Complexity of Path-Forming Games

Abstract For a number of two-player games where players alternately choose the next vertex of a simple or an elementary path in a graph, we consider the problem to determine whether, for a given game instance, there is a winning strategy for the first player. We show several of these problems to be PSPACE-complete. In some special cases, we obtain polynomial-time algorithms, based on graph rewriting or an intricate form of dynamic programming, e.g. we show GENERALIZED GEOGRAPHY and some other PSPACE-complete problems to be linear-time-solvable on graphs with constant bounded treewidth.

[1]  David S. Johnson.  The NP-Completeness Column: An Ongoing Guide , 1986, J. Algorithms.

[2]  Aviezri S. Fraenkel,et al.  PSPACE-hardness of some combinatorial games , 1987, J. Comb. Theory, Ser. A.

[3]  J. Lagergren.  Efficient parallel algorithms for tree-decomposition and related problems , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[4]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[5]  Stephen T. Hedetniemi,et al.  Linear algorithms on k-terminal graphs , 1987 .

[6]  Hans L. Bodlaender.  On the Complexity of Some Coloring Games , 1991, Int. J. Found. Comput. Sci..

[7]  John Michael Robson,et al.  N by N Checkers is Exptime Complete , 1984, SIAM J. Comput..

[8]  Shigeki Iwata,et al.  Some combinatorial game problems require Ω(nk) time , 1984, JACM.

[9]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[10]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[11]  Kurt Mehlhorn,et al.  On the complexity of a game related to the dictionary problem , 1989, 30th Annual Symposium on Foundations of Computer Science.

[12]  Robert E. Tarjan,et al.  A Combinatorial Problem Which Is Complete in Polynomial Space , 1976, JACM.

[13]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[14]  Hans L. Bodlaender,et al.  Some Classes of Graphs with Bounded Treewidth , 1988, Bull. EATCS.

[15]  Hans L. Bodlaender,et al.  NC-Algorithms for Graphs with Small Treewidth , 1988, WG.

[16]  Hans L. Boblaender.  Polynomial algorithms for graph isomorphism and chromatic index on partial k -trees , 1990 .

[17]  Hans L. Bodlaender,et al.  Improved Self-reduction Algorithms for Graphs with Bounded Treewidth , 1990, Discret. Appl. Math..

[18]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[19]  Eugene L. Lawler,et al.  Linear-Time Computation of Optimal Subgraphs of Decomposable Graphs , 1987, J. Algorithms.

[20]  Robin Thomas,et al.  Algorithms Finding Tree-Decompositions of Graphs , 1991, J. Algorithms.

[21]  Michael R. Fellows,et al.  An analogue of the Myhill-Nerode theorem and its use in computing finite-basis characterizations , 1989, 30th Annual Symposium on Foundations of Computer Science.

[22]  Clemens Lautemann,et al.  Efficient Algorithms on Context-Free Graph Grammars , 1988, ICALP.

[23]  Thomas Lengauer,et al.  Efficient Algorithms for Finding Minimum Spanning Forests of Hierarchically Defined Graphs , 1987, J. Algorithms.

[24]  Stefan Arnborg,et al.  Linear time algorithms for NP-hard problems restricted to partial k-trees , 1989, Discret. Appl. Math..

[25]  Thomas J. Schaefer,et al.  On the Complexity of Some Two-Person Perfect-Information Games , 1978, J. Comput. Syst. Sci..

[26]  S. Arnborg,et al.  Characterization and recognition of partial 3-trees , 1986 .

[27]  Stefan Arnborg,et al.  Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey , 1985, BIT.

[28]  Hans L. Bodlaender,et al.  Fast algorithms for the Tron game on trees , 1990 .

[29]  H. Bodlaender.  Planar graphs with bounded treewidth , 1988 .

[30]  John Michael Robson,et al.  Alternation with Restrictions on Looping , 1986, Inf. Control..

[31]  Ton Kloks,et al.  Better Algorithms for the Pathwidth and Treewidth of Graphs , 1991, ICALP.

[32]  Bruno Courcelle,et al.  An algebraic theory of graph reduction , 1990, JACM.

[33]  Larry J. Stockmeyer,et al.  Provably Difficult Combinatorial Games , 1979, SIAM J. Comput..

[34]  Thomas Lengauer,et al.  Efficient Solution of Connectivity Problems on Hierarchically Defined Graphs , 1988, SIAM J. Comput..

[35]  David Lichtenstein,et al.  GO Is Polynomial-Space Hard , 1980, JACM.

[36]  Detlef Seese,et al.  Easy Problems for Tree-Decomposable Graphs , 1991, J. Algorithms.

[37]  Aviezri S. Fraenkel,et al.  Computing a Perfect Strategy for n x n Chess Requires Time Exponential in n , 1981, J. Comb. Theory, Ser. A.