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Abstract
This paper introduces graph polynomials based on a concept from the game of Go. Suppose that, for each vertex of a graph, we either leave it uncoloured or choose a colour uniformly at random from a set of available colours, with the choices for the vertices being independent and identically distributed. We ask for the probability that the resulting partial assignment of colours has the following property: for every colour class, each component of the subgraph it induces has a vertex that is adjacent to an uncoloured vertex. In Go terms, we are requiring that every group is uncaptured. This definition leads to Go polynomials for a graph. Although these polynomials are based on properties that are less “local” in nature than those used to define more traditional graph polynomials such as the chromatic polynomial, we show that they satisfy recursive relations based on local modifications similar in spirit to the deletion–contraction relation for the chromatic polynomial. We then show that they are #P-hard to compute in general, using a result on linear forms in logarithms from transcendental number theory. We also briefly record some correlation inequalities.
... Two Go polynomials based on these concepts were introduced in [31], studied further in [42,36,39], and mentioned by János in his first inventory of the Zoo [69,70]. One of them simply counts legal positions: ...
... For example, it can be shown that Go # (C 4 ; λ) = 1 + 14λ 2 . The other Go polynomial from [31] is based on a simple probability model. Let p ≤ 1 2 and construct a random partial 2-assignment f as follows. ...
... This suggests the problem of finding combinatorial interpretations at other values of p and λ. We suggest λ = −1 as one that might be worth exploring, since the chromatic polynomial has an interesting combinatorial interpretation at λ = −1, namely the number of acyclic orientations [100], and Go polynomials can be expressed naturally as sums of chromatic polynomials [31]. ...
We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction relations (simple linear recursions based on local operations), perhaps in a wider class of combinatorial objects? How many levels of reduction relations does a graph polynomial need in order to express it in terms of trivial base cases? For a graph polynomial, how are properties such as equivalence and factorisation reflected in the structure of a graph? We illustrate our discussion with a variety of graph polynomials and other invariants. This leads us to reflect on the historical origins of graph polynomials. We also introduce some new polynomials based on partial colourings of graphs and establish some of their basic properties.
... In this paper we are concerned with the number of legal Go positions on grids of various types and sizes. This follows earlier work by one of us [8] on polynomials that give the numbers of legal Go positions when the game is played on general graphs with arbitrary numbers of players. Recursive schemes for computing these polynomials were given, and they were shown to be #P-hard to compute in general, even for just two players. ...
... The number of such positions (or assignments) is denoted by Go # (G; λ). This is one of the Go polynomials of [8]. In that paper, it was shown to be a polynomial in λ (for any given G), and that computation of it could be reduced to computation of chromatic polynomials of graphs constructed from G by local modifications. ...
... Unfortunately, exponentially many such modified graphs are used. A polynomial time algorithm for computing Go # (G; λ) for general graphs is unlikely to exist, since this Go polynomial is #P-hard to compute for all integers λ ≥ 2 (proved in [8] using a result on linear forms in logarithms from transcendental number theory). In this paper we are interested in much more constrained classes of graphs: planar lattice graphs of various kinds, especially square lattice graphs since it is on such a graph that Go is normally played. ...
We use transfer matrix methods to determine bounds for the numbers of legal Go positions for various numbers of players on some planar lattice graphs, including square lattice graphs such as those on which the game is normally played. We also Þnd bounds on limiting constants that describe the behaviour of the number of legal positions on these lattice graphs as the dimensions of the lattices tend to inÞnity. These results amount to giving bounds for some speciÞc evaluations of Go polynomials on these graphs.
... A relatively new topic of interest in combinatorial game theory is the enumeration of positions. Early work has focused on specific types of positions, such as Go end positions [6,5,16] and second-player win position for some lesser known games [7,11]. Recently, Domineering positions, as well as specific types of positions, were counted in [9]. ...
