuming that l lines are already constructed using colors {1, 2,..., c} and that all vertices in a column share the same color, construct the vertices in line l + 1 and give them all different colors within {c + 1, c + 2,..., 2c}. Connect vertex with color c+i with vertex with color c+j for all i j. Connect vertex with color c + i to vertices with color i. Rename color c + i to i. From Lemma 3.1 and using standard dynamic programming techniques, we can solve the single player version of UNO in time polynomial in the size of the input (assuming that the number of colors is a small constant). Theorem 3.2. The single player parameterized version of UNO where c is a parameter is in XP. An interesting question which was posed in [13] is whether the above parameterized problem is in FPT. Unfortunately, Hamilton cycle is known to be W-hard for clique-width (see [18]), so this observation can t be used in order to provide an FPT algorithm. On the other hand, one can consider 19

22 the equivalent problem Edge Hamiltonian Path (find a Hamilton path on the line graph) on a bipartite graph. If the number of colors (or numbers) is small, then this graph has a small vertex cover. The reason this fact might be helpful is the following observation: the large part can be divided into a small (bounded by a function of c) number of groups of vertices which are equivalent in terms of neighborhood. This fact could be helpful in order to provide a kernel for the problem. We are not aware of Edge Hamiltonian Path being solvable for graphs of small vertex cover, but this might be an interesting problem to tackle. Furthermore, the UNO graph itself has a very interesting structure that one could possibly explore in order to provide an FPT algorithm. 3.2 Many-player uncooperative version As it was suggested in the introduction of this section, the uncooperative version of UNO is equivalent to playing Generalized Geography on the card graph. The card graph is a directed p partite graph. However, in the 2-player case the graph is undirected since two cards u C 1 and v C 2 can be discarded either in the order u, v or in the order v, u. As it was stated in section 2, Generalized Geography is in P for undirected graphs. As a result, 2-player uncooperative UNO is also in P. An interesting question is the case of p 3. We conjecture that this problem is PSPACE-Complete. Namely, one can try to provide a reduction from Edge Generalized Geography on directed bipartite graphs, which is a known PSPACE-Complete problem (see [20]), following similar proof ideas as the ones that appear in [13]. Last, yet another interesting variation to consider (given that the p-player game is indeed proven to be PSPACE-hard for p 3) is the game parameterized by the number of rounds played. It can be shown that it this problem 20

23 is in XP by the trivial algorithm that creates the game DAG for all possible paths of length up to k with k being the number of rounds (size O(n k ) ) and analyzing the outcome starting from the leaves and working all the way to the root. It is interesting to explore whether there exists an FPT algorithm for this problem. 4 The Game of SET The game of SET is a card game in which players seek to form Sets of cards from a special deck. Each card from this deck has a picture with 4 attributes (shape, color, number, shading), and each attribute can take one of 3 values (for example the shape can be oval, squiggle, or diamond, the color can be blue, green, or purple, etc). To create a Set 2, the player needs to identify 3 cards in which, for each attribute independently, either all cards agree on the value, or they constitute a rainbow of all possible values. In a single round of the normal play, 12 cards are dealt and the players seek (simultaneously) a Set. The first player to find a Set wins the 3 cards constituting it. Then 3 new cards are dealt in the old ones places and the game continues with the next round. In the unlucky event that no Set exists among the 12 cards, players deal 3 more from the stack. The game finishes when there are no more cards to deal. The player acquiring the most cards wins. For more information regarding the game and its rules as well as for other variations see the official website of the game The game of SET has gained remarkable attention and popularity (especially among mathematicians) as well as many awards. The game has been the subject of both educational and technical research. A broad set of educational activities has been suggested, a collection of which can be found 2 The first letter of Set is capitalized to avoid a mix-up with the notion of mathematical set 21

