DROPS

Volume

LIPIcs, Volume 244

30th Annual European Symposium on Algorithms (ESA 2022)

ESA 2022, September 5-9, 2022, Berlin/Potsdam, Germany

Editors: Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman

Document

DOI: 10.4230/LIPIcs.STACS.2026.42

Computing Twin-Width via Treedepth and Vertex Integrity

Authors: Robert Ganian and Mathis Rocton

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract

Twin-width is a graph parameter that has become central to explaining the fixed-parameter tractability of first-order model checking across many graph classes. Despite its algorithmic importance, computing twin-width remains poorly understood: even recognizing graphs of twin-width at most four is NP-hard, and no fixed-parameter approximations parameterized by twin-width itself are known. A recent approach towards breaking this barrier focuses on first developing fixed-parameter algorithms for computing or approximating twin-width under parameterizations distinct from twin-width. Our first result establishes that approximating twin-width is fixed-parameter tractable when parameterized by treedepth, thereby breaking the long-standing barrier that all previous tractable parameterizations were based on deletion distance. The proof proceeds via oriented twin-width, yielding the first constructive evidence that this variant may be easier to handle algorithmically. As our second main result, we show that computing twin-width exactly is fixed-parameter tractable with respect to vertex integrity. This constitutes the first non-trivial parameterized algorithm for computing optimal contraction sequences.

Cite as

Robert Ganian and Mathis Rocton. Computing Twin-Width via Treedepth and Vertex Integrity. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ganian_et_al:LIPIcs.STACS.2026.42,
  author =	{Ganian, Robert and Rocton, Mathis},
  title =	{{Computing Twin-Width via Treedepth and Vertex Integrity}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{42:1--42:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle 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.2026.42},
  URN =		{urn:nbn:de:0030-drops-255318},
  doi =		{10.4230/LIPIcs.STACS.2026.42},
  annote =	{Keywords: twin-width, fixed-parameter algorithms, treedepth, vertex integrity}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.40

A Simple and Robust Protocol for Distributed Counting

Authors: Edith Cohen, Moshe Shechner, and Uri Stemmer

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We revisit the distributed counting problem, where a server must continuously approximate the total number of events occurring across k sites while minimizing communication. The communication complexity of this problem is known to be Θ(k/(ε)log N) for deterministic protocols. Huang, Yi, and Zhang (2012) showed that randomization can reduce this to Θ((√k)/ε log N), but their analysis is restricted to the oblivious setting, where the stream of events is independent of the protocol’s outputs. Xiong, Zhu, and Huang (2023) presented a robust protocol for distributed counting that removes the oblivious assumption. However, their communication complexity is suboptimal by a polylog(k) factor and their protocol is substantially more complex than the oblivious protocol of Huang et al. (2012). This left open a natural question: could it be that the simple protocol of Huang et al. (2012) is already robust? We resolve this question with two main contributions. First, we show that the protocol of Huang et al. (2012) is itself not robust by constructing an explicit adaptive attack that forces it to lose its accuracy. Second, we present a new, surprisingly simple, robust protocol for distributed counting that achieves the optimal communication complexity of O((√k)/ε log N). Our protocol is simpler than that of Xiong et al. (2023), perhaps even simpler than that of Huang et al. (2012), and is the first to match the optimal oblivious complexity in the adaptive setting.

