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

by   Dániel Marx, et al.

In the general AntiFactor problem, a graph G is given with a set X_v⊆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· 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)· 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 ϵ>0 such that #X-AntiFactor can be solved in time (max X+2-ϵ)^k· n^O(1) on a tree decomposition of width k, then the Counting Strong Exponential-Time Hypothesis (#SETH) fails.



There are no comments yet.


page 1

page 3

page 5

page 9

page 19

page 21

page 25

page 32


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

For the General Factor problem we are given an undirected graph G and fo...

On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the t...

Exact exponential algorithms for two poset problems

Partially ordered sets (posets) are fundamental combinatorial objects wi...

A-Discriminants for Complex Exponents, and Counting Real Isotopy Types

We extend the definition of A-discriminant varieties, and Kapranov's par...

Counting list homomorphisms from graphs of bounded treewidth: tight complexity bounds

The goal of this work is to give precise bounds on the counting complexi...

Tree Inference: Response Time in a Binary Multinomial Processing Tree, Representation and Uniqueness of Parameters

A Multinomial Processing Tree (MPT) is a directed tree with a probabilit...

Filling Crosswords is Very Hard

We revisit a classical crossword filling puzzle which already appeared i...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.