arXiv subjects

Laura Sanità

Publications and source records attributed to Laura Sanità.

26 records · Page 2Linked to original sources

arXiv2017

Single-sink Fractionally Subadditive Network Design

We study a generalization of the Steiner tree problem, where we are given a weighted network $G$ together with a collection of $k$ subsets of its vertices and a root $r$. We wish to construct a minimum cost network such that the network supports one unit of flow to the root from every node in a subset simultaneously. The network constructed does not need to support flows from all the subsets simultaneously. We settle an open question regarding the complexity of this problem for $k=2$, and give a $\frac{3}{2}$-approximation algorithm that improves over a (trivial) known 2-approximation. Furthermore, we prove some structural results that prevent many well-known techniques from doing better than the known $O(\log n)$-approximation. Despite these obstacles, we conjecture that this problem should have an $O(1)$-approximation. We also give an approximation result for a variant of the problem where the solution is required to be a path.

cs.DS↗

arXiv2017

A $\frac{3}{2}$-Approximation Algorithm for Tree Augmentation via Chvátal-Gomory Cuts

The weighted tree augmentation problem (WTAP) is a fundamental network design problem. We are given an undirected tree $G = (V,E)$, an additional set of edges $L$ called links and a cost vector $c \in \mathbb{R}^L_{\geq 1}$. The goal is to choose a minimum cost subset $S \subseteq L$ such that $G = (V, E \cup S)$ is $2$-edge-connected. In the unweighted case, that is, when we have $c_\ell = 1$ for all $\ell \in L$, the problem is called the tree augmentation problem (TAP). Both problems are known to be APX-hard, and the best known approximation factors are $2$ for WTAP by (Frederickson and JáJá, '81) and $\tfrac{3}{2}$ for TAP due to (Kortsarz and Nutov, TALG '16). In the case where all link costs are bounded by a constant $M$, (Adjiashvili, SODA '17) recently gave a $\approx 1.96418+\varepsilon$-approximation algorithm for WTAP under this assumption. This is the first approximation with a better guarantee than $2$ that does not require restrictions on the structure of the tree or the links. In this paper, we improve Adjiashvili's approximation to a $\frac{3}{2}+\varepsilon$-approximation for WTAP under the bounded cost assumption. We achieve this by introducing a strong LP that combines $\{0,\frac{1}{2}\}$-Chvátal-Gomory cuts for the standard LP for the problem with bundle constraints from Adjiashvili. We show that our LP can be solved efficiently and that it is exact for some instances that arise at the core of Adjiashvili's approach. This results in the improved guarantee of $\frac{3}{2}+\varepsilon$. For TAP, this is the best known LP-based result, and matches the bound of $\frac{3}{2}+\varepsilon$ achieved by the best SDP-based algorithm due to (Cheriyan and Gao, arXiv '15).

cs.DM↗

arXiv2016

Fast Approximation Algorithms for the Generalized Survivable Network Design Problem

In a standard $f$-connectivity network design problem, we are given an undirected graph $G=(V,E)$, a cut-requirement function $f:2^V \rightarrow {\mathbb{N}}$, and non-negative costs $c(e)$ for all $e \in E$. We are then asked to find a minimum-cost vector $x \in {\mathbb{N}}^E$ such that $x(δ(S)) \geq f(S)$ for all $S \subseteq V$. We focus on the class of such problems where $f$ is a proper function. This encodes many well-studied NP-hard problems such as the generalized survivable network design problem. In this paper we present the first strongly polynomial time FPTAS for solving the LP relaxation of the standard IP formulation of the $f$-connectivity problem with general proper functions $f$. Implementing Jain's algorithm, this yields a strongly polynomial time $(2+ε)$-approximation for the generalized survivable network design problem (where we consider rounding up of rationals an arithmetic operation).

math.OC↗

arXiv2015

On the existence of compact ε-approximated formulations for knapsack in the original space

