Fractional covers and matchings in families of weighted $d$-intervals

TL;DR

This paper explores bounds on fractional and weighted covers and matchings in families of $d$-intervals, extending previous topological and combinatorial methods to these variants with tight bounds.

Contribution

It introduces tight upper bounds for weighted and fractional versions of the problem, utilizing topological and combinatorial approaches, including a weighted Turán's theorem.

Findings

Bounded the fractional and weighted transversal number in terms of matching number and $d$.

Extended topological methods to weighted and fractional settings.

Provided a more direct proof of a known upper bound.

Abstract

A $d$-{\em interval} is a union of at most $d$ disjoint closed intervals on a fixed line. Tardos [Combinatorica 15 (1995), 123-134] and the second author [Disc. Comput. Geom. 18 (1997), 195-203] used topological tools to bound the transversal number $\tau$ of a family $H$ of $d$-intervals in terms of $d$ and the matching number $\nu$ of $H$. We investigate the weighted and fractional versions of this problem and prove upper bounds that are tight up to constant factors. We apply both the topological method and an approach of Alon [Disc. Comput. Geom. 19 (1998), 333-334]. For the use of the latter, we prove a weighted version of Tur\'{a}n's theorem. We also provide a proof of the second author's upper bound that is more direct than the original proof.