Iterative Packing for Demand and Hypergraph Matching

TL;DR

This paper introduces a simple iterative packing technique inspired by iterative rounding, achieving improved approximation algorithms for demand and hypergraph matching problems, effectively matching their integrality gaps.

Contribution

The paper presents a novel iterative packing method that simplifies and enhances approximation algorithms for demand matching and $k$-column-sparse packing problems.

Findings

Deterministic 2k-approximation for $k$-CS-PIP.

Deterministic 3-approximation for demand matching.

Effectively matches the integrality gaps for these problems.

Abstract

Iterative rounding has enjoyed tremendous success in elegantly resolving open questions regarding the approximability of problems dominated by covering constraints. Although iterative rounding methods have been applied to packing problems, no single method has emerged that matches the effectiveness and simplicity afforded by the covering case. We offer a simple iterative packing technique that retains features of Jain's seminal approach, including the property that the magnitude of the fractional value of the element rounded during each iteration has a direct impact on the approximation guarantee. We apply iterative packing to generalized matching problems including demand matching and $k$-column-sparse column-restricted packing ($k$-CS-PIP) and obtain approximation algorithms that essentially settle the integrality gap for these problems. We present a simple deterministic $2k$-approximation for $k$-CS-PIP, where an $8k$-approximation was the best deterministic algorithm previously known. The integrality gap in this case is at least $2(k-1+1/k)$. We also give a deterministic $3$-approximation for a generalization of demand matching, settling its natural integrality gap.