24 in [24]. Furthermore, the game has been studied extensively from a more technical mathematical point of view, considering questions like what is the maximum number of cards with n attributes and 3 values that can be laid such that no Sets are formed [10], or for fixed n, how many non-isomorphic collections of n cards are there [5]). In [33], many other similar questions are posed. In addition to the game s popularity, one motivation for this intense study is that the problem has a very natural alternative mathematical formulation: if one describes the cards as four-dimensional vectors over the set {0, 1, 2}, then a Set is exactly a collection of three collinear points, that is, three points whose vectors add up to 0(mod3). Nevertheless, the first and - to the best of our knowledge - only attempt to consider the game s computational complexity was made by Chaudhuri et al [3] in 2003, who showed that a generalization of the game is NP-complete. In order to study a game from the viewpoint of computational complexity theory, one needs to define a natural generalization of the game in question (as the original constant size game always has constant time and space complexity). In a round of SET, there are 3 parameters to consider: the number of cards m, the number of attributes n and the number of values k (in the original game m = 12, n = 4 and k = 3). A subset of k cards will be considered to be a Set if for all attributes, values either all agree or all differ. Of course these three parameters are not totally independent as the number of cards m is upper-bounded by k n. In any multi-round version of the game, an extra parameter r being the number or rounds is added. 4.1 One round of SET Chaudhuri et al. in [3] consider a single-round version of SET. We are dealt m cards, each with n attributes that can take one of k values and we need to find a set of size k. We call this problem k-value 1-Set. Their main insight is 22

25 Figure 4.1: An example of an n dimensional perfect matching and of a Set of cards with n attributes and k values each. that this problem can be seen as a hypergraph problem. Specifically, one may construct a hypergraph on n k vertices, each representing an attribute-value pair. Now, cards can be represented as hyperedges, by including in each hyperedge the k values that describe the corresponding card s attributes. See figure 4.1. It is not hard to see that a perfect matching in this n-partite hypergraph corresponds to a Set in the original instance. On the other hand, some Sets do not correspond to perfect matchings, because all cards may share the same value for some attributes. Nevertheless, Chaudhuri et al. have established that the two problems have the same complexity and finding a Set is essentially algorithmically equivalent to find a perfect matching in this hypergraph. Theorem 4.1. Perfect Multi-Dimensional Matching is polynomially reducible to k-value 1-Set. Proof. Given an instance of Perfect Multi-Dimensional Matching namely an n partite graph G(V 1 V 2... V n, E) with each part of equal size k, we construct an n partite graph G (V 1 V 2... V n, E ) as follows: we add a k + 1 th value (vertex) v ik+1 in each set V i to create V Also E = E e, where e = {v 1k+1, v 2k+1,..., v nk+1 } is a multiedge that spans through the newly added vertices v ik+1. Now, if G has a perfect multidimensional matching M then G also has a multidimensional matching M = M e. Since M is a multidimensional 23 i.

26 matching, it is also a Set in G. On the other hand, if G has a Set M, then this has to be a multidimensional matching since no other multiedge apart from e goes through the vertices v ik+1. So M \ e should be a multidimensional perfect matching in G. In what follows we will exploit this connection between k-value 1-Set and Perfect Multi-Dimensional Matching to analyze the complexity of finding a Set with respect to the three relevant parameters m, n, and k. If k is unbounded, Theorem 4.1 implies that k-value 1-Set is NP-hard even for just 3 attributes. If the cards have only 2 attributes, the game is in P as this problem is equivalent with finding a perfect matching or a star in a bipartite graph. On the other hand, if n is unbounded but the number of values k is considered as a parameter, the problem is in XP (by the trivial algorithm that enumerates all size-k sets of cards and checking whether any of them constitutes a Set). An interesting open question is whether the trivial algorithm can be improved to an FPT algorithm. In [27], this question was answered negatively, by proving that the problem is W[1]-hard. This W-hardness proof applies to Perfect Multi-Dimensional Matching as well, proving that Perfect Multi-Dimensional Matching parameterized by the size of the dimensions k (while the number of dimensions n is unbounded) is W[1]-hard. This result may be of independent interest, as this is a natural parameterization of a classic problem that has not been considered before. The only relevant parameterized result known is that Maximum Multi-Dimensional Matching parameterized by the size of the matching and the number of dimensions is FPT (first established in [14] and further improved in [4]. 24