Cite as

Edith Cohen, Moshe Shechner, and Uri Stemmer. A Simple and Robust Protocol for Distributed Counting. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 40:1-40:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cohen_et_al:LIPIcs.ITCS.2026.40,
  author =	{Cohen, Edith and Shechner, Moshe and Stemmer, Uri},
  title =	{{A Simple and Robust Protocol for Distributed Counting}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{40:1--40:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.40},
  URN =		{urn:nbn:de:0030-drops-253272},
  doi =		{10.4230/LIPIcs.ITCS.2026.40},
  annote =	{Keywords: Distributed Streaming, Adversarial Streaming}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.107

New Greedy Spanners and Applications

Authors: Elizaveta Popova and Elad Tzalik

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We present a simple greedy procedure to compute an (α,β)-spanner for a graph G. We then show that this procedure is useful for building fault-tolerant spanners, as well as spanners for weighted graphs. Our first main result is an algorithm that, given a multigraph G, outputs an f edge fault-tolerant (k,k-1)-spanner H of size O(fn^{1+1/k}) which is tight. To our knowledge, this is the first tight result concerning the price of fault tolerance in spanners which are not multiplicative, in any model of faults. Our second main result is a new construction of a spanner for weighted graphs. We show that any weighted graph G has a subgraph H with O(n^{1+1/k}) edges such that any path P of hop-length 𝓁 in G has a replacement path P' in H of weighted length ≤ w(P)+(2k-2)w^(1/2)(P) where w(P) is the total edge weight of P, and w^(1/2) denotes the sum of the largest ⌈𝓁/2⌉ edge weights along P. Moreover, we show such approximation is optimal for shortest paths of hop-length 2. To our knowledge, this is the first construction of a "spanner" for weighted graphs that strictly improves upon the stretch of multiplicative (2k-1)-spanners for all non-adjacent vertex pairs, while maintaining the same size bound. Our technique is based on using clustering and ball-growing, which are methods commonly used in designing spanner algorithms, to analyze simple greedy algorithms. This allows us to combine the flexibility of clustering approaches with the unique properties of the greedy algorithm to get improved bounds. In particular, our methods give a very short proof that the parallel greedy spanner adds O(kn^{1+1/k}) edges, improving upon known bounds.

Cite as

Elizaveta Popova and Elad Tzalik. New Greedy Spanners and Applications. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 107:1-107:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{popova_et_al:LIPIcs.ITCS.2026.107,
  author =	{Popova, Elizaveta and Tzalik, Elad},
  title =	{{New Greedy Spanners and Applications}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{107:1--107:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.107},
  URN =		{urn:nbn:de:0030-drops-253945},
  doi =		{10.4230/LIPIcs.ITCS.2026.107},
  annote =	{Keywords: Graph Spanners, Greedy Algorithms}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.5

Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs

Authors: Mridul Ahi, Keerti Choudhary, Shlok Pande, Pushpraj, and Lakshay Saggi

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

This paper addresses the problem of designing fault-tolerant data structures for the (s,t)-max-flow and (s,t)-min-cut problems in unweighted directed graphs. Given a directed graph G = (V, E) with a designated source s, sink t, and an (s,t)-max-flow of value λ, we present constructions for max-flow and min-cut sensitivity oracles, and introduce the concept of a fault-tolerant flow family, which may be of independent interest. Our main contributions are as follows. 1) Fault-Tolerant Flow Family: We construct a family ℬ of 2λ+1 (s,t)-flows such that for every edge e, ℬ contains an (s,t)-max-flow of G-e. This covering property is tight up to constants for single failures and provably cannot extend to comparably small families for k ≥ 2, where we show an Ω(n) lower bound on the family size, independent of λ. 2) Max-Flow Sensitivity Oracle: Using the fault-tolerant flow family, we construct a single as well as dual-edge sensitivity oracle for (s,t)-max-flow that requires only O(λ n) space. Given any set F of up to two failing edges, the oracle reports the updated max-flow value in G-F in O(n) time. Additionally, for the single-failure case, the oracle can determine in constant time whether the flow through an edge x changes when another edge e fails. 3) Min-Cut Sensitivity Oracle for Dual Failures: Recently, Baswana et al. (ICALP’22) designed an O(n²)-sized oracle for answering (s,t)-min-cut size queries under dual edge failures in constant time, along with a matching lower bound. We extend this by focusing on graphs with small min-cut values λ, and present a more compact oracle of size O(λ n) that answers such min-cut size queries in constant time and reports the corresponding (s,t)-min-cut partition in O(n) time. We also show that the space complexity of our oracle is asymptotically optimal in this setting. 4) Min-Cut Sensitivity Oracle for Multiple Failures: We extend our results to the general case of k edge failures. For any graph with (s,t)-min-cut of size λ, we construct a k-fault-tolerant min-cut oracle with space complexity O_{λ,k}(n log n) that answers min-cut size queries in O_{λ,k}(log n) time. This also leads to improved fault-tolerant (s,t)-reachability oracles, achieving O(n log n) space and O(log n) query time for up to k = O(1) edge failures.

