We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of ...

We present practical algorithms for constructing partitions of graphs into a fixed number of vertex-disjoint subgraphs that satisfy particular degree constraints. We use this in particular to find k-cuts of graphs of maximum degree \Delta that cut at least a k\Gamma1 k (1 + 1 2\Delta+k\Gamma1 ) fraction of the edges, improving previous bounds known. The partitions also apply to constraint networks, for which we give a tight analysis of natural local search heuristics for the maximum constraint satisfaction problem. These partitions also imply efficient approximations for several problems on weighted bounded-degree graphs. In particular, we improve the best performance ratio for the weighted independent set problem to 3 \Delta+2 , and obtain an efficient algorithm for coloring 3-colorable graphs with at most 3\Delta+2 4 colors.

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There is a number of VLSI problems that have a common structure. We investigate such a structure that leads to a unified approach for three independent VLSI layout problems: partitioning, placement and via minimization. Along the line, we first propose a linear-time approximation algorithm on maxcut and two closely related problems: k- coloring and maximal k-color ordering problem. The k-coloring is a generalization of the maxcut and the maximal k-color ordering is a generalization of the k-coloring. For a graph G with e edges and n vertices, our maxcut approximation algorithm runs in O(e + n) sequential time yielding a node-balanced maxcut with size at least (w(E) + w(E)=n)=2, improving the time complexity of O(e log e) known before. Building on the proposed maxcut technique and employing a height-balanced binary decomposition, we devise an O((e + n) log k) time algorithm for the k-coloring problem which always finds a k-partition of vertices such that the number of bad (or "defec...

ization problems. Nonlinear programming did not receive as much attention in this respect until the recent work by Goemans and Williamson [62]. They use semidefinite programs, which are nonlinear programs, to obtain approximation solutions with much better approximation factors. For example, the best previously known approximation algorithm for MAXCUT, which was invented twenty years ago, has approximation factor 0.5 [137]. The algorithm of Goemans and Williamson dramatically improves the approximation factor to 0.878. Inspired by the work on MAXCUT, Karger, Motwani, and Sudan [86] adapt the same technique and obtain the currently best approximation algorithm for coloring a k-colorable graph with the fewest possible number of colors. The approximation ratio is improved by a factor of \Omega\Gamma n 2=k ) over the best previously known result [29]. Later Karger and Blum give the best known approximation algorithm for color

. For a graph G with e edges and n vertices, and w(E) as a total edge weight, a maximum cardinality matching (MCM) (resp. maximum weighted matching (MWM)) of G is a maximum subset M of edges (resp. a subset M of edges with a maximum edge weight) such that no two edges of M are incident at a common vertex. The best known algorithm for solving the MCM problem in general graphs (resp. the MWM problem in bipartite graphs) requires O(n 5=2 ) time (resp. O(n(e + n log n)) time). We first propose an approximate MCM algorithm that runs in O(e + n) sequential time yielding a matching of size at least e n\Gamma1 . Next, the proposed MCM algorithm is extended to the weighted case running in O(e + n) time, yielding the size of at least w(E) n\Gamma1 , when n is even. When n is odd, the lower bound obtained is w(E)\Gammaw(I v ) n\Gamma2 , where w(Iv) is the weight on edges incident to vertex v which is minimum over considering all vertices. The results improve the bound known before. The ...

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