Distance games are games played on graphs in which the players alternately colour vertices, and which vertices can be coloured only depends on the distance to previously coloured vertices. The polynomial profile encodes the number of positions with a fixed number of vertices from each player. We extend previous work on finding the polynomial profile of several distance games (Col, Snort, and Cis) played on paths. We give recursions and generating functions for the polynomial profiles of generalizations of these three games when played on paths. We also find the polynomial profile of Cis played on cycles and the total number of positions of Col and Snort on cycles, as well as pose a conjecture about the number of positions when playing Col and Snort on complete bipartite graphs.
... Enumeration of positions has been studied, directly or indirectly, for several combinatorial games. The first few papers on counting game positions considered the problem of enumerating specific types of positions-Go end positions in [7,6,16] and second-player win positions for two games in [8,9]. In the game Node Kayles, played on a graph, the two players alternate choosing vertices not adjacent to any previously chosen ones, thus forming an independent set. ...
Domineering is a two player game played on a checkerboard in which one player places dominoes vertically and the other places them horizontally. We give bivariate generating polynomials enumerating Domineering positions by the number of each player's pieces. We enumerate all positions, maximal positions, and positions where one player has no move. Using these polynomials we count the number of positions that occur during alternating play. Our method extends to enumerating positions from mid-game positions and we include an analysis of a tournament game.
... Authors: Introduced by G.E. Farr in [Far03a]. ...
... The motivation for game polynomials came from Farr [6] in 2003 where the number of end positions and some polynomials of the game Go were considered, and work in this area was continued by Tromp and Farnebäck [10] in 2007 and by Farr and Schmidt [7] in 2008. Even though Go is not a placement game since pieces are removed, it shares many properties with this class of games. ...
Placement games are a subclass of combinatorial games which are played on
graphs. We will demonstrate that one can construct simplicial complexes
corresponding to a placement game, and this game could be considered as a game
played on these simplicial complexes. These complexes are constructed using
square-free monomials.
... [34, 57]. There are Farrell polynomials [34], clique and independent set polynomials [42] , dependence polynomials [35], Martin polynomials [32], Penrose polynomials [1], Go-polynomials [33], and many more. It is worth searching for all these at scholar.google.com. ...
We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular,
the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and
the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and
outline a research program for their classification in terms of definability and complexity considerations, and various notions
of reducibilities.
... [22, 39]. There are interlace polynomials [5], Go-polynomials [21], Penrose polynomials [1], and many more. It is worth searching for all these at scholar.google.com. ...
We outline a general theory of graph polynomials which cov- ers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce the class of (hyper)graph polynomials de- nable in second order logic, and outline a research program for their classication in terms of denabilit y and complexity considerations, and various notions of reducibilities.
With increased public interest in the ancient game of Go since 2016, it is an espe- cially good time to use it in teaching. The game is an excellent source of exercises in the theory of computation. We give some exercises developed during our research on Go which were then used when teaching this subject at Monash University. These are based on One-Dimensional Go (1D-Go) which uses a path graph as its board. They are about determining whether or not a position is legal and counting the number of legal positions. Curriculum elements that may be illustrated and practised using 1D-Go include: regular expressions, linear recurrences, proof by induction, nite automata, regular grammars, context-free grammars and languages, pushdown automata, and Turing machines.
In this paper we analyze a recently proposed impartial combinatorial ruleset that is played on a permutation of the set (Formula presented.). We call this ruleset Stirling Shave. A procedure utilizing the ordinal sum operation is given to determine the nim value of a given normal play position. Additionally, we enumerate the number of permutations of (Formula presented.) which are (Formula presented.)-positions. The formula given involves the Stirling numbers of the first-kind. We also give a complete analysis of the Misère version of Stirling Shave using Conway’s genus theory. An interesting by-product of this analysis is insight into how the ordinal sum operation behaves in Misère Play.
In the game of Go, the question of whether a ladder—a method of capturing stones-works, is shownto be PSPACE-complete. Our
reduction closely follows that of Lichtenstein and Sipser [2], who first showed PSPACE-hardness of Go by letting the outcome of a game depend on the capture of a large group of stones.
A greater simplicity is achieved by avoiding the need for pipes and crossovers.