We show that there exists a family of Knapsack polytopes such that, for each polytope P from this family and each ε > 0, any ε-approximated formulation of P in the original space R^n requires a number of inequalities that is super-polynomial in n. This answers a question by Bienstock and McClosky (2012). We also prove that, for any down-monotone polytope, an ε-approximated formulation in the original space can be obtained with inequalities using at most O(min{log(n/ε),n}/ε) different coefficients.

math.OC↗

arXiv2013

An LMP O(log n)-Approximation Algorithm for Node Weighted Prize Collecting Steiner Tree

In the node-weighted prize-collecting Steiner tree problem (NW-PCST) we are given an undirected graph $G=(V,E)$, non-negative costs $c(v)$ and penalties $π(v)$ for each $v \in V$. The goal is to find a tree $T$ that minimizes the total cost of the vertices spanned by $T$ plus the total penalty of vertices not in $T$. This problem is well-known to be set-cover hard to approximate. Moss and Rabani (STOC'01) presented a primal-dual Lagrangean-multiplier-preserving $O(\ln |V|)$-approximation algorithm for this problem. We show a serious problem with the algorithm, and present a new, fundamentally different primal-dual method achieving the same performance guarantee. Our algorithm introduces several novel features to the primal-dual method that may be of independent interest.

cs.DS↗

arXiv2010

Set Covering with Ordered Replacement -- Additive and Multiplicative Gaps

We consider set covering problems where the underlying set system satisfies a particular replacement property w.r.t. a given partial order on the elements: Whenever a set is in the set system then a set stemming from it via the replacement of an element by a smaller element is also in the set system. Many variants of BIN PACKING that have appeared in the literature are such set covering problems with ordered replacement. We provide a rigorous account on the additive and multiplicative integrality gap and approximability of set covering with replacement. In particular we provide a polylogarithmic upper bound on the additive integrality gap that also yields a polynomial time additive approximation algorithm if the linear programming relaxation can be efficiently solved. We furthermore present an extensive list of covering problems that fall into our framework and consequently have polylogarithmic additive gaps as well.

cs.DM↗

arXiv2008

The Virtual Private Network Design Problem with Concave Costs (Oberwolfach abstract)

The symmetric Virtual Private Network Design (VPND) problem is concerned with buying capacity on links (edges) in a communication network such that certain traffic demands can be met. We investigate a natural generalization of VPND where the cost per unit of capacity may decrease if a larger amount of capacity is reserved (economies of scale principle). The growth of the cost of capacity is modelled by a non-decreasing concave function $f$. We call the problem the concave symmetric Virtual Private Network Design (cVPND) problem. After showing that a generalization of the so-called Pyramidal Routing problem and hence also the cVPND have the so-called tree routing property, we study approximation algorithms for cVPND. For general $f$, using known results on the so-called Single Source Buy at Bulk problem by Grandoni and Italiano, we give a randomized 24.92-approximation algorithm.

math.OC↗

arXiv2008

The VPN Tree Routing Conjecture for Outerplanar Networks

The VPN Tree Routing Conjecture is a conjecture about the Virtual Private Network Design problem. It states that the symmetric version of the problem always has an optimum solution which has a tree-like structure. In recent work, Hurkens, Keijsper and Stougie (Proc. IPCO XI, 2005; SIAM J. Discrete Math., 2007) have shown that the conjecture holds when the network is a ring. A shorter proof of the VPN Conjecture for rings was found a few months ago by Grandoni, Kaibel, Oriolo and Skutella (to appear in Oper. Res. Lett., 2008). In their paper, Grandoni et al. introduce another conjecture, called the Pyramidal Routing Conjecture (or simply PR Conjecture), which implies the VPN Conjecture. Here we consider a strengthened version of the PR Conjecture. First we establish several general tools which can be applied in arbitrary networks. Then we use them to prove that outerplanar networks satisfy the PR Conjecture.

math.OC↗