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Improved Online Hitting Set Algorithms for Structured and Geometric Set Systems

Sujoy Bhore, Anupam Gupta, Amit Kumar

Abstract

In the online hitting set problem, sets arrive over time, and the algorithm has to maintain a subset of elements that hit all the sets seen so far. Alon, Awerbuch, Azar, Buchbinder, and Naor (SICOMP 2009) gave an algorithm with competitive ratio $O(\log n \log m)$ for the (general) online hitting set and set cover problems for $m$ sets and $n$ elements; this is known to be tight for efficient online algorithms. Given this barrier for general set systems, we ask: can we break this double-logarithmic phenomenon for online hitting set/set cover on structured and geometric set systems? We provide an $O(\log n \log\log n)$-competitive algorithm for the weighted online hitting set problem on set systems with linear shallow-cell complexity, replacing the double-logarithmic factor in the general result by effectively a single logarithmic term. As a consequence of our results we obtain the first bounds for weighted online hitting set for natural geometric set families, thereby answering open questions regarding the gap between general and geometric weighted online hitting set problems.

Improved Online Hitting Set Algorithms for Structured and Geometric Set Systems

Abstract

In the online hitting set problem, sets arrive over time, and the algorithm has to maintain a subset of elements that hit all the sets seen so far. Alon, Awerbuch, Azar, Buchbinder, and Naor (SICOMP 2009) gave an algorithm with competitive ratio for the (general) online hitting set and set cover problems for sets and elements; this is known to be tight for efficient online algorithms. Given this barrier for general set systems, we ask: can we break this double-logarithmic phenomenon for online hitting set/set cover on structured and geometric set systems? We provide an -competitive algorithm for the weighted online hitting set problem on set systems with linear shallow-cell complexity, replacing the double-logarithmic factor in the general result by effectively a single logarithmic term. As a consequence of our results we obtain the first bounds for weighted online hitting set for natural geometric set families, thereby answering open questions regarding the gap between general and geometric weighted online hitting set problems.
Paper Structure (18 sections, 11 theorems, 20 equations, 1 figure, 4 algorithms)

This paper contains 18 sections, 11 theorems, 20 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Our Techniques
  4. Related Work
  5. Notation and Definitions
  6. Shallow Cell Complexity
  7. Uniform Set Systems
  8. Unweighted Online Hitting Set
  9. The NetFinder Algorithm
  10. Implementing NetFinder Online
  11. The Final Algorithm
  12. Online Algorithm for Weighted Hitting Sets
  13. Element Levels
  14. The Weighted Hitting Set Algorithm
  15. Backup Elements and Consistent Rounding
  16. ...and 3 more sections

Key Result

Theorem 1.1

There is a randomized online algorithm for (unweighted) hitting set that has competitive ratio $O(\log n \cdot (d + \log (d\cdot \varphi(d\, {\sf OPT}\, \log n, d))))$, where $\varphi(\cdot, \cdot)$ is the shallow cell complexity (SCC) and $d$ the VC dimension of the set system.

Figures (1)

  • Figure 1: Bounds on the SCC and VC dimensions of some primal/dual set systems from mustafa2022samplingmustafa2017epsilon. Primal bounds give results for hitting sets, and dual bounds give results for set cover. A triangle is $\alpha$-fat if the ratio of the radius of the smallest enclosing ball to that of the largest inscribed ball is at most $\alpha$.

Theorems & Definitions (29)

  • Theorem 1.1: Unweighted Hitting Set
  • Theorem 1.2: Weighted Hitting Set
  • Definition 2.1: Shallow Cell Complexity
  • Definition 2.2: Well-Behaved SCC
  • Definition 3.1: Projection
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • ...and 19 more