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

Sophie Boileau, Andrew Hong, David Liben-Nowell, Alistair Pattison, Anna N. Rafferty, Charlie Roslansky

TL;DR

This paper introduces the Target Approximation Problem (TAP), a margin-based covering framework that balances rewarding desirable elements against penalizing undesirable ones. It establishes TAP is NP-hard in general and hard to approximate, then identifies tractable regimes under restricted weight or restricted occurrence, including exact polynomial-time algorithms for 1-weight or 1-occurrence cases. It further derives a tight 0.5-approximation for 2-weight TAP via a reduction to one-red TAP and connects TAP to classical problems like Red-Blue Set Cover, Set Cover, and MAX-k-SAT through various reductions. The results map the boundary between tractable and intractable instances, highlight structural reductions to vertex cover on special graphs, and open questions about intermediate parameter regimes and potential improvements via alternative algorithms or SDP approaches.

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.

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

TL;DR

This paper introduces the Target Approximation Problem (TAP), a margin-based covering framework that balances rewarding desirable elements against penalizing undesirable ones. It establishes TAP is NP-hard in general and hard to approximate, then identifies tractable regimes under restricted weight or restricted occurrence, including exact polynomial-time algorithms for 1-weight or 1-occurrence cases. It further derives a tight 0.5-approximation for 2-weight TAP via a reduction to one-red TAP and connects TAP to classical problems like Red-Blue Set Cover, Set Cover, and MAX-k-SAT through various reductions. The results map the boundary between tractable and intractable instances, highlight structural reductions to vertex cover on special graphs, and open questions about intermediate parameter regimes and potential improvements via alternative algorithms or SDP approaches.

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.

Paper Structure

This paper contains 6 sections, 11 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. TAP with Restricted Weight
  4. TAP with Restricted Occurrence
  5. Restricted Weight and Restricted Occurrence
  6. Greedy Approximation of 2-Weight TAP

Key Result

theorem thmcountertheorem

The case of TAP in which every subset has either zero blue-weight or zero red-weight (which includes $1$-weight TAP) is solvable in polynomial time.

Figures (1)

  • Figure 1: Summary of results for $k$-occurrence, $w$-weight TAP: each of the given subsets contains $\mathop{\le}w$ elements, and each element appears in $\mathop{\le}k$ of the given subsets.

Theorems & Definitions (20)

  • definition thmcounterdefinition: Target Approximation Problem [TAP]
  • remark thmcounterremark
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • theorem thmcountertheorem
  • remark thmcounterremark
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • proof : sketch; see \ref{['sec:proofs:restricted-occurrence']}
  • theorem thmcountertheorem
  • ...and 10 more