Abstract|We present an object recognition system based

This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

The k-dimensional Weisfeiler-Leman algorithm, for k 1, is a natural and simple combinatorial algorithm attempting to decide whether two given graphs are isomorphic. In this paper, we show that for every surface S (orientable or non-orientable) there is a k 1 such that the k-dimensional WL-algorithm succeeds to decide isomorphism of graphs embeddable in S. To prove this, we use a close connection between the WL-algorithm and denability in certain nite variable logics that has been established by Cai, Furer, and Immerman [7]. 1. INTRODUCTION The graph isomorphism problem asks whether two given graphs are isomorphic. While complexity theoretic results indicate that the isomorphism problem is not NP-complete (if it was, the polynomial hierarchy would collapse to its second level [6; 29]), no polynomial time algorithm for the general problem is known. However, there is a number of important classes of graphs on which the isomorphism problem is known to be solvable in polynomial t...

We describe the first algorithms to compute minimum cuts in surface-embedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)-cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)-time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimum-cut algorithm computes a minimum-cost subgraph in every Z2-homology class. We also prove that finding a minimum-cost subgraph homologous to a single input cycle is NP-hard.

We describe the first algorithms to compute maximum flows in surface-embedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)-flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities. Except for the special case of planar graphs, for which an O(n log n)-time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surface-embedded graphs follow from algorithms for general sparse graphs. Our key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself. A dual formulation of our algorithm computes the minimum-cost cycle or circulation in a given (real or integer) homology class.

. We give a short introduction to an heuristic to find automorphisms in a graph such as axial, central or rotational symmetries. Using technics of factorial analysis, we embed the graph in an Euclidean space and try to detect and interpret the geometric symmetries of of the embedded graph. 1. Introduction Testing whether a graph has any axial (rotational, central, respectively) symmetry is a NP-complete problem [9]. Some restrictions (central symmetry with exactly one fixed vertex and no fixed edge) are polynomialy equivalent to the graph isomorphism test. Notice that this latter problem is not known to be either polynomial or NP-complete in general. But several heuristics are known (e.g. [3]) and several restrictions leads to efficient algorithms: linear time isomorphism test for planar graphs [6] and interval graphs [8], polynomial time isomorphism test for fixed genus [10, 5], k-contractible graphs [12] and pairwise k-separable graphs [11], linear axial symmetry detection for plana...

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In computational complexity theory, a function f is called b(n)-enumerable if there exists a polynomial-time function which can restrict the output of f(x) to one of b(n) possible values. This paper investigates #GA, the function which computes the number of automorphisms of an undirected graph, and GI, the set of pairs of isomorphic graphs. The results in this paper show the following connections between the enumerability of #GA and the computational complexity of GI. 1. #GA is exp(O ( √ n log n))-enumerable. 2. If #GA is polynomially enumerable then GI ∈ R. 3. For ɛ < 1 2, if #GA is nɛ-enumerable then GI ∈ P.

Abstract—We prove that fixed-point logic with counting captures polynomial time on all classes of graphs with excluded minors. That is, for every class C of graphs such that some graph H is not a minor of any graph in C, a property P of graphs in C is decidable in polynomial time if and only if it is definable in fixed-point logic with counting. Furthermore, we prove that for every class C of graphs with excluded minors there is a k such that the k-dimensional Weisfeiler-Lehman algorithm decides isomorphism of graphs in C in polynomial time. The Weisfeiler-Lehman algorithm is a combinatorial algorithm for testing isomorphism. It generalises the basic colour refinement algorithm and is much simpler than the known group-theoretic algorithms for deciding isomorphism of graphs with excluded minors. The main technical theorem behind these two results is a “definable structure theorem ” for classes of graphs with excluded minors. It states that graphs with excluded minors can be decomposed into pieces arranged in a treelike structure, together with a linear order of each of the pieces. Furthermore, the decomposition and the linear orders on the pieces are definable in fixed-point logic (without counting). Index Terms—descriptive complexity; graph minor theory; fixed-point logic; graph canonisation I.

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