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<p><b>Abstract</b>—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 <it>maxcut</it> and two closely related problems: <it>k-coloring</it> and <it>maximal k-color ordering</it> problem. The <it>k</it>-coloring is a generalization of the maxcut and the maximal <it>k</it>-color ordering is a generalization of the <it>k</it>-coloring. For a graph <it>G</it> with <it>e</it> edges and <it>n</it> vertices, our maxcut approximation algorithm runs in <it>O</it>(<it>e</it> + <it>n</it>) sequential time yielding a node-balanced maxcut with size at least (<it>w</it>(<it>E</it>) + <it>w</it>(<it>E</it>)/<it>n</it>)/2, improving the time complexity of <it>O</it>(<it>e</it> log <it>e</it>) known before. Building on the proposed maxcut technique and employing a height-balanced binary decomposition, we devise an <it>O</it>((<it>e</it> + <it>n</it>)log <it>k</it>) time algorithm for the <it>k</it>-coloring problem which always finds a <it>k</it>-partition of vertices such that the number of bad (or "defected") edges does not exceed (<it>w</it>(<it>E</it>)/<it>k</it>)((<it>n</it><tmath>$-$</tmath> 1)/<it>n</it>)<super>log <it>k</it></super>, thus improving both the time complexity <it>O</it>(<it>enk</it>) and the bound <it>e/k</it> known before. The other related problem is the <it>maximal k-color ordering problem</it> that has been an open problem [<ref rid="bibt125316" type="bib">16</ref>]. We show the problem is NP-complete, then present an approximation algorithm building on our <it>k</it>-coloring structure. A performance bound on maximal <it>k</it>-color ordering cost, 2<it>kw</it>(<it>E</it>)/3 is attained in <it>O</it>(<it>ek</it>) time. The solution quality of this algorithm is also tested experimentally and found to be effective.</p>
VLSI partitioning, maximum cut, k-Coloring, k-Color Ordering, approximation algorithm.
Jun-Dong Cho, Majid Sarrafzadeh, Salil Raje, "Fast Approximation Algorithms on Maxcut, k-Coloring, and k-Color Ordering for VLSI Applications", IEEE Transactions on Computers, vol. 47, no. , pp. 1253-1266, November 1998, doi:10.1109/12.736440
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