A sublinear-time randomized parallel algorithm for the maximum clique problem in perfect graphs

We will show that Lovasz number of graphs maybe computed using interior-point methods. This technique will require 0(~) 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 alg~ rithm 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\, the size of the maximum clique, w(G'), is equal to the size of the minimum vertex coloring x(G'). 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' of G, the size of the largest independent set, cr(G'), is equal to the size of the minimum clique cover, p(G'). Grotschel, Lovasz and Schrijver have shown that for perfect graphs one can compute the largest clique and the largest independent set in polynomial time; see [6], [7], [8], and in particular , their elaborate book [9]. Their idea is based on computing an invariant known as the Lovasz number of graphs c9(G). Lovasz has shown that for all graphs u(G) < O(G) < x(G). As will be seen in the next section, 6(G) is defined as the minimum of some convex function derived from the graph. Grotschel, Lovasz and Schrijver use the ellipsoid method for convex pr~ gramming to establish polynomial time computability of O(G). For perfect graphs, since u(G) = (?(G) = x(G), it follows that u(G) and x(G) (and also a(G) and P(G)) are polynomial time computable. The process of self-reducibility then yields a polynomial time algorithm for construction of maximum cliques and maximum independent sets in these graphs. It should be noticed that in general Lovasz number of a graph is not even a rational number (it is algebraic though), and its polynomial time computability must be qualified by the required number of significant digits. The ellipsoid method, though of polynomial time complexity, is in practice an inefficient and numerically unstable algorithm. Indeed, Grotschel, Lovasz and Schrijver report that they had difficulty making the algorithm converge for …