PSPACE-Complete Two-Color Placement Games

We show that three placement games, Col, NoGo, and Fjords, are PSPACE-complete on planar graphs. The hardness of Col and Fjords is shown via a reduction from Bounded 2-Player Constraint Logicand NoGo is shown to be hard directly from Col. 1 Background 1.1 Combinatorial Game Theory Combinatorial Game Theory is the study of games with: • Two players alternating turns, • No randomness, and • Perfect information for both players. A ruleset is a pair of functions that determines which moves each player can make from some position. Most games in this paper use normal play rules, meaning if a player can’t make a move on their turn, they lose the game (i.e. the last player to move wins). The two players are commonly known as Left and Right. The rulesets discussed here include players assigned to different colors (Blue vs Red or Black vs White). We use the usual method of distinguishing between them: Left will play as Blue and Black; Right plays Red and White. For more information on combinatorial game theory, the reader is encouraged to look at [2] and [1]. 1.2 Algorithmic Combinatorial Game Theory Algorithmic Combinatorial Game Theory is the application of algorithms to combinatorial games. The difficulty of a ruleset is analyzed by studying the computational complexity of determining whether the current player has a winning strategy. In this paper, we show that many games are PSPACE-complete, which means that no polynomial-time algorithm exists to determine the winnability of all positions unless such an algorithm exists for all PSPACE problems. Usually determining the winnability of a ruleset is considered as the computational problem of the same name. We use that language here, e.g. saying Bounded 2-Player Constraint Logic is PSPACE-complete means that the associated winnability problem is PSPACE-complete. All games considered in this paper exist in PSPACE due to the max height of the game tree being polynomial []. Thus, by showing that any of these games are PSPACE-hard, we also show that they are PSPACE-complete. For more on algorithmic combinatorial game theory, the reader is encouraged to reference [4]. 1 ar X iv :1 60 2. 06 01 2v 1 [ cs .C C ] 1 9 Fe b 20 16 1.3 Bounded 2-Player Constraint Logic Bounded 2-Player Constraint Logic is a combinatorial ruleset played on a directed graph where each arc has three properties: • Color: which of the two players is allowed to flip it. • Flipped: a boolean flag indicating whether it has already been flipped. Each arc may be flipped only once. • Weight: one of {1, 2}.1 An orientation of the arc is legal if each vertex in the graph has at total weight of incoming edges of at least 2. A move consists of a player choosing an arc, (v, w), to flip where: • The arc is that player’s color, and • The arc has not yet been flipped, and • Flipping the arc (meaning the graph with (w, v) replacing (v, w)) results in a legal orientation. The goal of Bounded 2-Player Constraint Logic is for Left to flip a goal edge. If they can flip this edge, then they win the game. Otherwise, Right wins. Bounded 2-Player Constraint Logic is PSPACE-complete, even when: • The graph is planar, and • Only six types of vertices exist in the graph. These six vertex types are: And, Or, Choice, Split, Variable, and Goal, named for the gadgets they represent in the proof of Bounded 2-Player Constraint Logic hardness [5]. The following is a description of each of these vertices. Diagrams for each may be found in [5]. • Variable: One of two edges (one of each color) may be flipped. Black’s edge corresponds to setting the variable to true, White sets it to false. • Goal: This is the Black edge that Left needs to flip to win the game. • And: A vertex with two outward-oriented “input” edges and one inward-oriented “output” edge. In order to flip the output edge, both input edges must first be flipped. • Or: Another vertex with two inputs and one output, but here only one of the inputs must be flipped in order for the output to be flipped. • Choice: One input edge which, when flipped to orient inwards, means one of two output edges may be flipped to orient outwards. • Split: One input edge which, when flipped to orient inwards, allows both output edges to be flipped orienting outwards. In order to use Bounded 2-Player Constraint Logic as the source problem for a proof of PSPACE-hardness, it is sufficient to show that gadgets that simulate each of the six Bounded 2-Player Constraint Logic vertex types. We use this to reduce directly to Col and GraphFjords to show both are PSPACE-complete. For more information about the structure of each of these gadgets, the interested reader may reference [5]. Warning: in [5], these weights are denoted by blue vs. red edges. These colors do not correspond to the identity of the player that may flip the arc.