Large Peg-Army Maneuvers

Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Cs\'ak\'any and Juh\'asz (Mathematics Magazine, 2000). To appreciate the combinatorial beauty of our solutions, we recommend to visit the gallery of animations provided at http://solitairearmy.isnphard.com.Comment: Conference versio

Parameterized Complexity of Graph Constraint Logic

Graph constraint logic is a framework introduced by Hearn and Demaine, which provides several problems that are often a convenient starting point for reductions. We study the parameterized complexity of Constraint Graph Satisfiability and both bounded and unbounded versions of Nondeterministic Constraint Logic (NCL) with respect to solution length, treewidth and maximum degree of the underlying constraint graph as parameters. As a main result we show that restricted NCL remains PSPACE-complete on graphs of bounded bandwidth, strengthening Hearn and Demaine's framework. This allows us to improve upon existing results obtained by reduction from NCL. We show that reconfiguration versions of several classical graph problems (including independent set, feedback vertex set and dominating set) are PSPACE-complete on planar graphs of bounded bandwidth and that Rush Hour, generalized to $k\times n$ boards, is PSPACE-complete even when $k$ is at most a constant

On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games

Peg Duotaire is a two-player version of the classical puzzle called Peg Solitaire. Players take turns making peg-jumping moves, and the first player which is left without available moves loses the game. Peg Duotaire has been studied from a combinatorial point of view and two versions of the game have been considered, namely the single- and the multi-hop variant. On the other hand, understanding the computational complexity of the game is explicitly mentioned as an open problem in the literature. We close this problem and prove that both versions of the game are PSPACE-complete. We also prove the PSPACE-completeness of other peg-jumping games where two players control pegs of different colors

Peg-Solitaire, String Rewriting Systems and Finite Automata

We consider a class of length-preserving string rewriting systems and show that the set of encodings of pairs of strings ! s; f ? such that f can be derived from s using the rewriting rules can be accepted by finite automata. As a consequence, we show the existence of a linear time algorithm for determining the solvability of a given k \Theta n peg-solitaire board, for any fixed k. This result is in contrast to the recent results of [UEHA] and [AVIS] that the same problem is NP-hard for n \Theta n boards. We look at some related string rewriting systems and find conditions under which the encodings of the pairs ! s; f ? where f can be derived from s is regular. 1 Introduction Peg Solitaire is one of the most popular solitaire board games. Its history dates back to at least seventeenth century. It has been sold as a board game in various shapes, sizes and names. A complete chapter of the well-known work on mathematical games due to Berlekamp et al. [BERL] is devoted to peg-solitaire...