Problems hard for treewidth but easy for stable gonality

TL;DR

This paper demonstrates that certain classical flow and optimization problems, hard for treewidth, become fixed-parameter tractable when parameterized by stable gonality, a new graph parameter, using novel algorithms and reductions.

Contribution

It introduces stable gonality as a new parameter that simplifies complex problems and develops algorithms based on weighted tree partitions and dynamic programming techniques.

Findings

Problems become FPT when parameterized by stable gonality

New algorithms use weighted tree partitions and treebreadth

Hardness results improve previous reductions

Abstract

We show that some natural problems that are XNLP-hard (which implies W[t]-hardness for all t) when parameterized by pathwidth or treewidth, become FPT when parameterized by stable gonality, a novel graph parameter based on optimal maps from graphs to trees. The problems we consider are classical flow and orientation problems, such as Undirected Flow with Lower Bounds (which is strongly NP-complete, as shown by Itai), Minimum Maximum Outdegree (for which W[1]-hardness for treewidth was proven by Szeider), and capacitated optimization problems such as Capacitated (Red-Blue) Dominating Set (for which W[1]-hardness was proven by Dom, Lokshtanov, Saurabh and Villanger). Our hardness proofs (that beat existing results) use reduction to a recent XNLP-complete problem (Accepting Non-deterministic Checking Counter Machine). The new easy parameterized algorithms use a novel notion of weighted tree partition with an associated parameter that we call treebreadth, inspired by Seese's notion of tree-partite graphs, as well as techniques from dynamical programming and integer linear programming.