Since the beginning of AI, mind games have been studied as relevant application fields. Nowadays, some programs are better than human players in most classical games. Their results highlight the efficiency of AI methods that are now quite standard. Such methods are very useful to Go programs, but they do not enable a strong Go program to be built. The problems related to Computer Go require new AI problem solving methods. Given the great number of problems and the diversity of possible solutions, Computer Go is an attractive research domain for AI. Prospective methods of programming the game of Go will probably be of interest in other domains as well. The goal of this paper is to present Computer Go by showing the links between existing studies on Computer Go and different AI related domains: evaluation function, heuristic search, machine learning, automatic knowledge generation, mathematical morphology and cognitive science. In addition, this paper describes both the practical aspects of Go programming, such as program optimization, and various theoretical aspects such as combinatorial game theory, mathematical morphology, and Monte Carlo methods.
This volume, the third in a sequence that began with The Theory of Matroids and Combinatorial Geometries, concentrates on the applications of matroid theory to a variety of topics from engineering (rigidity and scene analysis), combinatorics (graphs, lattices, codes and designs), topology and operations research (the greedy algorithm). As with its predecessors, the contributors to this volume have written their articles to form a cohesive account so that the result is a volume which will be a valuable reference for research workers.
This paper features a problem that was composed to illustrate the power of combinatorial game theory applied to Go endgame positions. The problem is the sum of many subproblems, over a dozen of which have temperatures significantly greater than one. One of the subproblems is a conspicuous four-point ko, and there are several overlaps among other subproblems. Even though the theory of such positions is far from complete, the paper demonstrates that enough mathematics is now known to obtain provably correct, counterintuitive, solutions to some very difficult Go endgame problems.
For a finite graph G with d vertices we define a homogeneous symmetric function XG of degree d in the variables x1, x2, ... . If we set x1 = ... = xn= 1 and all other xi = 0, then we obtain χG(n), the chromatic polynomial of G evaluated at n. We consider the expansion of XG in terms of various symmetric function bases. The coefficients in these expansions are related to partitions of the vertices into stable subsets, the Möbius function of the lattice of contractions of G, and the structure of the acyclic orientations of G. The coefficients which arise when XG is expanded in terms of elementary symmetric functions are particularly interesting, and for certain graphs are related to the theory of Hecke algebras and Kazhdan-Lusztig polynomials.
Computer Go is one of the biggest challenges faced by game programmers. This survey describes the typical components of a Go program, and discusses knowledge representation, search methods and techniques for solving specific subproblems in this domain. Along with a summary of the development of computer Go in recent years, areas for future research are pointed out.
These notes are based on a series of lectures given at the Advanced Research Institute of Discrete Applied Mathematics held at Rutgers University. Their aim is to link together algorithmic problems arising in knot theory, statistical physics and classical combinatorics. Apart from the theory of computational complexity concerned with enumeration problems, introductions are given to several of the topics treated, such as combinatorial knot theory, randomised approximation algorithms, percolation and random cluster models. To researchers in discrete mathematics, computer science and statistical physics, this book will be of great interest, but any non-expert should find it an appealing guide to a very active area of research.
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Motivated by the work of Chmutov, Duzhin and Lando on Vassiliev invariants, we define a polynomial on weighted graphs which contains as specialisations the weighted chromatic invariants but also contains many other classical invariants including the Tutte and matching polynomials. The paper also gives the symmetric function generalisation of the chromatic polynomial introduced by Stanley. We study its complexity and prove hardness results for very restricted classes of graphs.
In a previous paper, the author extended the Whitney rank generating function (or Tutte polynomial) from binary matroids to arbitrary functions f:2S→R, where the binary matroid special case is obtained by letting f be the indicator function of the row space of a matrix over GF(2). This paper continues that work in two directions. Firstly, a natural generalisation of the partition function of the statistical mechanical Potts model of a graph is shown to be a partial evaluation of this generalised Whitney function. Secondly, a continuum of minor operations for functions on 2S is introduced, in which deletion and contraction are distinct points, and the theory of these operations is developed. A related construction of rank-like functions is given, its properties are investigated, and a corresponding continuum of Whitney-type functions is introduced. These functions are shown to contain weight enumerators of general codes over a subset of their domains. We also discuss what these new operations mean at the level of binary matroids and graphs.
