4 Search Results for "Vaz, Daniel"

Document
DOI: 10.4230/LIPIcs.MFCS.2024.45
Nearly-Tight Bounds for Flow Sparsifiers in Quasi-Bipartite Graphs

Authors: Syamantak Das, Nikhil Kumar, and Daniel Vaz

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Flow sparsification is a classic graph compression technique which, given a capacitated graph G on k terminals, aims to construct another capacitated graph H, called a flow sparsifier, that preserves, either exactly or approximately, every multicommodity flow between terminals (ideally, with size as a small function of k). Cut sparsifiers are a restricted variant of flow sparsifiers which are only required to preserve maximum flows between bipartitions of the terminal set. It is known that exact cut sparsifiers require 2^Ω(k) many vertices [Krauthgamer and Rika, SODA 2013], with the hard instances being quasi-bipartite graphs, where there are no edges between non-terminals. On the other hand, it has been shown recently that exact (or even (1+ε)-approximate) flow sparsifiers on networks with just 6 terminals require unbounded size [Krauthgamer and Mosenzon, SODA 2023, Chen and Tan, SODA 2024]. In this paper, we construct exact flow sparsifiers of size 3^k³ and exact cut sparsifiers of size 2^k² for quasi-bipartite graphs. In particular, the flow sparsifiers are contraction-based, that is, they are obtained from the input graph by (vertex) contraction operations. Our main contribution is a new technique to construct sparsifiers that exploits connections to polyhedral geometry, and that can be generalized to graphs with a small separator that separates the graph into small components. We also give an improved reduction theorem for graphs of bounded treewidth [Andoni et al., SODA 2011], implying a flow sparsifier of size O(k⋅w) and quality O((log w)/log log w), where w is the treewidth.
Cite as

Syamantak Das, Nikhil Kumar, and Daniel Vaz. Nearly-Tight Bounds for Flow Sparsifiers in Quasi-Bipartite Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 45:1-45:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{das_et_al:LIPIcs.MFCS.2024.45,
  author =	{Das, Syamantak and Kumar, Nikhil and Vaz, Daniel},
  title =	{{Nearly-Tight Bounds for Flow Sparsifiers in Quasi-Bipartite Graphs}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{45:1--45:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.45},
  URN =		{urn:nbn:de:0030-drops-206018},
  doi =		{10.4230/LIPIcs.MFCS.2024.45},
  annote =	{Keywords: Graph Sparsification, Cut Sparsifiers, Flow Sparsifiers, Quasi-bipartite Graphs, Bounded Treewidth}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2022.10
On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

Authors: Antonios Antoniadis, Sándor Kisfaludi-Bak, Bundit Laekhanukit, and Daniel Vaz

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Abstract
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 ℝ^d, with d ≥ 3, are NP-hardness and an O(log³ n)-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in ℝ^d is APX-hard for any d ≥ 3. More generally, this implies that TSP with k-dimensional flats does not admit a PTAS for any 1 ≤ k ≤ d-2 unless P = NP, which gives a complete classification regarding the existence of polynomial time approximation schemes for 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(log² n)-approximation algorithm for the problem, albeit with a running time of n^{O(log log n)}.
Cite as

Antonios Antoniadis, Sándor Kisfaludi-Bak, Bundit Laekhanukit, and Daniel Vaz. On the Approximability of the Traveling Salesman Problem with Line Neighborhoods. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{antoniadis_et_al:LIPIcs.SWAT.2022.10,
  author =	{Antoniadis, Antonios and Kisfaludi-Bak, S\'{a}ndor and Laekhanukit, Bundit and Vaz, Daniel},
  title =	{{On the Approximability of the Traveling Salesman Problem with Line Neighborhoods}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{10:1--10:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.10},
  URN =		{urn:nbn:de:0030-drops-161706},
  doi =		{10.4230/LIPIcs.SWAT.2022.10},
  annote =	{Keywords: Traveling Salesman with neighborhoods, Group Steiner Tree, Geometric approximation algorithms}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.39
On Approximating Degree-Bounded Network Design Problems

