This paper is a chapter of the forthcoming Handbook of Combinatorics, to be published by North-Holland. It surveys the basic techniques and methods in combinatorial optimization. We organize our material according to the fundamental algorithmic techniques and illustrate them on problems to which these methods have been applied successfully. Special attention is given to approximation algorithms and fast (primal and dual) heuristics.

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Abstract. We consider the problem of designing efficient algorithms for computing certain matchings in a bipartite graph G = (A ∪ P, E), with a partition of the edge set as E = E1 ˙ ∪ E2... ˙ ∪ Er. A matching is a set of (a, p) pairs, a ∈ A, p ∈ P such that each a and each p appears in at most one pair. We first consider the popular matching problem; an O(m √ n) algorithm to solve the popular matching problem was given in [3], where n is the number of vertices and m is the number of edges in the graph. Here we present an O(n ω) randomized algorithm for this problem, where ω < 2.376 is the exponent of matrix multiplication. We next consider the rank-maximal matching problem; an O(min(mn, Cm √ n)) algorithm was given in [7] for this problem. Here we give an O(Cn ω) randomized algorithm, where C is the largest rank of an edge used in such a matching. We also consider a generalization of this problem, called the weighted rank-maximal matching problem, where vertices in A have positive weights. 1

This paper gives output sensitive parallel algorithms whose performance

Abstract. In this note, we give tighter bounds on the complexity of the bipartite unique perfect matching problem, bipartite-UPM. We show that the problem is in C=L and in NL ⊕L, both subclasses of NC 2. We also consider the (unary) weighted version of the problem. We show that testing uniqueness of the minimum-weight perfect matching problem for bipartite graphs is in L C=L and in NL ⊕L. Furthermore, we show that bipartite-UPM is hard for NL. 1

Abstract. For planar graphs, counting the number of perfect matchings (and hence determining whether there exists a perfect matching) can be done in NC [4, 10]. For planar bipartite graphs, finding a perfect matching when one exists can also be done in NC [8, 7]. However in general planar graphs (when the bipartite condition is removed), no NC algorithm for constructing a perfect matching is known. We address a relaxation of this problem. We consider the fractional matching polytope P(G) of a planar graph G. Each vertex of this polytope is either a perfect matching, or a half-integral solution: an assignment of weights from the set {0, 1/2, 1} to each edge of G so that the weights of edges incident on each vertex of G add up to 1 [6]. We show that a vertex of this polytope can be found in NC, provided G has at least one perfect matching to begin with. If, furthermore, the graph is bipartite, then all vertices are integral, and thus our procedure actually finds a perfect matching without explicitly exploiting the bipartiteness of G. 1

One of the fundamental problems of computational algebra is to classify polynomials according to the hardness of computing them. Recently, this problem has been related to another important problem: Polynomial identity testing. Informally, the problem is to decide if a certain succinct representation of a polynomial is zero or not. This problem has been extensively studied owing to its connections with various areas in theoretical computer science. Several efficient randomized algorithms have been proposed for the identity testing problem over the last few decades but an efficient deterministic algorithm is yet to be discovered. It is known that such an algorithm will imply hardness of computing an explicit polynomial. In the last few years, progress has been made in designing deterministic algorithms for restricted circuits, and also in understanding why the problem is hard even for small depth. In this article, we survey important results for the polynomial identity testing problem and its connection with classification of polynomials. 1

y interested in the maximum matching problem; that is, th problem of nding a matching of maximum cardinality. For simplicity, we refer to a matching of maximum cardinality as a maximum matching, and let (G) denote the size of a maximum matching in G. Let M be a matching in G, and let v be a vertex of G. If v is the end of an edge in M , then we say that M saturates v. The set of all vertices saturated by a particular matching is called a matchable set of G. Note that, since each edge saturates two vertices, matchable sets have even cardinality. A matching that saturates every vertex is called perfect. The perfect matching problem is the problem of deciding whether a graph has a perfect matching. Obviously, G

In this paper we present a min-max theorem for a factorization problem in directed graphs. This extends the Berge-Tutte formula on matchings as well as formulas for the maximum even factor in weakly symmetric directed graphs and a factorization problem in undirected graphs. We also prove an extension to the structural theorem of Gallai and Edmonds about a canonical set attaining minimum in the formula. The matching matroid can be generalized to this context: we get a matroidal description of the coverable node sets.

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