27 4.2 Multi-round versions In order to study the game from a more realistic point of view, it is necessary to study multi-round variations. The game is already NP-Complete for one round for n 3. On the other hand, if k = 2 and n is unbounded, the answer is trivial as any two cards form a Set. So it makes sence to focus our attention to the case where the number of values k is 3 and n is unbounded (k = 3 is also interesting, since this is the value of the parameter k in the actual game). We remind the reader that there is a polynomial time algorithm to find whether there exists at least one Set, in other words to play just one round. The complexity stays the same even if we consider the question of enumerating all Sets. This generalizes the daily puzzles found either on the official website of SET or in the New York Times. In these puzzles we are given m cards and need to find the maximum number of Sets assuming that we don t remove any cards from the table after finding a Set. It becomes interesting to ask the same question for a multi-round game, where cards are gradually removed. This corresponds to the CO-OP version of the game, where players have to cooperate in order to find the maximum number of available Sets given that cards of found Sets are removed from the table (see rules for CO-OP SET for the rules of this variation). We call this variation Max 3-Value r-set. Another interesting variation is the corresponding minimization version, where we are looking for the minimum number of Sets that once picked destroy all existing Sets. We call this variation Min 3-Value r-set. Both problems can be seen as special cases of more general packing and covering problems. In the maximization version, one is looking for a maximum 3-set packing, while in the minimization version one is looking for a minimum independent edge dominating set (or equivalently a Minimum Maximal Matching) in a 3-uniform hypergraph. It is interesting to examine whether these problems remain NP-Hard even on instances that correspond to the SET game. This question is answered 25

28 positively in [27]. For the latter problem, it is easy to show a reduction from Independent Edge Dominating Set on graphs. Theorem 4.2. Min 3-Value r-set is polynomially reducible to Independent Edge Dominating Set. Proof. Given an instance of Independent Edge Dominating Set (a graph G(V, E) and a number r), we create an instance of Min 3-Value r-set of V + E cards with V dimensions each, such that if G has an edge dominating set of size at most r then there exist at most r Sets which once picked up destroy all other Sets. Cards will be represented by vectors in F V 3. The construction is as follows: For each vertex i V we create a card where all coordinates are 0 except from the value of the i th coordinate which is equal to 1. Furthermore, for each edge (i, j) E we create a card where all coordinates are 0 except from the values of coordinates i and j which are equal to 2. Observe that the only Sets formed correspond directly to edges in G. Picking a Set corresponding to edge (i, j) eliminates the cards corresponding to vertices i, j (together with the card corresponding to edge (i, j)). This move causes the elimination of any potential Set containing cards corresponding to vertices i and j. Thus an edge dominating set of size at most r in G corresponds to an equal number of Sets overlapping all other Sets. On the other hand the smallest number of Sets that overlap all other Sets is equal to the minimum edge dominating set. From the parameterized point of view, if one considers as the parameter the number of rounds played r, natural parameterizations of the two problems are to ask whether there exist at least r mutually disjoint Sets, or whether there exist at most r Sets to destroy all Sets. The former is shown to be 26

29 Fixed Parameter Tractable by Chen et al. [4]. The latter is an interesting open question, and it is tackled in [27] through the close connection with Independent Edge Dominating Set on graphs that was presented. 4.3 A two-player game In this section, we analyse a potentially interesting two-player game which we call 2P 3-Value Set. Suppose that an arbitrary set of cards is on the table and two opposing players take turns playing. Each player may select three cards that form a Set and remove them from play. No additional cards are dealt. The game goes on until a player is unable to find a Set, in which case she loses. Unlike the solitaire games Max 3-Value r-set and Min 3-Value r- Set, here players must exercise some strategic thinking: each player is trying not only to maximize the number of Sets she will collect but also to prevent the opponent from forming a set. This game can be seen as a restriction of the game Arc Kayles in 3- uniform hypergraphs (where hyperedges should be valid Sets). Arc Kayles is defined below. Definition 4.1. Problem: Arc Kayles. Input: A (hyper-)graph G(V, E). Description: Players take turns picking edges from G. When an edge e is picked, e together with all incident vertices to e are removed. First player unable to pick an edge loses. Question: Is there a winning strategy for player A? The complexity of Arc Kayles is currently unknown even on graphs and it has been a long-standing open question since the PSPACE-Completeness of its sibling problem Node Kayles was established in [30]. The multiround 2-player version of SET is at least as hard as Arc Kayles, following 27