Cite as

Mridul Ahi, Keerti Choudhary, Shlok Pande, Pushpraj, and Lakshay Saggi. Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 5:1-5:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ahi_et_al:LIPIcs.ITCS.2026.5,
  author =	{Ahi, Mridul and Choudhary, Keerti and Pande, Shlok and Pushpraj and Saggi, Lakshay},
  title =	{{Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{5:1--5:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.5},
  URN =		{urn:nbn:de:0030-drops-252920},
  doi =		{10.4230/LIPIcs.ITCS.2026.5},
  annote =	{Keywords: Fault tolerance, Data structures, Minimum cuts, Maximum flows}
}

Document

DOI: 10.4230/LIPIcs.IPEC.2025.17

Treedepth Inapproximability and Exponential ETH Lower Bound

Authors: Édouard Bonnet, Daniel Neuen, and Marek Sokołowski

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract

Treedepth is a central parameter to algorithmic graph theory. The current state-of-the-art in computing and approximating treedepth consists of a 2^{O(k²)} n-time exact algorithm and a polynomial-time O(OPT log^{3/2} OPT)-approximation algorithm, where the former algorithm returns an elimination forest of height k (witnessing that treedepth is at most k) for the n-vertex input graph G, or correctly reports that G has treedepth larger than k, and OPT is the actual value of the treedepth. On the complexity side, exactly computing treedepth is NP-complete, but the known reductions do not rule out a polynomial-time approximation scheme (PTAS), and under the Exponential Time Hypothesis (ETH) only exclude a running time of 2^o(√n) for exact algorithms. We show that 1.0003-approximating Treedepth is NP-hard, and that exactly computing the treedepth of an n-vertex graph requires time 2^Ω(n), unless the ETH fails. We further derive that there exist absolute constants δ, c > 0 such that any (1+δ)-approximation algorithm requires time 2^Ω(n/log^c n). We do so via a simple direct reduction from Satisfiability to Treedepth, inspired by a reduction recently designed for Treewidth [STOC '25].

Cite as

Édouard Bonnet, Daniel Neuen, and Marek Sokołowski. Treedepth Inapproximability and Exponential ETH Lower Bound. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 17:1-17:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bonnet_et_al:LIPIcs.IPEC.2025.17,
  author =	{Bonnet, \'{E}douard and Neuen, Daniel and Soko{\l}owski, Marek},
  title =	{{Treedepth Inapproximability and Exponential ETH Lower Bound}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{17:1--17:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.17},
  URN =		{urn:nbn:de:0030-drops-251494},
  doi =		{10.4230/LIPIcs.IPEC.2025.17},
  annote =	{Keywords: treedepth, lower bounds, approximation}
}

