Parameterized Problems Complete for Nondeterministic FPT time and Logarithmic Space

TL;DR

This paper introduces a class of parameterized problems solvable in nondeterministic FPT time and logarithmic space, providing many examples and establishing their computational hardness, including resolving a long-standing open problem.

Contribution

It defines the class XNLP, proves many problems are XNLP-complete, and demonstrates their W[t]-hardness, including resolving the complexity of the Bandwidth problem.

Findings

Many problems are XNLP-complete.

All these problems are W[t]-hard for all t.

Resolved the parameterized complexity of the Bandwidth problem.

Abstract

Let XNLP be the class of parameterized problems such that an instance of size n with parameter k can be solved nondeterministically in time $f(k)n^{O(1)}$ and space $f(k)\log(n)$ (for some computable function f). We give a wide variety of XNLP-complete problems, such as List Coloring and Precoloring Extension with pathwidth as parameter, Scheduling of Jobs with Precedence Constraints, with both number of machines and partial order width as parameter, Bandwidth and variants of Weighted CNF-Satisfiability. In particular, this implies that all these problems are W[t]-hard for all t. This also answers a long standing question on the parameterized complexity of the Bandwidth problem.