### Bisection of bounded treewidth graphs by convolutions

Eduard Eiben, Daniel Lokshtanov, Amer E. Mouawad
2021 Journal of computer and system sciences (Print)
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
more » ... 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  . 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  .