Document

DOI: 10.4230/LIPIcs.IPEC.2025.4

On the Complexity of Secluded Path Problems

Authors: Tesshu Hanaka and Daisuke Tsuru

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract

This paper investigates the complexity of finding secluded paths in graphs. We focus on the Short Secluded Path problem and a natural new variant we introduce, Shortest Secluded Path. Formally, given an undirected graph G = (V, E), two vertices s,t ∈ V, and two integers k,l, the Short Secluded Path problem asks whether there exists an s-t path of length at most k with at most l neighbors. This problem is known to be computationally hard: it is W[1]-hard when parameterized by the path length k or by cliquewidth, and para-NP-complete when parameterized by the number l of neighbors. The fixed-parameter tractability is known for k+l or treewidth. In this paper, we expand the parameterized complexity landscape by designing (1) an XP algorithm parameterized by cliquewidth and (2) fixed-parameter algorithms parameterized by neighborhood diversity and twin cover number, respectively. As a byproduct, our results also provide parameterized algorithms for the classic s-t k-Path problem. Furthermore, we introduce the Shortest Secluded Path problem, which seeks a shortest s-t path with the minimum number of neighbors. In contrast to the hardness of the original problem, we reveal that this variant is solvable in polynomial time on unweighted graphs. We complete this by showing that for edge-weighted graphs, the problem becomes W[1]-hard yet remains in XP when parameterized by the shortest path distance between s and t.

Cite as

Tesshu Hanaka and Daisuke Tsuru. On the Complexity of Secluded Path Problems. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hanaka_et_al:LIPIcs.IPEC.2025.4,
  author =	{Hanaka, Tesshu and Tsuru, Daisuke},
  title =	{{On the Complexity of Secluded Path Problems}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{4:1--4:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.4},
  URN =		{urn:nbn:de:0030-drops-251361},
  doi =		{10.4230/LIPIcs.IPEC.2025.4},
  annote =	{Keywords: Secluded path, Parameterized complexity, Polynomial-time algorithm}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2025.27

Fault-Tolerant Approximate Distance Oracles with a Source Set

Authors: Dipan Dey and Telikepalli Kavitha

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract

Our input is an undirected weighted graph G = (V,E) on n vertices along with a source set S ⊆ V. The problem is to preprocess G and build a compact data structure such that upon query Qu(s,v,f) where (s,v) ∈ S×V and f is any faulty edge, we can quickly find a good estimate (i.e., within a small multiplicative stretch) of the s-v distance in G-f. We use a fault-tolerant ST-distance oracle from the work of Bilò et al. (STACS 2018) to construct an S×V approximate distance oracle or sourcewise approximate distance oracle of size Õ(|S|n + n^{3/2}) with multiplicative stretch at most 5. We construct another fault-tolerant sourcewise approximate distance oracle of size Õ(|S|n + n^{4/3}) with multiplicative stretch at most 13. Both the oracles have O(1) query answering time.

Cite as

Dipan Dey and Telikepalli Kavitha. Fault-Tolerant Approximate Distance Oracles with a Source Set. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dey_et_al:LIPIcs.FSTTCS.2025.27,
  author =	{Dey, Dipan and Kavitha, Telikepalli},
  title =	{{Fault-Tolerant Approximate Distance Oracles with a Source Set}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{27:1--27:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.27},
  URN =		{urn:nbn:de:0030-drops-251081},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.27},
  annote =	{Keywords: Weighted graphs, approximate distances, fault-tolerant data structures}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.43

