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Bisection of bounded treewidth graphs by convolutions

Eduard Eiben, Daniel Lokshtanov, Amer E. Mouawad

2021
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Journal of computer and system sciences (Print)
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In the Bisection problem, we are given as input an edge-weighted graph G. The task is to find a partition of V (G) into two parts A and B such that ||A| − |B|| ≤ 1 and the sum of the weights of the edges with one endpoint in A and the other in B is minimized. We show that the complexity of the Bisection problem on trees, and more generally on graphs of bounded treewidth, is intimately linked to the (min, +)-Convolution problem. Here the input consists of two sequences (a assuming a tree
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... ition of width t is provided as input. Plugging in the naive O(n 2 ) time algorithm for (min, +)-Convolution yields a O(8 t t O(1) n 2 log n) time algorithm for Bisection. This improves over the (dependence on n of the) O(2 t n 3 ) time algorithm of Jansen et al. [SICOMP 2005] at the cost of a worse dependence on t. "Conversely", we show that if Bisection can be solved in time O(β(n)) on edge weighted trees, then (min, +)-Convolution can be solved in O(β(n)) time as well. Thus, obtaining a sub-quadratic algorithm for Bisection on trees is extremely challenging, and could even be impossible. On the other hand, for unweighted graphs of treewidth t, by making use of a recent algorithm for Bounded Difference (min, +)-Convolution of Chan and Lewenstein [STOC 2015], we obtain a sub-quadratic algorithm for Bisection with running time O(8 t t O(1) n 1.864 log n). ACM Subject Classification Theory of computation → Design and analysis of algorithms; Theory of computation → Graph algorithms analysis A bisection of a graph G is a partition of V (G) into two parts A and B such that ||A|−|B|| ≤ 1. The weight of a bisection (A, B) of an edge-weighted graph G is the sum of the weights of all edges with one endpoint in A and the other in B. In the Bisection problem the task is to find a minimum weight bisection in an edge-weighted graph G given as input. The problem can be seen as a version of Minimum Cut with a balance constraint on the sizes of two sides of the cut. While Minimum Cut is solvable in polynomial time, Bisection is one of the classic examples of NP-complete problems [15] . Bisection has been studied extensively from the perspective of approximation algorithms [14, 13, 18, 21] , parameterized algorithms [7, 11, 22] heuristics [6, 8] and average case complexity [5] .

doi:10.1016/j.jcss.2021.02.002
fatcat:uph4ndk24nafjonawszjvy2fj4