Edge-Cut Width: An Algorithmically Driven Analogue of Treewidth Based on Edge Cuts

TL;DR

This paper introduces edge-cut width, a new graph parameter based on edge cuts, which can be computed efficiently and enables fixed-parameter algorithms for problems where existing parameters like tree-cut width do not.

Contribution

The paper develops a novel edge-cut width parameter that overcomes limitations of tree-cut width and provides fixed-parameter algorithms for certain NP-hard problems.

Findings

Edge-cut width can be computed by a fixed-parameter algorithm.

It enables fixed-parameter algorithms for problems hard for tree-cut width.

Edge-cut width measures cycle density through a spanning tree.

Abstract

Decompositional parameters such as treewidth are commonly used to obtain fixed-parameter algorithms for NP-hard graph problems. For problems that are W[1]-hard parameterized by treewidth, a natural alternative would be to use a suitable analogue of treewidth that is based on edge cuts instead of vertex separators. While tree-cut width has been coined as such an analogue of treewidth for edge cuts, its algorithmic applications have often led to disappointing results: out of twelve problems where one would hope for fixed-parameter tractability parameterized by an edge-cut based analogue to treewidth, eight were shown to be W[1]-hard parameterized by tree-cut width. As our main contribution, we develop an edge-cut based analogue to treewidth called edge-cut width. Edge-cut width is, intuitively, based on measuring the density of cycles passing through a spanning tree of the graph. Its benefits include not only a comparatively simple definition, but mainly that it has interesting algorithmic properties: it can be computed by a fixed-parameter algorithm, and it yields fixed-parameter algorithms for all the aforementioned problems where tree-cut width failed to do so.