Recent results and questions in combinatorial game complexities

Abstract Aim: Present a systematic development of part of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Isolate the various difficulties separating hard from easy games, and attack them individually. Presentation: Informal; examples of games sampled from various levels illustrate the theory, with emphasis on formulating and motivating new and old research problems.

[1]  Edward M. Reingold,et al.  The Complexity of Pursuit on a Graph , 1995, Theor. Comput. Sci..

[2]  Aviezri S. Fraenkel,et al.  The generalized Sprague-Grundy function and its invariance under certain mappings , 1986, J. Comb. Theory, Ser. A.

[3]  Robert E. Tarjan,et al.  A combinatorial problem which is complete in polynomial space , 1975, STOC.

[4]  E. Berlekamp,et al.  Winning Ways for Your Mathematical Plays , 1983 .

[5]  Ahiezer S. Shaki,et al.  Algebraic solutions of partizan games with cycles , 1979, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  F. L. Morris.  Playing disjunctive sums is polynomial space complete , 1981 .

[7]  Edward R. Scheinerman,et al.  A deletion game on hypergraphs , 1991, Discret. Appl. Math..

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

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

[10]  Elwyn R. Berlekamp,et al.  Mathematical Go - chilling gets the last point , 1994 .

[11]  Shuo-Yen Robert Li,et al.  Sums of Zuchswang Games , 1976, J. Comb. Theory, Ser. A.

[12]  Aviezri S. Fraenkel,et al.  Nonhomogeneous spectra of numbers , 1981, Discret. Math..

[13]  D. Gale,et al.  Nim-type games , 1982 .

[14]  Aviezri S. Fraenkel.  Heap games, numeration systems and sequences , 1998 .

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

[16]  L. Yedwab.  ON PLAYING WELL IN A SUM OF GAMES , 1985 .

[17]  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.

[18]  J. Úlehla,et al.  A complete analysis of Von Neumann's hackendot , 1980 .

[19]  J. Conway.  On Numbers and Games , 1976 .

[20]  Jaroslav Nesetril,et al.  Epidemiography II. Games with a dozing yet winning player , 1988, J. Comb. Theory, Ser. A.

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

[22]  David S. Johnson,et al.  The complexity of checkers on an N × N board , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[23]  Aviezri S. Fraenkel,et al.  Systems of numeration , 1983, IEEE Symposium on Computer Arithmetic.

[24]  Aviezri S. Fraenkel,et al.  Computing a Perfect Strategy for n*n Chess Requires Time Exponential in N , 1981, ICALP.

[25]  Aviezri S. Fraenkel,et al.  Strategies for compounds of partizan games , 1982, Mathematical Proceedings of the Cambridge Philosophical Society.

[26]  David Wolfe.  The Gamesman's Toolkit , 1996 .

[27]  Aviezri S. Fraenkel,et al.  Complexity of problems in games, graphs and algebraic equations , 1979, Discret. Appl. Math..

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

[29]  John Michael Robson,et al.  The Complexity of Go , 1983, IFIP Congress.

[30]  J. H. Conway,et al.  All Games Bright and Beautiful , 1977 .

[31]  John H. Conway,et al.  A Gamut of Game Theories , 1978 .

[32]  John Michael Robson,et al.  Combinatorial Games with Exponential Space Complete Decision Problems , 1984, MFCS.

[33]  Shigeki Iwata,et al.  Classes of Pebble Games and Complete Problems , 1979, SIAM J. Comput..

[34]  Aviezri S. Fraenkel,et al.  Theory of annihilation games , 1976 .

[35]  Richard J. Nowakowski,et al.  Games of No Chance 3: Surveys , 1998 .

[36]  Michael O. Rabin,et al.  6. EFFECTIVE COMPUTABILITY OF WINNING STRATEGIES , 1958 .

[37]  Aviezri S. Fraenkel,et al.  Combinatorial games with an annihilation rule , 1974 .

[38]  David Gale,et al.  A Curious Nim-Type Game , 1974 .

[39]  Cedric A. B. Smith.  Graphs and composite games , 1966 .

[40]  Aviezri S. Fraenkel,et al.  Strategy for a class of games with dynamic ties , 1975 .

[41]  Thomas S. Ferguson,et al.  Misère Annihilation Games , 1984, J. Comb. Theory, Ser. A.