Efficient parameterized algorithms on graphs with heterogeneous structure: Combining tree-depth and modular-width

TL;DR

This paper introduces a unified framework for designing efficient parameterized algorithms on graphs with complex heterogeneous structures by combining properties like modular-width and tree-depth, broadening applicability while maintaining fast runtimes.

Contribution

It presents a novel algebraic expression-based approach to combine different graph structural properties, enabling the development of efficient algorithms on more general graph classes.

Findings

Framework successfully combines modular-width and tree-depth.

Algorithms for Negative Cycle Detection, All-Pairs Shortest Paths, and Triangle Counting are efficient.

Achieves competitive runtimes similar to homogeneous graph cases.

Abstract

Many computational problems admit fast algorithms on special inputs, however, the required properties might be quite restrictive. E.g., many graph problems can be solved much faster on interval or cographs, or on graphs of small modular-width or small tree-width, than on general graphs. One challenge is to attain the greatest generality of such results, i.e., being applicable to less restrictive input classes, without losing much in terms of running time. Building on the use of algebraic expressions we present a clean and robust way of combining such homogeneous structure into more complex heterogeneous structure, and we show-case this for the combination of modular-width, tree-depth, and a natural notion of modular tree-depth. We give a generic framework for designing efficient parameterized algorithms on the created graph classes, aimed at getting competitive running times that match the homogeneous cases. To show the applicability we give efficient parameterized algorithms for Negative Cycle Detection, Vertex-Weighted All-Pairs Shortest Paths, and Triangle Counting.