Min-Max Connected Multiway Cut

TL;DR

This paper introduces the min-max connected multiway cut problem, a variant of multiway cut with connectivity and max-boundary objectives, explores its computational hardness, and provides algorithms for special cases.

Contribution

It defines and studies the min-max connected multiway cut problem, establishing hardness results and offering algorithms for graphs with bounded treewidth.

Findings

Proves weak NP-hardness on graphs with treewidth two.

Shows W[1]-hardness when parameterized by treewidth.

Provides a pseudopolynomial algorithm and an FPTAS for bounded treewidth graphs.

Abstract

We introduce a variant of the multiway cut that we call the min-max connected multiway cut. Given a graph $G=(V,E)$ and a set $\Gamma\subseteq V$ of $t$ terminals, partition $V$ into $t$ parts such that each part is connected and contains exactly one terminal; the objective is to minimize the maximum weight of the edges leaving any part of the partition. This problem is a natural modification of the standard multiway cut problem and it differs from it in two ways: first, the cost of a partition is defined to be the maximum size of the boundary of any part, as opposed to the sum of all boundaries, and second, the subgraph induced by each part is required to be connected. Although the modified objective function has been considered before in the literature under the name min-max multiway cut, the requirement on each component to be connected has not been studied as far as we know. Our main motivation for studying this problems is its close connection with the spanning tree congestion problem that has been extensively studied but on which little progress has been made. We show various hardness results for this problem, including a proof of weak NP-hardness of the weighted version of the problem on graphs with treewidth two, and $W[1]$-hardness for the problem when parameterized by the treewidth of the graph. Complementing our lower bounds tightly, we also provide a pseudopolynomial time algorithm as well as an FPTAS for the weighted problem on graphs of bounded treewidth. As a consequence of our investigation we also show that the (unconstrained) min-max multiway cut problem is NP-hard even for three terminals, strengthening the known results.