Maximizing the Margin between Desirable and Undesirable Elements in a Covering Problem

TL;DR

This paper introduces the Target Approximation Problem (TAP), a new combinatorial problem balancing desirable and undesirable elements in set cover scenarios, analyzes its computational complexity, and offers algorithms with approximation guarantees.

Contribution

It formalizes TAP, proves its hardness in general, and provides exact algorithms for restricted cases along with a 0.5-approximation algorithm for specific instances.

Findings

TAP is NP-hard and hard to approximate in general.

Exact polynomial-time algorithms exist for certain restricted cases.

A 0.5-approximation algorithm is developed for elements appearing at most twice.

Abstract

In many covering settings, it is natural to consider the presence both of elements that we seek to include and of elements that we seek to avoid. This paper introduces a novel combinatorial problem formalizing this tradeoff: from a collection of sets containing both "desirable" and "undesirable" items, pick the subcollection that maximizes the margin between the number of desirable and undesirable elements covered. We call this the Target Approximation Problem (TAP) and argue that many real-world scenarios are naturally modeled via this objective. We first show that TAP is hard, even when restricted to cases where the given sets are small or where elements appear in only a small number of sets. In a large swath of these cases, we show that TAP is hard even to approximate. We then exhibit exact polynomial-time algorithms for other restricted cases and provide an efficient 0.5-approximation for the case where elements occur at most twice, derived through a tight connection to the greedy algorithm for Unweighted Set Cover.