The Whitney quasi-rank generating function, which generalizes the Whitney rank generating function (or Tutte polynomial) of a graph, is introduced. It is found to include as special cases the weight enumerator of a (not necessarily linear) code, the percolation probability of an arbitrary clutter and a natural generalization of the chromatic polynomial. The crucial construction, essentially equivalent to one of Kung, is a means of associating, to any function, a rank-like function with suitable properties. Some of these properties, including connections with the Hadamard transform, are discussed.
We show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the Tutte polynomial T(M; x, y) of a matroid M except for a few listed special points and curves of the (x, y)-plane. In particular the problem of evaluating the Tutte polynomial of a graph at a point in the (x, y)-plane is #P-hard except when (x − 1)(y − 1) = 1 or when (x, y) equals (1, 1), (−1, −1), (0, −1), (−1, 0), (i, −i), (−i, i), (j, j2), (j2, j) where j = e2πi/3
The concept of a matroid is generalized by imitating the generalization of a graph to a hypergraph.
Two basic examples of such hypermatroids are given. The first type models the coloring problem for hypergraphs whereas the
second one is related to coverings of hypergraphs. This second example, covering hypermatroids, is the proper extension of
trivial matroids. Covering hypermatroids are characterized by a generalized semimodularity condition.
It is shown how all hypermatroids can be naturally constructed from suitable matroids.
A Poincaré polynomial is defined for hypermatroids. The coloring polynomial for hypergraphs is seen to be a special case.
For covering hypermatroids the Poincaré polynomial yields a polynomial identity which gives an enumerative relationship between
the covering sets and the transversal sets of a hypergraph.
This paper is a sequel to an earlier paper dealing with a symmetric function generalization XG of the chromatic polynomial of a finite graph G. We consider the question of when the expansion of XG in terms of Schur functions has nonnegative coefficients and give a number of applications, including new conditions on the f-vector of a flag complex and a new class of polynomials with real zeros. Some generalizations of XG are also considered related to the Tutte polynomial, directed graphs, and hypergraphs.
The class of # P-complete problems is a class of computationally eqivalent counting problems (defined by the author in a previous paper) that are at least as difficult as the NP-complete problems. Here we show, for a large number of natural counting problems for which there was no previous indication of intractability, that they belong to this class. The technique used is that of polynomial time reduction with oracles via translations that are of algebraic or arithmetic nature.
It is shown that, given an arbitrary GO position on an n × n board, the problem of determining the winner is Pspace hard. New techniques are exploited to overcome the difficulties arising from the planar nature of board games. In particular, it is proved that GO is Pspace hard by reducing a Pspace-complete set, TQBF, to a game called generalized geography, then to a planar version of that game, and finally to GO.
Suppose that each vertex of a graph independently chooses a colour uniformly from the set f1;:::;kg; and let Si be the random set of vertices coloured i. Farr shows that the probability that each set Si is stable (so that the colouring is proper) is at most the product of the k probabilities that the sets Si separately are stable. We give here a simple proof of an extension of this result.
Suppose each vertex of a graph G is chosen with probability p, these choices being independent. Let A(G, p) be the probability that no two chosen vertices are adjacent. This is essentially the clique polynomial of the complement of G which has been extensively studied in a variety of incarnations. We use the Ahlswede-Daykin Theorem to prove that, for all G, and all positive integers λ, P(G, λ)/λn ≤ A(G, λ−1)λ, where P(G, λ) is the chromatic polynomial of G.
This paper develops a theory of Tutte invariants for 2-polymatroids that parallels the corresponding theory for matroids. It is shown that such 2-polymatroid Invariants arise in the enumeration of a wide variety of combinatorial structures including matchings and perfect matchings in graphs, weak colourings in hypergraphs, and common bases in pairs of matroids. The main result characterizes all such invariants proving that, with some trivial exceptions, every 2-polymatroid Tutte invariant can be easily expressed in terms of a certain two-variable polynomial that is closely related to the Tutte polynomial of a matroid.
