Deciding the Winner of an Arbitrary Finite Poset Game is PSPACE-Complete

A poset game is a two-player game played over a partially ordered set (poset) in which the players alternate choosing an element of the poset, removing it and all elements greater than it. The first player unable to select an element of the poset loses. Polynomial time algorithms exist for certain restricted classes of poset games, such as the game of Nim. However, until recently the complexity of arbitrary finite poset games was only known to exist somewhere between NC^1 and PSPACE. We resolve this discrepancy by showing that deciding the winner of an arbitrary finite poset game is PSPACE-complete. To this end, we give an explicit reduction from Node Kayles, a PSPACE-complete game in which players vie to chose an independent set in a graph

Strategy-Stealing Is Non-Constructive

In many combinatorial games, one can prove that the first player wins under best play using a simple but non-constructive argument called strategy-stealing. This work is about the complexity behind these proofs: how hard is it to actually find a winning move in a game, when you know by strategy-stealing that one exists? We prove that this problem is PSPACE-Complete already for Minimum Poset Games and Symmetric Maker-Maker Games, which are simple classes of games that capture two of the main types of strategy-stealing arguments in the current literature

The Computational Complexity of the Game of Set and its Theoretical Applications

The game of SET is a popular card game in which the objective is to form Sets using cards from a special deck. In this paper we study single- and multi-round variations of this game from the computational complexity point of view and establish interesting connections with other classical computational problems. Specifically, we first show that a natural generalization of the problem of finding a single Set, parameterized by the size of the sought Set is W-hard; our reduction applies also to a natural parameterization of Perfect Multi-Dimensional Matching, a result which may be of independent interest. Second, we observe that a version of the game where one seeks to find the largest possible number of disjoint Sets from a given set of cards is a special case of 3-Set Packing; we establish that this restriction remains NP-complete. Similarly, the version where one seeks to find the smallest number of disjoint Sets that overlap all possible Sets is shown to be NP-complete, through a close connection to the Independent Edge Dominating Set problem. Finally, we study a 2-player version of the game, for which we show a close connection to Arc Kayles, as well as fixed-parameter tractability when parameterized by the number of rounds played

Easy to Win, Hard to Master: Optimal Strategies in Parity Games with Costs

The winning condition of a parity game with costs requires an arbitrary, but fixed bound on the distance between occurrences of odd colors and the next occurrence of a larger even one. Such games quantitatively extend parity games while retaining most of their attractive properties, i.e, determining the winner is in NP and co-NP and one player has positional winning strategies. We show that the characteristics of parity games with costs are vastly different when asking for strategies realizing the minimal such bound: the solution problem becomes PSPACE-complete and exponential memory is both necessary in general and always sufficient. Thus, playing parity games with costs optimally is harder than just winning them. Moreover, we show that the tradeoff between the memory size and the realized bound is gradual in general

The Computational Complexity of Some Games and Puzzles With Theoretical Applications

The subject of this thesis is the algorithmic properties of one- and two-player games people enjoy playing, such as Sudoku or Chess. Questions asked about puzzles and games in this context are of the following type: can we design efficient computer programs that play optimally given any opponent (for a two-player game), or solve any instance of the puzzle in question? We examine four games and puzzles and show algorithmic as well as intractability results. First, we study the wolf-goat-cabbage puzzle, where a man wants to transport a wolf, a goat, and a cabbage across a river by using a boat that can carry only one item at a time, making sure that no incompatible items are left alone together. We study generalizations of this puzzle, showing a close connection with the Vertex Cover problem that implies NP-hardness as well as inapproximability results. Second, we study the SET game, a card game where the objective is to form sets of cards that match in a certain sense using cards from a special deck. We study single- and multi-round variations of this game and establish interesting con- nections with other classical computational problems, such as Perfect Multi- Dimensional Matching, Set Packing, Independent Edge Dominating Set, and Arc Kayles. We prove algorithmic and hardness results in the classical and the parameterized sense. Third, we study the UNO game, a game of colored numbered cards where players take turns discarding cards that match either in color or in number. We extend results by Demaine et. al. (2010 and 2014) that connected one- and two-player generaliza- tions of the game to Edge Hamiltonian Path and Generalized Geography, proving that a solitaire version parameterized by the number of colors is fixed param- eter tractable and that a k-player generalization for k greater or equal to 3 is PSPACE-hard. Finally, we study the Scrabble game, a word game where players are trying to form words in a crossword fashion by placing letter tiles on a grid board. We prove that a generalized version of Scrabble is PSPACE-hard, answering a question posed by Demaine and Hearn in 2008

Parity and Streett Games with Costs

We consider two-player games played on finite graphs equipped with costs on edges and introduce two winning conditions, cost-parity and cost-Streett, which require bounds on the cost between requests and their responses. Both conditions generalize the corresponding classical omega-regular conditions and the corresponding finitary conditions. For parity games with costs we show that the first player has positional winning strategies and that determining the winner lies in NP and coNP. For Streett games with costs we show that the first player has finite-state winning strategies and that determining the winner is EXPTIME-complete. The second player might need infinite memory in both games. Both types of games with costs can be solved by solving linearly many instances of their classical variants.Comment: A preliminary version of this work appeared in FSTTCS 2012 under the name "Cost-parity and Cost-Streett Games". The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreements 259454 (GALE) and 239850 (SOSNA

Taking and Breaking Games

V této práci analyzujeme několik otevřených problémů v oblasti nestranných i stranných her typu Taking and Breaking. Pro nestranné odčítací hry dokážeme existenci hry s aperiodickou nim-sekvencí a periodickou sekvencí výhra-prohra. Analyzujeme ekvivalenční třídy těchto her a nalézáme řešení jedné z těchto tříd. Také představujeme novou hru typu Taking and Breaking, kterou z velké části vyřešíme. V oblasti stranných her provedeme analýzu několika odčítacích her a her typu Pure Breaking. Pro tyto hry také představíme obecnou techniku testování aritmetické periodicity. Pro automatické řešení nestranných her typu Taking and Breaking navrhujeme několik algoritmů. Práci uzavíráme důkazem PSPACE-těžkosti nestranné zobecněné odčítací hry a EXPTIME-těžkosti této hry ve stranné variantě.In this thesis, we examine several open problems in taking and breaking games under the impartial and partizan setting. We prove the existence of an impartial subtraction game with aperiodic nim-sequence and periodic outcome sequence. We also analyze the equivalence classes of subtraction games and provide a solution to one of these classes. We introduce a new taking and breaking game and partially solve it. Then we solve several partizan subtraction games and partizan pure breaking games and describe a general technique for testing arithmetic periodicity of these games. Moreover, we design some game solving algorithms for impartial taking and breaking games. We prove PSPACE-hardness for a generalized subtraction game under the impartial setting and EXPTIME-hardness under the partizan setting

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems