Maximizing Weighted Dominance in the Plane

TL;DR

This paper addresses the problem of selecting a subset of unweighted points in the plane to maximize the weighted dominance coverage of a set of weighted points, providing efficient algorithms with improved bounds under certain conditions.

Contribution

The authors introduce a new algorithm with improved time complexity for maximizing weighted dominance coverage, and also present a simpler algorithm with competitive bounds.

Findings

The first algorithm operates in O(k min{n+m, n/k + m^2} log m) time.

The second, simpler algorithm runs in O(k m^2 + n log m) time.

Both algorithms are efficient when m = o(√n).

Abstract

Let P be a set of n weighted points, Q be a set of m unweighted points in the plane, and k a non-negative integer. We consider the problem of computing a subset $Q'\subseteq Q$ with size at most k such that the sum of the weights of the points of P dominated by at least one point in the set Q' is maximized. A point q in the plane dominates another point p if and only if $x(q)\ge x(p)$ and $y(q)\ge y(p)$, and at least one inequality is strict. We present a solution to the problem that takes O(n + m)-space and $O(k \min\{n+m, \frac{n}{k}+m^2\}\log m)$-time. We (conditionally) improve upon the existing result (the bounds of our solution are interesting when $m= o(\sqrt{n}))$. Moreover, we also present a simple algorithm solving the problem in $O(km^2+n\log m)$-time and $O(n+m)$-space. The bounds of the algorithm are interesting when $m= o(\sqrt{n})$.