The Minimum Feasible Tileset problem

TL;DR

This paper introduces the Minimum Feasible Tileset problem, proves its computational hardness, and provides a 4/3-approximation algorithm along with fixed-parameter tractability results.

Contribution

It formally defines the problem, establishes its APX-hardness and NP-hardness, and offers a novel approximation algorithm and fixed-parameter algorithms.

Findings

The problem is APX-hard and NP-hard even with small scenarios.

A 4/3-approximation algorithm is developed for the general case.

The problem is fixed-parameter tractable with respect to scenarios and symbols.

Abstract

We introduce and study the Minimum Feasible Tileset problem: Given a set of symbols and subsets of these symbols (scenarios), find a smallest possible number of pairs of symbols (tiles) such that each scenario can be formed by selecting at most one symbol from each tile. We show that this problem is APX-hard and that it is NP-hard even if each scenario contains at most three symbols. Our main result is a 4/3-approximation algorithm for the general case. In addition, we show that the Minimum Feasible Tileset problem is fixed-parameter tractable both when parameterized with the number of scenarios and with the number of symbols.