2 Search Results for "Nimavat, Rachit"

Document
DOI: 10.4230/LIPIcs.STACS.2025.17
Tight Approximation and Kernelization Bounds for Vertex-Disjoint Shortest Paths

Authors: Matthias Bentert, Fedor V. Fomin, and Petr A. Golovach

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
We examine the possibility of approximating Maximum Vertex-Disjoint Shortest Paths. In this problem, the input is an edge-weighted (directed or undirected) n-vertex graph G along with k terminal pairs (s_1,t_1),(s_2,t_2),…,(s_k,t_k). The task is to connect as many terminal pairs as possible by pairwise vertex-disjoint paths such that each path is a shortest path between the respective terminals. Our work is anchored in the recent breakthrough by Lochet [SODA '21], which demonstrates the polynomial-time solvability of the problem for a fixed value of k. Lochet’s result implies the existence of a polynomial-time ck-approximation for Maximum Vertex-Disjoint Shortest Paths, where c ≤ 1 is a constant. (One can guess 1/c terminal pairs to connect in k^O(1/c) time and then utilize Lochet’s algorithm to compute the solution in n^f(1/c) time.) Our first result suggests that this approximation algorithm is, in a sense, the best we can hope for. More precisely, assuming the gap-ETH, we exclude the existence of an o(k)-approximation within f(k) ⋅ poly(n) time for any function f that only depends on k. Our second result demonstrates the infeasibility of achieving an approximation ratio of m^{1/2-ε} in polynomial time, unless P = NP. It is not difficult to show that a greedy algorithm selecting a path with the minimum number of arcs results in a ⌈√𝓁⌉-approximation, where 𝓁 is the number of edges in all the paths of an optimal solution. Since 𝓁 ≤ n, this underscores the tightness of the m^{1/2-ε}-inapproximability bound. Additionally, we establish that Maximum Vertex-Disjoint Shortest Paths is fixed-parameter tractable when parameterized by 𝓁 but does not admit a polynomial kernel. Our hardness results hold for undirected graphs with unit weights, while our positive results extend to scenarios where the input graph is directed and features arbitrary (non-negative) edge weights.
Cite as

Matthias Bentert, Fedor V. Fomin, and Petr A. Golovach. Tight Approximation and Kernelization Bounds for Vertex-Disjoint Shortest Paths. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bentert_et_al:LIPIcs.STACS.2025.17,
  author =	{Bentert, Matthias and Fomin, Fedor V. and Golovach, Petr A.},
  title =	{{Tight Approximation and Kernelization Bounds for Vertex-Disjoint Shortest Paths}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{17:1--17:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.17},
  URN =		{urn:nbn:de:0030-drops-228422},
  doi =		{10.4230/LIPIcs.STACS.2025.17},
  annote =	{Keywords: Inapproximability, Fixed-parameter tractability, Parameterized approximation}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.38
Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary

Authors: Julia Chuzhoy, David H. K. Kim, and Rachit Nimavat

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract
We study the classical Node-Disjoint Paths (NDP) problem: given an undirected n-vertex graph G, together with a set {(s_1,t_1),...,(s_k,t_k)} of pairs of its vertices, called source-destination, or demand pairs, find a maximum-cardinality set {P} of mutually node-disjoint paths that connect the demand pairs. The best current approximation for the problem is achieved by a simple greedy O(sqrt{n})-approximation algorithm. Until recently, the best negative result was an Omega(log^{1/2-epsilon}n)-hardness of approximation, for any fixed epsilon, under standard complexity assumptions. A special case of the problem, where the underlying graph is a grid, has been studied extensively. The best current approximation algorithm for this special case achieves an O~(n^{1/4})-approximation factor. On the negative side, a recent result by the authors shows that NDP is hard to approximate to within factor 2^{Omega(sqrt{log n})}, even if the underlying graph is a subgraph of a grid, and all source vertices lie on the grid boundary. In a very recent follow-up work, the authors further show that NDP in grid graphs is hard to approximate to within factor Omega(2^{log^{1-epsilon}n}) for any constant epsilon under standard complexity assumptions, and to within factor n^{Omega(1/(log log n)^2)} under randomized ETH. In this paper we study the NDP problem in grid graphs, where all source vertices {s_1,...,s_k} appear on the grid boundary. Our main result is an efficient randomized 2^{O(sqrt{log n}* log log n)}-approximation algorithm for this problem. Our result in a sense complements the 2^{Omega(sqrt{log n})}-hardness of approximation for sub-graphs of grids with sources lying on the grid boundary, and should be contrasted with the above-mentioned almost polynomial hardness of approximation of NDP in grid graphs (where the sources and the destinations may lie anywhere in the grid). Much of the work on approximation algorithms for NDP relies on the multicommodity flow relaxation of the problem, which is known to have an Omega(sqrt n) integrality gap, even in grid graphs, with all source and destination vertices lying on the grid boundary. Our work departs from this paradigm, and uses a (completely different) linear program only to select the pairs to be routed, while the routing itself is computed by other methods.
Cite as

Julia Chuzhoy, David H. K. Kim, and Rachit Nimavat. Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chuzhoy_et_al:LIPIcs.ICALP.2018.38,
  author =	{Chuzhoy, Julia and Kim, David H. K. and Nimavat, Rachit},
  title =	{{Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{38:1--38:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.38},
  URN =		{urn:nbn:de:0030-drops-90423},
  doi =		{10.4230/LIPIcs.ICALP.2018.38},
  annote =	{Keywords: Node-disjoint paths, approximation algorithms, routing and layout}
}
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