A constructive proof of Swap Local Search worst-case instances for the Maximum Coverage Problem

Kerkkamp, R.B.O.; Aardal, Karen

doi:10.1016/j.orl.2016.03.001

Cited by 3 publications

(1 citation statement)

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“…In this paper, we study the effectiveness of local search for geometric MC problems. In the general case, b-swap local search is known to yield a tight approximation ratio of 1/2 [23]. However, for special cases such as geometric MC problems local search is a promising candidate for beating the barrier 1 − 1/e.…”

Section: Introductionmentioning

confidence: 99%

Approximation Schemes for Geometric Coverage Problems

Chaplick,

De,

Ravsky

et al. 2016

Preprint

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In their seminal work, Mustafa and Ray [29] showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search -this is one of the most general approaches known for such problems. Their result applies if a naturally defined "exchange graph" for two feasible solutions is planar and is based on subdividing this graph via a planar separator theorem due to Frederickson [17]. Obtaining similar results for the related maximum k-coverage problem (MC) seems non-trivial due to the hard cardinality constraint. In fact, while Badanidiyuru, Kleinberg, and Lee [4] have shown(via a different analysis) that local search yields a PTAS for two-dimensional real halfspaces, they only conjectured that the same holds true for dimension three. Interestingly, at this point it was already known that local search provides a PTAS for the corresponding set cover case and this followed directly from the approach of Mustafa and Ray.In this work we provide a way to address the above-mentioned issue. First, we propose a color-balanced version of the planar separator theorem. The resulting subdivision approximates locally in each part the global distribution of the colors. Second, we show how this roughly balanced subdivision can be employed in a more careful analysis to strictly obey the hard cardinality constraint. More specifically, we obtain a PTAS for any "planarizable" instance of MC and thus essentially for all cases where the corresponding SC instance can be tackled via the approach of Mustafa and Ray. As a corollary, we confirm the conjecture of Badanidiyuru, Kleinberg, and Lee [4] regarding real half spaces in dimension three. We feel that our ideas could also be helpful in other geometric settings involving a cardinality constraint.

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Section: Introductionmentioning

confidence: 99%