Authors: Xiangyu Guo, Guy Kortsarz, Bundit Laekhanukit, Shi Li, Daniel Vaz, and Jiayi Xian

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
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 ∈ ℝ_{≥ 0}^E, a root r ∈ V and k terminals K ⊆ V, we need to output a minimum-cost arborescence in G that contains an rrightarrow t path for every t ∈ K. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time O(log²k/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 ∈ 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(log² n))-bicriteria approximation: The algorithm produces a solution with cost at most O(log nlog k) times the cost of the optimum solution that violates the degree constraints by at most a factor of O(log²n). This is the first non-trivial result for the problem. While our cost-guarantee is nearly optimal, the degree violation factor of O(log²n) 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 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 nlog k), O(log n))-bicriteria approximation algorithm for DB-GST-T.
Cite as

Xiangyu Guo, Guy Kortsarz, Bundit Laekhanukit, Shi Li, Daniel Vaz, and Jiayi Xian. On Approximating Degree-Bounded Network Design Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 39:1-39:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{guo_et_al:LIPIcs.APPROX/RANDOM.2020.39,
  author =	{Guo, Xiangyu and Kortsarz, Guy and Laekhanukit, Bundit and Li, Shi and Vaz, Daniel and Xian, Jiayi},
  title =	{{On Approximating Degree-Bounded Network Design Problems}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{39:1--39:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.39},
  URN =		{urn:nbn:de:0030-drops-126420},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.39},
  annote =	{Keywords: Directed Steiner Tree, Group Steiner Tree, degree-bounded}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.8
Survivable Network Design for Group Connectivity in Low-Treewidth Graphs

Authors: Parinya Chalermsook, Syamantak Das, Guy Even, Bundit Laekhanukit, and Daniel Vaz

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Abstract
In the Group Steiner Tree problem (GST), we are given a (edge or vertex)-weighted graph G=(V,E) on n vertices, together with 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 minimum-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 N, and we wish to find a minimum-cost subgraph H subseteq G such that, for each group S_i, there is a vertex in the group that is connected to the root via k_i (vertex or edge) disjoint paths. While GST admits O(log^2 n log h) approximation, its higher connectivity variants are known to be Label-Cover hard, and for the vertex-weighted version, the hardness holds even when k=2 (it is widely believed that there is no subpolynomial approximation for the Label-Cover problem [Bellare et al., STOC 1993]). More precisely, the problem admits no 2^{log^{1-epsilon}n}-approximation unless NP subseteq DTIME(n^{polylog(n)}). 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 k_i disjoint paths from r may end at different vertices in a group [Chalermsook et al., SODA 2015], for which the authors gave a bicriteria approximation. For k >= 3, there is no non-trivial approximation algorithm known for edge-weighted Restricted Group SNDP, except for the special case of the relaxed variant on trees (folklore). Our main result is an O(log n log h) approximation algorithm for Restricted Group SNDP that runs in time n^{f(k, w)}, where w is the treewidth of the input graph. Our algorithm works for both edge and vertex weighted variants, and the approximation ratio nearly matches the lower bound when k and w are constants. The key to achieving this result is a non-trivial extension of a framework introduced in [Chalermsook et al., SODA 2017]. This framework first 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.
Cite as

Parinya Chalermsook, Syamantak Das, Guy Even, Bundit Laekhanukit, and Daniel Vaz. Survivable Network Design for Group Connectivity in Low-Treewidth Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chalermsook_et_al:LIPIcs.APPROX-RANDOM.2018.8,
  author =	{Chalermsook, Parinya and Das, Syamantak and Even, Guy and Laekhanukit, Bundit and Vaz, Daniel},
  title =	{{Survivable Network Design for Group Connectivity in Low-Treewidth Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{8:1--8:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.8},
  URN =		{urn:nbn:de:0030-drops-94127},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.8},
  annote =	{Keywords: Approximation Algorithms, Hardness of Approximation, Survivable Network Design, Group Steiner Tree}
}
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