Approximate Minimum-Weight Matching with Outliers under Translation

TL;DR

This paper presents efficient algorithms for approximating minimum-weight matchings with outliers under translation, applicable to various Lp norms, and constructs compact diagrams representing approximate solutions across translation space.

Contribution

It introduces novel algorithms for approximate matchings under translation for any Lp norm and develops small-size diagrams for efficient approximation over translation space.

Findings

Algorithms work for all Lp norms, including Euclidean.

Constructs small diagrams partitioning translation space.

Achieves efficient approximate solutions for matching problems.

Abstract

Our goal is to compare two planar point sets by finding subsets of a given size such that a minimum-weight matching between them has the smallest weight. This can be done by a translation of one set that minimizes the weight of the matching. We give efficient algorithms (a) for finding approximately optimal matchings, when the cost of a matching is the $L_p$-norm of the tuple of the Euclidean distances between the pairs of matched points, for any $p\in [1,\infty]$, and (b)~for constructing small-size approximate minimization (or matching) diagrams: partitions of the translation space into regions, together with an approximate optimal matching for each region.