Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1305.5998

Integrality gaps for strengthened LP relaxations of Capacitated and Lower-Bounded Facility Location

Stavros G. Kolliopoulos·Yannis Moysoglou
View original source ↗Read source document ↗Export metadata ↓

Abstract

The metric uncapacitated facility location problem (UFL) enjoys a special stature in approximation algorithms as a testbed for various techniques. Two generalizations of UFL are capacitated facility location (CFL) and lower-bounded facility location (LBFL). In the former, every facility has a capacity which is the maximum demand that can be assigned to it, while in the latter, every open facility is required to serve a given minimum amount of demand. Both CFL and LBFL are approximable within a constant factor but their respective natural LP relaxations have an unbounded integrality gap. According to Shmoys and Williamson, the existence of a relaxation-based algorithm for CFL is one of the top 10 open problems in approximation algorithms. In this paper we give the first results on this problem. We provide substantial evidence against the existence of a good LP relaxation for CFL by showing unbounded integrality gaps for two families of strengthened formulations. The first family we consider is the hierarchy of LPs resulting from repeated applications of the lift-and-project Lovász-Schrijver procedure starting from the standard relaxation. We show that the LP relaxation for CFL resulting after $Ω(n)$ rounds, where $n$ is the number of facilities in the instance, has unbounded integrality gap. Note that the Lovász-Schrijver procedure is known to yield an exact formulation for CFL in at most $n$ rounds. We also introduce the family of proper relaxations which generalizes to its logical extreme the classic star relaxation, an equivalent form of the natural LP. We characterize the integrality gap of proper relaxations for both LBFL and CFL and show a threshold phenomenon under which it decreases from unbounded to 1.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Stavros G. Kolliopoulos, Yannis Moysoglou. 2013-07-18. Integrality gaps for strengthened LP relaxations of Capacitated and Lower-Bounded Facility Location. https://arxiv.org/abs/1305.5998

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