Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation

TL;DR

This paper investigates the computational complexity and approximation algorithms for the Upper Dominating Set problem, establishing tight bounds and algorithms based on ETH and SETH assumptions, and providing a complete characterization of its sub-exponential approximability.

Contribution

It improves existing algorithms for pathwidth parameterization, proves tight lower bounds under ETH and SETH, and introduces a sub-exponential approximation algorithm with matching lower bounds.

Findings

An $O^*(6^{pw})$ algorithm for pathwidth parameterization, improving previous $O^*(7^{pw})$.

No $r$-approximation algorithm runs in time $O(n^{k^{1- ext{ε}}})$ under ETH.

A sub-exponential approximation algorithm with ratio $r$ in time $n^{O(n/r)}$, tight under ETH.

Abstract

An upper dominating set is a minimal dominating set in a graph. In the \textsc{Upper Dominating Set} problem, the goal is to find an upper dominating set of maximum size. We study the complexity of parameterized algorithms for \textsc{Upper Dominating Set}, as well as its sub-exponential approximation. First, we prove that, under ETH, \textsc{$k$-Upper Dominating Set} cannot be solved in time $O(n^{o(k)})$ (improving on $O(n^{o(\sqrt{k})})$), and in the same time we show under the same complexity assumption that for any constant ratio $r$ and any $\varepsilon > 0$, there is no $r$-approximation algorithm running in time $O(n^{k^{1-\varepsilon}})$. Then, we settle the problem's complexity parameterized by pathwidth by giving an algorithm running in time $O^*(6^{pw})$ (improving the current best $O^*(7^{pw})$), and a lower bound showing that our algorithm is the best we can get under the SETH. Furthermore, we obtain a simple sub-exponential approximation algorithm for this problem: an algorithm that produces an $r$-approximation in time $n^{O(n/r)}$, for any desired approximation ratio $r < n$. We finally show that this time-approximation trade-off is tight, up to an arbitrarily small constant in the second exponent: under the randomized ETH, and for any ratio $r > 1$ and $\varepsilon > 0$, no algorithm can output an $r$-approximation in time $n^{(n/r)^{1-\varepsilon}}$. Hence, we completely characterize the approximability of the problem in sub-exponential time.