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

2025-07-04

AI Summary

This paper introduces the Target Approximation Problem (TAP): given a family of sets containing both “desirable” and “undesirable” elements, select a subfamily to maximize the difference between the numbers of covered desirable and undesirable elements. TAP is the first formal model capturing the trade-off between covering desirable versus avoiding undesirable elements in set coverage, with a rigorous analysis of its computational complexity. We prove that TAP is inapproximable in general—and remains strongly inapproximable even under several structural restrictions. For the special case where each element appears in at most two sets, we design a tight 0.5-approximation greedy algorithm. We further uncover deep connections between TAP and unweighted set cover, and devise polynomial-time exact algorithms under bounded frequency and tree-structured set families. Our core contributions are: (i) establishing TAP as a novel, theoretically grounded problem formulation; (ii) characterizing its hardness boundaries; and (iii) providing algorithms that are both practically efficient and theoretically optimal.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In many covering settings, it is natural to consider the simultaneous presence of desirable elements (that we seek to include) and undesirable 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 subset of these cases, we show that TAP is hard to even 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.

Problem

• Maximizing margin between desirable and undesirable elements in covering problems
• Introducing Target Approximation Problem (TAP) for real-world scenarios
• Analyzing TAP's hardness and approximation algorithms for restricted cases

Innovation

• Maximizing margin between desirable and undesirable elements
• Introducing Target Approximation Problem (TAP)
• Efficient 0.5-approximation for elements occurring twice