New Approximate Distance Oracles and Their Applications

Authors: Avi Kadria and Liam Roditty

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

Let G = (V, E) be an undirected graph with n vertices and m edges, and let μ = m/n. A distance oracle is a data structure designed to answer approximate distance queries, with the goal of achieving low stretch, efficient space usage, and fast query time. While much of the prior work focused on distance oracles with constant query time, this paper presents a comprehensive study of distance oracles with non-constant query time. We explore the tradeoffs between space, stretch, and query time of distance oracles in various regimes. Specifically, we consider both weighted and unweighted graphs in the regimes of stretch < 2 and stretch ≥ 2. In addition, we demonstrate several applications of our new distance oracles to the n-Pairs Shortest Paths (n-PSP) problem and the All Nodes Shortest Cycles (ANSC) problem. Our main contributions are: - Weighted graphs: We present a new three-way trade-off between stretch, space, and query time, offering a natural extension of the classical Thorup–Zwick distance oracle [STOC’01 and JACM’05] to regimes with larger query time. Specifically, for any 0 < r < 1/2 and integer k ≥ 1, we construct a (2k(1 - 2r) - 1)-stretch distance oracle with Õ(m + n^{1 + 1/k}) space and Õ(μ n^r) query time. This construction provides an asymptotic improvement over the classical (2k - 1)-stretch and O(n^{1 + 1/k})-space tradeoff of Thorup and Zwick in sparse graphs, at the cost of increased query time. We also improve upon a result of Dalirrooyfard et al. [FOCS’22], who presented a (2k - 2)-stretch distance oracle with O(m + n^{1 + 1/k}) space and O(μ n^{1/k}) query time. In our oracle we reduce the stretch from (2k - 2) to (2k - 5) while preserving the same space and query time. - Unweighted graphs: We present a (2k - 5, 4 + 2_{odd})-approximation distance oracle with O(n^{1 + 1/k}) space and O(n^{1/k}) query time. This improves upon a (2k - 2, 2_{odd})-approximation distance oracle of Dalirrooyfard et al. [FOCS’22] while maintaining the same space and query time. We also present a distance oracle that given u,v ∈ V returns an estimate d̂(u,v) ≤ d(u,v) + 2⌈ d(u,v) / 3 ⌉ + 2, using O(n^{4/3 + 2ε}) space and O(n^{1 - 3ε}) query time. To the best of our knowledge, this is the first distance oracle that simultaneously achieves a multiplicative stretch < 2, and a space complexity O(n^{1.5 - α}), for some α > 0. - Applications for n-PSP and ANSC: We present an Õ(m^{1 - 1/(k+1)} n)-time algorithm for the n-PSP problem, that for every input pair ⟨s_i,t_i⟩, where i ∈ [n], returns an estimate d̂(s_i, t_i) such that d̂(s_i,t_i) ≤ d(s_i,t_i) + 2⌈d(s_i,t_i)/2k⌉. By allowing a small additive error, this result circumvents the conditional running time lower bound of Ω(m^{2 - 2/(k+1)} ⋅ n^{1/(k+1) - o(1)}), established by Dalirrooyfard et al. [FOCS’22] for achieving (1 + 1/k)-stretch. Additionally, we present an Õ(mn^{1 - 1/k})-time algorithm for the ANSC problem that computes, for every u ∈ V, an estimate ĉ_u such that ĉ_u ≤ SC(u) + 2⌈SC(u)/2(k - 1)⌉, where SC(u) denotes the length of the shortest cycle containing u. This improves upon the Õ(m^{2 - 2/k}n^{1/k})-time algorithm of Dalirrooyfard et al. [FOCS'22], while achieving the same approximation guarantee. We obtain our results by developing several new techniques, among them are the borderline vertices technique and the middle vertex technique, which may be of independent interest.

Cite as

Avi Kadria and Liam Roditty. New Approximate Distance Oracles and Their Applications. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 43:1-43:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kadria_et_al:LIPIcs.ISAAC.2025.43,
  author =	{Kadria, Avi and Roditty, Liam},
  title =	{{New Approximate Distance Oracles and Their Applications}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{43:1--43:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.43},
  URN =		{urn:nbn:de:0030-drops-249514},
  doi =		{10.4230/LIPIcs.ISAAC.2025.43},
  annote =	{Keywords: Distance oracles, Fine-grained algorithms, Graph algorithms, Data structures}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.46

