arXiv subjects

Daniel Vaz

Publications and source records attributed to Daniel Vaz.

24 records · Page 2Linked to original sources

arXiv2020

On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in $\mathbb{R}^d$, with $d\ge 3$, are $\mathrm{NP}$-hardness and an $O(\log^3 n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in $\mathbb{R}^d$ is APX-hard for any $d\ge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1\le k \leq d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a complete classification of the approximability of these problems, as there are known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to give a stronger inapproximability factor for $d=O(\log n)$ by showing that TSP with lines does not admit a $(2-ε)$-approximation in $d$ dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an $O(\log^2 n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(\log\log n)}$.

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arXiv2020

Vertex Sparsification for Edge Connectivity

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $(1+ε)$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we study a thresholded version of the problem: for a given parameter $c$, find a smaller graph, which we call connectivity-$c$ mimicking network, which preserves connectivity among $k$ terminals exactly up to the value of $c$. We show that connectivity-$c$ mimicking networks with $O(kc^4)$ edges exist and can be found in time $m(c\log n)^{O(c)}$. We also give a separate algorithm that constructs such graphs with $k \cdot O(c)^{2c}$ edges in time $mc^{O(c)}\log^{O(1)}n$. These results lead to the first data structures for answering fully dynamic offline $c$-edge-connectivity queries for $c \ge 4$ in polylogarithmic time per query, as well as more efficient algorithms for survivable network design on bounded treewidth graphs.

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arXiv2020

On Approximating Degree-Bounded Network Design Problems

Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph $G=(V, E)$ with edge costs $c \in \mathbb{R}_{\geq 0}^E$, a root $r \in V$ and $k$ terminals $K\subseteq V$, we need to output the minimum-cost arborescence in $G$ that contains an $r$\textrightarrow $t$ path for every $t \in K$. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time $O(\log^2k/\log \log k)$-approximation algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general Degree-Bounded Directed Steiner Tree (DB-DST) problem, where we are additionally given a degree bound $d_v$ on each vertex $v \in V$, and we require that every vertex $v$ in the output tree has at most $d_v$ children. We give a quasi-polynomial time $(O(\log n \log k), O(\log^2 n))$-bicriteria approximation: The algorithm produces a solution with cost at most $O(\log n\log k)$ times the cost of the optimum solution that violates the degree constraints by at most a factor of $O(\log^2n)$. This is the first non-trivial result for the problem. While our cost-guarantee is nearly optimal, the degree violation factor of $O(\log^2n)$ is an $O(\log n)$-factor away from the approximation lower bound of $Ω(\log n)$ from the set-cover hardness. The hardness result holds even on the special case of the {\em Degree-Bounded Group Steiner Tree} problem on trees (DB-GST-T). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an $(O(\log n\log k), O(\log n))$-bicriteria approximation algorithm for DB-GST-T.

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arXiv2019

Mimicking Networks Parameterized by Connectivity

Given a graph $G=(V,E)$, capacities $w(e)$ on edges, and a subset of terminals $\mathcal{T} \subseteq V: |\mathcal{T}| = k$, a mimicking network for $(G,\mathcal{T})$ is a graph $(H,w')$ that contains copies of $\mathcal{T}$ and preserves the value of minimum cuts separating any subset $A, B \subseteq \mathcal{T}$ of terminals. Mimicking networks of size $2^{2^k}$ are known to exist and can be constructed algorithmically, while the best known lower bound is $2^{Ω(k)}$; therefore, an exponential size is required if one aims at preserving cuts exactly. In this paper, we study mimicking networks that preserve connectivity of the graph exactly up to the value of $c$, where $c$ is a parameter. This notion of mimicking network is sufficient for some applications, as we will elaborate. We first show that a mimicking of size $3^c \cdot k$ exists, that is, we can preserve cuts with small capacity using a network of size linear in $k$. Next, we show an algorithm that finds such a mimicking network in time $2^{O(c^2)} \operatorname{poly}(m)$.

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arXiv2018

Survivable Network Design for Group Connectivity in Low-Treewidth Graphs

In the Group Steiner Tree problem (GST), we are given a (vertex or edge)-weighted graph $G=(V,E)$ on $n$ vertices, a root vertex $r$ and a collection of groups $\{S_i\}_{i\in[h]}: S_i\subseteq V(G)$. The goal is to find a min-cost subgraph $H$ that connects the root to every group. We consider a fault-tolerant variant of GST, which we call Restricted (Rooted) Group SNDP. In this setting, each group $S_i$ has a demand $k_i\in[k],k\in\mathbb N$, and we wish to find a min-cost $H\subseteq G$ such that, for each group $S_i$, there is a vertex in $S_i$ connected to the root via $k_i$ (vertex or edge) disjoint paths. While GST admits $O(\log^2 n\log h)$ approximation, its high connectivity variants are Label-Cover hard, and for the vertex-weighted version, the hardness holds even when $k=2$. Previously, positive results were known only for the edge-weighted version when $k=2$ [Gupta et al., SODA 2010; Khandekar et al., Theor. Comput. Sci., 2012] and for a relaxed variant where the disjoint paths may end at different vertices in a group [Chalermsook et al., SODA 2015]. Our main result is an $O(\log n\log h)$ approximation for Restricted Group SNDP that runs in time $n^{f(k, w)}$, where $w$ is the treewidth of $G$. This nearly matches the lower bound when $k$ and $w$ are constant. The key to achieving this result is a non-trivial extension of the framework in [Chalermsook et al., SODA 2017], which embeds all feasible solutions to the problem into a dynamic program (DP) table. However, finding the optimal solution in the DP table remains intractable. We formulate a linear program relaxation for the DP and obtain an approximate solution via randomized rounding. This framework also allows us to systematically construct DP tables for high-connectivity problems. As a result, we present new exact algorithms for several variants of survivable network design problems in low-treewidth graphs.

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arXiv2016

New Integrality Gap Results for the Firefighters Problem on Trees

The firefighter problem is NP-hard and admits a $(1-1/e)$ approximation based on rounding the canonical LP. In this paper, we first show a matching integrality gap of $(1-1/e+ε)$ on the canonical LP. This result relies on a powerful combinatorial gadget that can be used to prove integrality gap results for many problem settings. We also consider the canonical LP augmented with simple additional constraints (as suggested by Hartke). We provide several evidences that these constraints improve the integrality gap of the canonical LP: (i) Extreme points of the new LP are integral for some known tractable instances and (ii) A natural family of instances that are bad for the canonical LP admits an improved approximation algorithm via the new LP. We conclude by presenting a $5/6$ integrality gap instance for the new LP.

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