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Fast Approximation Algorithms on Maxcut, k-Coloring, and k-Color Ordering for VLSI Applications
November 1998 (vol. 47 no. 11)
pp. 1253-1266

Abstract—There are 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 "defected") edges does not exceed (w(E)/k)((n$-$ 1)/n)log k, thus improving both the time complexity O(enk) and the bound e/k known before. The other related problem is the maximal k-color ordering problem that has been an open problem [16]. We show the problem is NP-complete, then present an approximation algorithm building on our k-coloring structure. A performance bound on maximal k-color ordering cost, 2kw(E)/3 is attained in O(ek) time. The solution quality of this algorithm is also tested experimentally and found to be effective.

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Index Terms:
VLSI partitioning, maximum cut, k-Coloring, k-Color Ordering, approximation algorithm.
Citation:
Jun-Dong Cho, Salil Raje, Majid Sarrafzadeh, "Fast Approximation Algorithms on Maxcut, k-Coloring, and k-Color Ordering for VLSI Applications," IEEE Transactions on Computers, vol. 47, no. 11, pp. 1253-1266, Nov. 1998, doi:10.1109/12.736440
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