Degrees and Gaps: Tight Complexity Results of General Factor Problems Parameterized by Treewidth and Cutwidth

TL;DR

This paper advances the understanding of the computational complexity of the General Factor problem, providing tight algorithms and bounds based on treewidth and cutwidth parameters, especially for cases with bounded maxgap.

Contribution

It introduces an optimal counting algorithm parameterized by treewidth, establishes tight complexity bounds, and extends these results to the cutwidth parameter for the General Factor problem.

Findings

New $O^*((M+1)^k)$ counting algorithm for graphs with bounded treewidth.

Proved that the algorithm is essentially optimal under SETH assumptions.

Provided a $O^*(2^k)$ algorithm for the problem parameterized by cutwidth, with matching lower bounds.

Abstract

For the General Factor problem we are given an undirected graph $G$ and for each vertex $v\in V(G)$ a finite set $B_v$ of non-negative integers. The task is to decide if there is a subset $S\subseteq E(G)$ such that $deg_S(v)\in B_v$ for all vertices $v$ of $G$. The maxgap of a finite integer set $B$ is the largest $d\ge 0$ such that there is an $a\ge 0$ with $[a,a+d+1]\cap B=\{a,a+d+1\}$. Cornu\'ejols (1988) showed that if the maxgap of all sets $B_v$ is at most 1, then the decision version of General Factor is poly-time solvable. Dudycz and Paluch (2018) extended this result for the minimization and maximization versions. Using convolution techniques from van Rooij (2020), we improve upon the previous algorithm by Arulselvan et al. (2018) and present an algorithm counting the number of solutions of a certain size in time $O^*((M+1)^k)$, given a tree decomposition of width $k$, where $M=\max_v \max B_v$. We prove that this algorithm is essentially optimal for all cases that are not polynomial time solvable for the decision, minimization or maximization versions. We prove that such improvements are not possible even for $B$-Factor, which is General Factor on graphs where all sets $B_v$ agree with the fixed set $B$. We show that for every fixed $B$ where the problem is NP-hard, our new algorithm cannot be significantly improved: assuming the Strong Exponential Time Hypothesis (SETH), no algorithm can solve $B$-Factor in time $O^*((\max B+1-\epsilon)^k)$ for any $\epsilon>0$. We extend this bound to the counting version of $B$-Factor for arbitrary, non-trivial sets $B$, assuming #SETH. We also investigate the parameterization of the problem by cutwidth. Unlike for treewidth, a larger set $B$ does not make the problem harder: Given a linear layout of width $k$ we give a $O^*(2^k)$ algorithm for any $B$ and provide a matching lower bound that this is optimal for the NP-hard cases.