Anti-Factor is FPT Parameterized by Treewidth and List Size (but Counting is Hard)

TL;DR

This paper studies the AntiFactor problem, showing fixed-parameter tractability when the number of forbidden degrees per vertex is small, but also proving counting versions are computationally hard under standard complexity assumptions.

Contribution

The paper introduces a fixed-parameter algorithm for AntiFactor with sparse forbidden degree sets and establishes hardness results for counting versions.

Findings

AntiFactor is FPT when each vertex has at most x forbidden degrees.

Counting AntiFactor (#AntiFactor) is #W[1]-hard even for simple cases.

Standard dynamic programming algorithms are essentially optimal for counting AntiFactor.

Abstract

In the general AntiFactor problem, a graph $G$ is given with a set $X_v\subseteq \mathbb{N}$ of forbidden degrees for every vertex $v$ and the task is to find a set $S$ of edges such that the degree of $v$ in $S$ is not in the set $X_v$. Standard techniques (dynamic programming + fast convolution) can be used to show that if $M$ is the largest forbidden degree, then the problem can be solved in time $(M+2)^k\cdot n^{O(1)}$ if a tree decomposition of width $k$ is given. However, significantly faster algorithms are possible if the sets $X_v$ are sparse: our main algorithmic result shows that if every vertex has at most $x$ forbidden degrees (we call this special case AntiFactor$_x$), then the problem can be solved in time $(x+1)^{O(k)}\cdot n^{O(1)}$. That is, the AntiFactor$_x$ is fixed-parameter tractable parameterized by treewidth $k$ and the maximum number $x$ of excluded degrees. Our algorithm uses the technique of representative sets, which can be generalized to the optimization version, but (as expected) not to the counting version of the problem. In fact, we show that #AntiFactor$_1$ is already #W[1]-hard parameterized by the width of the given decomposition. Moreover, we show that, unlike for the decision version, the standard dynamic programming algorithm is essentially optimal for the counting version. Formally, for a fixed nonempty set $X$, we denote by $X$-AntiFactor the special case where every vertex $v$ has the same set $X_v=X$ of forbidden degrees. We show the following lower bound for every fixed set $X$: if there is an $\epsilon>0$ such that #$X$-AntiFactor can be solved in time $(\max X+2-\epsilon)^k\cdot n^{O(1)}$ on a tree decomposition of width $k$, then the Counting Strong Exponential-Time Hypothesis (#SETH) fails.