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Greedy Heuristics and Linear Relaxations for the Random Hitting Set Problem
Authors
Gabriel Arpino 
,
Daniil Dmitriev ,
-
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-09-16
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Acknowledgements
The authors thank Dylan J. Altschuler, Afonso S. Bandeira, Raphaël Barboni, and Anastasia Kireeva for helpful discussions. DD is supported by ETH AI Center doctoral fellowship and ETH Foundations of Data Science initiative. GA is supported by the Cambridge Trust. NG is grateful for the funding received from Elizaveta Rebrova.
Cite AsGet BibTex
Gabriel Arpino, Daniil Dmitriev, and Nicolo Grometto. Greedy Heuristics and Linear Relaxations for the Random Hitting Set Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 30:1-30:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.30
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.30
Abstract
Consider the Hitting Set problem where, for a given universe 𝒳 = {1, ..., n} and a collection of subsets 𝒮₁, ..., 𝒮_m, one seeks to identify the smallest subset of 𝒳 which has a nonempty intersection with every element in the collection. We study a probabilistic formulation of this problem, where the underlying subsets are formed by including each element of the universe independently with probability p. We rigorously analyze integrality gaps between linear programming and integer programming solutions to the problem. In particular, we prove the absence of an integrality gap in the sparse regime mp ≲ log(n) and the presence of a non-vanishing integrality gap in the dense regime mp ≫ log{n}. Moreover, for large enough values of n, we look at the performance of Lovász’s celebrated Greedy algorithm [Lovász, 1975] with respect to the chosen input distribution, and prove that it finds optimal solutions up to multiplicative constants. This highlights separation of Greedy performance between average-case and worst-case settings.
Subject Classification
ACM Subject Classification
- Mathematics of computing → Combinatorial optimization
- Theory of computation → Approximation algorithms analysis
- Theory of computation → Randomness, geometry and discrete structures
Keywords
- Hitting Set
- Random Hypergraph
- Integrality Gap
- Greedy Algorithm
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References
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