On Competitive Permutations for Set Cover by Intervals

TL;DR

This paper analyzes the performance of greedy algorithms for set cover by intervals, providing approximation guarantees, a new algorithm, and insights into the properties of optimal solutions.

Contribution

It introduces a 3/4-competitive permutation for greedy interval cover, an efficient approximation algorithm, and demonstrates the non-monotonicity of marginal benefits in set cover solutions.

Findings

Greedy permutation is 3/4-competitive for prefix coverage.

An $O(n + m/psilon)$ algorithm approximates optimal cover.

Optimal set cover solutions lack diminishing returns property.

Abstract

We revisit the problem of computing an optimal partial cover of points by intervals. We show that the greedy algorithm computes a permutation $\Pi = \pi_1, \pi_2,\ldots$ of the intervals that is $3/4$-competitive for any prefix of $k$ intervals. That is, for any $k$, the intervals $\pi_1 \cup \cdots \cup \pi_k$ covers at least $3/4$-fraction of the points covered by the optimal solution using $k$ intervals. We also provide an approximation algorithm that, in $O(n + m/\varepsilon)$ time, computes a cover by $(1+\varepsilon)k$ intervals that is as good as the optimal solution using $k$ intervals, where $n$ is the number of input points, and $m$ is the number of intervals (we assume here the input is presorted). Finally, we show a counter example illustrating that the optimal solutions for set cover do not have the diminishing return property -- that is, the marginal benefit from using more sets is not monotonically decreasing. Fortunately, the diminishing returns does hold for intervals.