30 the reduction of Min 3-Value r-set to Independent Edge Dominating Set. A slightly more general version of Arc Kayles is mentioned to be PSPACE-complete in [30], while the natural generalization of Arc Kayles to hypergraphs with unbounded hyperedge size is PSPACE-hard by the complexity of poset games [23]. The 2-player SET problem on graphs is a natural restriction of Arc Kayles, though this version of SET, unlike its hypergraph counterpart turns out to be trivial: if the size of the Sets (i.e. the number of different values) is 2 then any 2 cards form a Set; thus the 2-player problem is equivalent to Arc Kayles on complete graphs and becomes a simple matter of parity of the number of nodes. A possible direction is to study 2P 3-Value Set from a parameterized point of view. An interesting question is whether player 1 has a winning strategy in r moves. In [1] it was established that the r-move parameterized version of Node Kayles is AW[*]-hard. This question is addressed in [27], where we prove that the problem is FPT. Settling that 2-player SET is FPT parameterized by r implies the same result for Arc Kayles on graphs. 28

31 References [1] Karl R. Abrahamson, Rodney G. Downey, and Michael R. Fellows. Fixed-parameter tractability and completeness iv: On completeness for w[p] and pspace analogues. Ann. Pure Appl. Logic, 73(3): , [2] Alan A Bertossi. The edge Hamiltonian path problem is NP-complete. Information Processing Letters, 13(4): , [3] K. Chaudhuri, B. Godfrey, D. Ratajczak, and H. Wee. On the complexity of the game of Set. Manuscript, [4] Jianer Chen, Qilong Feng, Yang Liu, Songjian Lu, and Jianxin Wang. Improved deterministic algorithms for weighted matching and packing problems. Theor. Comput. Sci., 412(23): , [5] Ben Coleman and Kevin Hartshorn. Game, set, math. Mathematics Magazine, 85(2):83 96, [6] S.A. Cook. The complexity of theorem-proving procedures. In Proceedings of the third annual ACM symposium on Theory of computing, pages ACM, [7] Derek G Corneil and Udi Rotics. On the relationship between cliquewidth and treewidth. SIAM Journal on Computing, 34(4): , [8] Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and computation, 85(1):12 75, [9] Bruno Courcelle, Johann A Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33(2): , [10] Benjamin Lent Davis, Davis, and Diane Maclagan. The card game set, [11] E. Demaine, S. Hohenberger, and D. Liben-Nowell. Tetris is hard, even to approximate. Computing and Combinatorics, pages ,

32 [12] Erik D Demaine, Martin L Demaine, Nicholas J. A Harvey, Ryuhei Uehara, Takeaki Uno, and Yushi Uno. UNO is hard, even for a single player. Manuscript, [13] Erik D Demaine, Martin L Demaine, Ryuhei Uehara, Takeaki Uno, and Yushi Uno. UNO is hard, even for a single player. In Fun with Algorithms, pages Springer, [14] Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Springer-Verlag, pp. [15] Wolfgang Espelage, Frank Gurski, and Egon Wanke. How to solve NPhard graph problems on clique-width bounded graphs in polynomial time. In Graph-theoretic concepts in computer science, pages Springer, [16] Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci., 410(1):53 61, [17] Michael R Fellows, Frances A Rosamond, Udi Rotics, and Stefan Szeider. Clique-width minimization is NP-hard. In Proceedings of the thirtyeighth annual ACM symposium on Theory of computing, pages ACM, [18] Fedor V Fomin, Petr A Golovach, Daniel Lokshtanov, and Saket Saurabh. Clique-width: on the price of generality. In Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages Society for Industrial and Applied Mathematics, [19] A.S. Fraenkel and D. Lichtenstein. Computing a perfect strategy for n n chess requires time exponential in n. Journal of Combinatorial Theory, Series A, 31(2): , [20] Aviezri S Fraenkel, Edward R Scheinerman, and Daniel Ullman. Undirected edge geography. Theoretical Computer Science, 112(2): , [21] Michael R Garey and David S Johnson. Computers and Intractability, volume 174. freeman New York,