Lali Barri`ere , Pierre Fraigniaud , Evangelos Kranakis , and Danny Krizanc Dept. de Matem`atica Aplicada i Telem`atica, Universitat Polit`ecnica de Catalunya.

A new parallel algorithm for the maximal independent set problem is constructed. It runs in O(log 4 n) time when implemented on a linear number of EREW-processors. This is the first deterministic algorithm for the maximal independent set problem (MIS) whose running time is polylogarithmic and whose processor-time product is optimal up to a polylogarithmic factor.

The algebraic framework introduced in [4] gives an algebraic condition for the feasibility of a set of multicast connections in a network, that is equivalent to the max-flow min-cut condition of [1]. The algebraic condition also checks the validity of a linear coding solution to a given multicast connection problem. We present two alternative formulations of this condition that are closely tied to network flow, providing further insights and connections with combinatorial optimization. They are also easier to work with in many cases as they do not involve matrix products and inversions. From these results we derive a substantially tighter upper bound on the coding field size required for a given connection problem than that in [4].

this paper and supplied many helpful comments. This research was supported in part by NSF grants DCR-85-11713, CCR-86-05353, and CCR-88-14977, and by DARPA contract N00039-84-C-0165.

We present a linear time approximation algorithm with a performance ratio of 1/2 for nding a maximum weight matching in an arbitrary graph. Such a result is already known and is due to Preis [7].

Abstract not found

Let H be a graph on h vertices, and let G be a graph on n vertices. An H-factor of G is a spanning subgraph of G consisting of n=h vertex disjoint copies of H. The fractional arboricity of H is a(H) = maxf jE 0 j jV 0 j\Gamma1 g, where the maximum is taken over all subgraphs (V 0 ; E 0 ) of H with jV 0 j ? 1. Let ffi (H) denote the minimum degree of a vertex of H. It is shown that if ffi (H) ! a(H) then n \Gamma1=a(H) is a sharp threshold function for the property that the random graph G(n; p) contains an H-factor. I.e., there are two positive constants c and C so that for p(n) = cn \Gamma1=a(H) , almost surely G(n; p(n)) does not have an H-factor, whereas for p(n) = Cn \Gamma1=a(H) , almost surely G(n; p(n)) contains an H-factor (provided h divides n). A special case of this answers a problem of Erdos. Research supported in part by a United States Israeli BSF grant 1 1 Introduction All graphs considered here are finite, undirected and simple. If G is a graph o...

A randomized (Las Vegas) algorithm is given for finding the Gallai–Edmonds decomposition of a graph. Let n denote the number of vertices, and let M(n) denote the number of arithmetic operations for multiplying two n × n matrices. The sequential running time (i.e., number of bit operations) is within a poly-logarithmic factor of M(n). The parallel complexity is O((log n) 2) parallel time using a number of processors within a poly-logarithmic factor of M(n). The same complexity bounds suffice for solving several other problems: (i) finding a minimum vertex cover in a bipartite graph, (ii) finding a minimum X→Y vertex separator in a directed graph, where X and Y are specified sets of vertices, (iii) finding the allowed edges (i.e., edges that occur in some maximum matching) of a graph, and (iv) finding the canonical partition of the vertex set of an elementary graph. The sequential algorithms for problems (i), (ii), and (iv) are Las Vegas, and the algorithm for problem (iii) is Monte Carlo. The new complexity bounds are significantly better than the best previous ones, e.g., using the best value of M(n) currently known, the new sequential running time is O(n2.38) versus the previous best O(n2.5 /(log n)) or more.

An undirected edge-weighted graph can have at most \Gamma n 2 \Delta edge connectivity cuts. A succinct and algorithmically useful representation for this set of cuts was given by [4], and an efficient sequential algorithm for obtaining it was given by [12]. In this paper, we present a fast parallel algorithm for obtaining this representation; our algorithm is an RNC algorithm in case the weights are given in unary. We also observe that for a unary weighted graph, the problems of counting and enumerating the connectivity cuts are in RNC.

We will show that Lovasz number of graphs may be computed using interior-point methods. This technique will require O( p jV j) iterations, each consisting of matrix operations which have polylog parallel time complexity. In case of perfect graphs Lovasz number equals the size of maximum clique in the graph and thus may be obtained in sublinear parallel time. By using the isolating lemma, we get a Las Vegas randomized parallel algorithm for constructing the maximum clique in perfect graphs. 1 Introduction. In this work, we will be studying algorithms for computation of maximum cliques and maximum independent sets in perfect graphs. A graph G = (V; E) is perfect when, for all of its induced subgraphs G 0 , the size of the maximum clique, !(G 0 ), is equal to the size of the minimum vertex coloring Ø(G 0 ). The celebrated perfect graph theorem of Lovasz [12] indicates that the complements of perfect graphs are also perfect; in other words, for all induced subgraphs G 0 of G, ...