On the Complexity of Solution Extension of Optimization Problems

TL;DR

This paper investigates the computational complexity of extending partial solutions in optimization problems, revealing NP-hardness even when the original problems are polynomial-time solvable, and explores parameterized complexity and algorithmic limits.

Contribution

It introduces a general framework for extension problems, analyzes their complexity, and studies the impact of various parameters and restrictions on their computational difficulty.

Findings

Extension problems are often NP-hard despite polynomial-time solvability of original problems.

Parameterization by partial solution size affects complexity significantly.

Simple algorithms may be optimal under the Exponential-Time Hypothesis in restricted scenarios.

Abstract

The question if a given partial solution to a problem can be extended reasonably occurs in many algorithmic approaches for optimization problems. For instance, when enumerating minimal dominating sets of a graph $G=(V,E)$, one usually arrives at the problem to decide for a vertex set $U \subseteq V$, if there exists a \textit{minimal} dominating set $S$ with $U\subseteq S$. We propose a general, partial-order based formulation of such extension problems and study a number of specific problems which can be expressed in this framework. Possibly contradicting intuition, these problems tend to be NP-hard, even for problems where the underlying optimisation problem can be solved in polynomial time. This raises the question of how fixing a partial solution causes this increase in difficulty. In this regard, we study the parameterised complexity of extension problems with respect to parameters related to the partial solution, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis. All complexity considerations are also carried out in very restricted scenarios, be it degree restrictions or topological restrictions (planarity) for graph problems or the size of the given partition for the considered extension variant of Bin Packing.