Realizing Metric Spaces with Convex Obstacles

Authors: Sándor Kisfaludi-Bak and Leonidas Theocharous

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

The presence of obstacles has a significant impact on distance computation, motion-planning, and visibility. These problems have been studied extensively in the planar setting, while our understanding of these problems in 3- and higher-dimensional spaces is still rudimentary. In this paper, we study the impact of different types of obstacles on the induced geodesic metric in 3-dimensional Euclidean space. We say that a finite metric space (X, dist_X) is approximately realizable by a collection 𝒯 of obstacles in ℝ³ if for any ε > 0 it can be embedded into (ℝ³⧵⋃_{T∈𝒯} T, dist_𝒯) with worst-case multiplicative distortion 1+ε, where dist_𝒯 denotes the geodesic distance in the free space induced by 𝒯. We focus on three key geometric properties of obstacles -convexity, disjointness, and fatness- and examine how dropping each one of them affects the existence of such embeddings. Our main result concerns dropping the fatness property: we demonstrate that any finite metric space is realizable with 1+ε worst-case multiplicative distortion using a collection of convex and pairwise disjoint obstacles in ℝ³, even if the obstacles are congruent and equilateral triangles. Based on the same construction, we can also show that if we require fatness but drop any of the other two properties instead, then we can still approximately realize any finite metric space. Our results have important implications on the approximability of tsp with obstacles, a natural variant of tsp introduced recently by Alkema et al. (ESA 2022). Specifically, we use the recent results of Banerjee et al. on tsp in doubling spaces (FOCS 2024) and of Chew et al. on distances among obstacles (Inf. Process. Lett. 2002) to show that tsp with obstacles admits a PTAS if the obstacles are convex, fat, and pairwise disjoint. If any of these three properties is dropped, then our results, combined with the APX-hardness of Metric tsp, demonstrate that tsp with obstacles is APX-hard.

Cite as

Sándor Kisfaludi-Bak and Leonidas Theocharous. Realizing Metric Spaces with Convex Obstacles. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 46:1-46:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kisfaludibak_et_al:LIPIcs.ISAAC.2025.46,
  author =	{Kisfaludi-Bak, S\'{a}ndor and Theocharous, Leonidas},
  title =	{{Realizing Metric Spaces with Convex Obstacles}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{46:1--46:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.46},
  URN =		{urn:nbn:de:0030-drops-249545},
  doi =		{10.4230/LIPIcs.ISAAC.2025.46},
  annote =	{Keywords: traveling salesman, geodesic distance}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.15

Parameterized Complexity of Directed Traveling Salesman Problem

Authors: Václav Blažej, Andreas Emil Feldmann, Foivos Fioravantes, Paweł Rzążewski, and Ondřej Suchý

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

The Directed Traveling Salesman Problem (DTSP) is a variant of the classical Traveling Salesman Problem in which the edges in the graph are directed and a vertex and edge can be visited multiple times. The goal is to find a directed closed walk of minimum length (or total weight) that visits every vertex of the given graph at least once. In a yet more general version, Directed Waypoint Routing Problem (DWRP), some vertices are marked as terminals and we are only required to visit all terminals. Furthermore, each edge has its capacity bounding the number of times this edge can be used by a solution. While both problems (and many other variants of TSP) were extensively investigated, mostly from the approximation point of view, there are surprisingly few results concerning the parameterized complexity. Our starting point is the result of Marx et al. [APPROX/RANDOM 2016] who proved that DTSP is W[1]-hard parameterized by distance to pathwidth 3. In this paper we aim to initiate the systematic complexity study of variants of Directed Traveling Salesman Problem with respect to various, mostly structural, parameters. We show that DWRP is FPT parameterized by the solution size, the feedback edge number and the vertex integrity of the underlying undirected graph. Furthermore, the problem is XP parameterized by treewidth. On the complexity side, we show that the problem is W[1]-hard parameterized by the distance to constant treedepth.