The critical problem of matroid theory can be posed in the more general context of finite relations. Given a relation R between the finite sets S and T, the critical problem is to determine the smallest number n such that there exists an n-tuple (u1,…, un) of elements from T such that for every x in S, there exists a u1 such that xRui. All the enumerative results, in particular, the Tutte decomposition and Möbius function formula, can be rephrased so that they still hold. In this way, we obtain a uniform approach to all the classical critical problems.
. The mean value of the matching polynomial is computed in the family of all labeled graphs with n vertices. We define the dominating polynomial of a graph whose coefficients enumerate the dominating sets for a graph and study some properties of the polynomial. The mean value of this polynomial is determined in a certain special family of bipartite digraphs. 1. Introduction The goal of this paper is to compute the average polynomials for the well-known matching polynomial and the dominating polynomial in certain classes of graphs. The matching polynomial first appeared in a paper by Heilman and Lieb [5] as a thermodynamic partition function. For a very interesting introduction to its combinatorial study as well as many of its properties we refer the reader to [2] and [3]. The notion of domination in graphs was introduced last century. This theory can be consulted in the books by Ore [10] and Berge [1]. The paper [7] shows recent developments of the theory and a large account of refere...
Go for Beginners, Ishi Press, 1972, and Penguin Books
- K Iwamoto
K. Iwamoto, Go for Beginners, Ishi Press, 1972, and Penguin Books, Harmondsworth, 1976.
- M Uller
- Computer Go
M. M uller, Computer Go, Artif. Intell. 134 (2002) 145-179.
- E R Berlekamp
- Y Kim-Dollar
- Ko
E.R. Berlekamp, Y. Kim, Where is the " Thousand-Dollar Ko " ? in: R.J. Nowakowski (Ed.), Games of
No Chance, Math. Sci. Res. Inst. Publ., Vol. 29, Berkeley, Ca, 11–21 July 1994, Cambridge University
Press, Cambridge, 1996, pp. 203–226.
- L Lovã Asz
- M D Plummer
- Matching Theory
L. Lovà asz, M.D. Plummer, Matching Theory, Annals of Discrete Mathematics, Vol. 29, North-Holland
Mathematical Studies, Vol. 121, North-Holland, Amsterdam, 1986.
Fel'dman, An improvement of the estimate of a linear form in the logarithms of algebraic numbers (Russian)
N.I. Fel'dman, An improvement of the estimate of a linear form in the logarithms of algebraic numbers
(Russian), Mat. Sbornik 77 (1968) 423-436 (English translation: Math. USSR Sbornik 6 (1968)
393-406).
Ladders are PSPACE-complete
- M Crâ Smaru
- J Tromp
M. Crâ smaru, J. Tromp, Ladders are PSPACE-complete, in: T.A. Marsland, I. Frank (Eds.), Computers
and Games: Second International Conference, CG 2000, Hamamatsu, Japan, 26 -28 October 2000,
Lecture Notes in Computer Science, Vol. 2063, Springer, Berlin, 2001, pp. 241-249.
- T Helgason
T. Helgason, Aspects of the theory of hypermatroids, in: Hypergraph Seminar, Lecture Notes in
Mathematics, Vol. 411, Springer, Berlin, 1974, pp. 191-213.
- D J A Welsh
D.J.A. Welsh, Complexity: Knots, Colourings and Counting, in: London Mathematical Society Lecture
Note Series, Vol. 186, Cambridge University Press, Cambridge, 1993.
- E R Berlekamp
- Y Kim-Dollar
- Ko
E.R. Berlekamp, Y. Kim, Where is the "Thousand-Dollar Ko"? in: R.J. Nowakowski (Ed.), Games of
No Chance, Math. Sci. Res. Inst. Publ., Vol. 29, Berkeley, Ca, 11-21 July 1994, Cambridge University
Press, Cambridge, 1996, pp. 203-226.
- M Uller
M. M uller, Review: computer Go 1984 -2000, in: T.A. Marsland, I. Frank (Eds.), Computers and Games:
Second International Conference, CG 2000, Hamamatsu, Japan, 26 -28 October 2000, Lecture Notes in
Computer Science, Vol. 2063, Springer, Berlin, 2001, pp. 405 -413.