The Lovász Local Lemma is a tool that enables one to show that certain events hold with pos-itive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NC 1 algorithms for several interesting algorithmic problems.

Given a 0-1 square matrix A, when can some of the 1’s be changed to −1’s in such a way that the permanent of A equals the determinant of the modified matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either have the same sign, or are both zero) is non-singular? When is a hypergraph with n vertices and n hyperedges minimally non-bipartite? When does a bipartite graph have a “Pfaffian orientation”? Given a digraph, does it have no directed circuit of even length? Given a digraph, does it have a subdivision with no even directed circuit? It is known that all the above problems are equivalent. We prove a structural char-acterization of the feasible instances, which implies a polynomial-time algorithm to solve all of the above problems. The structural characterization says, roughly speaking, that a bipartite graph has a Pfaffian orientation if and only if it can be obtained by piecing together (in a specified way) planar bipartite graphs and one sporadic non-planar bipartite graph.

A square real matrix is sign-nonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of sign-nonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determinant are equal. The structural characterizations, which are topological in nature, imply polynomial algorithms. 1

A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3-space, Hadwiger’s conjecture on t-colorability of graphs with no Kt+1 minor, Tutte’s edge 3-coloring conjecture on edge 3-colorability of 2-connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2-colorability of hypergraphs, and sign-nonsingular matrices.

We give a polynomial-time algorithm for the following problem of Pólya. Given an n × n 0-1 matrix, either find a matrix obtained from it by changing some of the 1’s to −1’s in such a way that the determinant of the new matrix equals the permanent of the old one, or determine that no such matrix exists. This is equivalent to finding Pfaffian orientations of bipartite graphs and to the even circuit problem for directed graphs. The algorithm is based on a structural characterization of bipartite graphs that admit a Pfaffian orientation.

Abstract not found

A digraph D is called noneven if it is possible to assign weights of 0,1 to its arcs so that D contains no cycle of even weight. A noneven digraph D corresponds to one or more nonsingular sign patterns. Given an n \Theta n sign pattern H, a symplectic pair in Q(H) is a pair of matrices (A; D) such that A 2 Q(H), D 2 Q(H), and A T D = I . An unweighted digraph D allows a matrix property P if at least one of the sign patterns whose digraph is D allows P . In [1], Thomassen characterized the noneven, 2-connected symmetric digraphs (i.e., digraphs for which the existence of arc (u; v) implies the existence of arc (v; u)). In the first part of our paper, we recall this characterization and use it to determine which strong symmetric digraphs allow symplectic pairs. A digraph D is called semi-complete if, for each pair of distinct vertices (u; v), at least one of the arcs (u; v) and (v; u) exists in D. Thomassen, again in [1], presented a characterization of two classes of strong, noneven...

We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3-coloring conjecture, and Pfaffian orientations of bipartite graphs.

This compendium is the result of reformatting and minor editing of the author's transparencies for his talk delivered at the conference. The talk covered Euler's Formula, Kuratowski's Theorem, linear planarity tests, Schnyder's Theorem and drawing on the grid, the two paths problem, Pfaffian orientations, linkless embeddings, and the Four Color Theorem.

Let a ⊕ b = max(a, b), a ⊗ b = a + b for a, b ∈ R: = R ∪ {−∞}. By max-algebra we understand the analogue of linear algebra developed for the pair of operations (⊕, ⊗) extended to matrices and vectors. Max-algebra, which has been studied for more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machine-scheduling, information technology and discrete-event dynamic systems. This paper focuses on presenting a number of links between basic max-algebraic problems like systems of linear equations, eigenvalue-eigenvector problem, linear independence, regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimization problems on the other hand. This indicates that max-algebra may be regarded as a linear-algebraic encoding of a class of combinatorial problems. The paper is intended for wider readership including researchers not familiar with max-algebra.