Cite as

Václav Blažej, Andreas Emil Feldmann, Foivos Fioravantes, Paweł Rzążewski, and Ondřej Suchý. Parameterized Complexity of Directed Traveling Salesman Problem. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blazej_et_al:LIPIcs.ISAAC.2025.15,
  author =	{Bla\v{z}ej, V\'{a}clav and Feldmann, Andreas Emil and Fioravantes, Foivos and Rz\k{a}\.{z}ewski, Pawe{\l} and Such\'{y}, Ond\v{r}ej},
  title =	{{Parameterized Complexity of Directed Traveling Salesman Problem}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{15:1--15:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.15},
  URN =		{urn:nbn:de:0030-drops-249231},
  doi =		{10.4230/LIPIcs.ISAAC.2025.15},
  annote =	{Keywords: Directed TSP, parameterized complexity, vertex integrity, treedepth}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.23

Precoloring Extension with Demands on Paths

Authors: Arun Kumar Das, Michal Opler, and Tomáš Valla

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

Let G be a graph with a set of precolored vertices, and let us be given an integer distance parameter d and a set of integer demands d₁,… ,d_c. The Distance Precoloring Extension with Demands (DPED) problem is to compute a vertex c-coloring of G such that the following three conditions hold: (i) the resulting coloring respects the colors of the precolored vertices, (ii) the distance of two vertices of the same color is at least d, and (iii) the number of vertices colored by color i is exactly d_i. This problem is motivated by a program scheduling in commercial broadcast channels with constraints on content repetition and placement, which leads precisely to the DPED problem for paths. In this paper, we study DPED on paths and present a polynomial time exact algorithm when precolored vertices are restricted to the two ends of the path and devise an approximation algorithm for DPED with an additive approximation factor polynomially bounded by d and the number of precolored vertices. Then, we prove that the Distance Precoloring Extension problem on paths, a less restrictive version of DPED without the demand constraints, and then DPED itself, is NP-complete. Motivated by this result, we further study the parameterized complexity of DPED on paths. We establish that the DPED problem on paths is W[1]-hard when parameterized by the number of colors and the distance. On the positive side, we devise a fixed parameter tractable (FPT) algorithm for DPED on paths when the number of colors, the distance, and the number of precolored vertices are considered as the parameters. Moreover, we prove that Distance Precoloring Extension is FPT parameterized by the distance. As a byproduct, we also obtain several results for the Distance List Coloring problem on paths.

Cite as

Arun Kumar Das, Michal Opler, and Tomáš Valla. Precoloring Extension with Demands on Paths. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{das_et_al:LIPIcs.ISAAC.2025.23,
  author =	{Das, Arun Kumar and Opler, Michal and Valla, Tom\'{a}\v{s}},
  title =	{{Precoloring Extension with Demands on Paths}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.23},
  URN =		{urn:nbn:de:0030-drops-249319},
  doi =		{10.4230/LIPIcs.ISAAC.2025.23},
  annote =	{Keywords: precoloring extension, distance coloring, FPT, approximation algorithms}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.13

Optimal Online Bipartite Matching in Degree-2 Graphs

Authors: Amey Bhangale, Arghya Chakraborty, and Prahladh Harsha

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

Online bipartite matching is a classical problem in online algorithms and we know that both the deterministic fractional and randomized integral online matchings achieve the same competitive ratio of 1-1/e. In this work, we study classes of graphs where the online degree is restricted to 2. As expected, one can achieve a competitive ratio of better than 1-1/e in both the deterministic fractional and randomized integral cases, but surprisingly, these ratios are not the same. It was already known that for fractional matching, a 0.75 competitive ratio algorithm is optimal. We show that the folklore Half-Half algorithm achieves a competitive ratio of η ≈ 0.717772… and more surprisingly, show that this is optimal by giving a matching lower-bound. This yields a separation between the two problems: deterministic fractional and randomized integral, showing that it is impossible to obtain a perfect rounding scheme.

Cite as

Amey Bhangale, Arghya Chakraborty, and Prahladh Harsha. Optimal Online Bipartite Matching in Degree-2 Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhangale_et_al:LIPIcs.ISAAC.2025.13,
  author =	{Bhangale, Amey and Chakraborty, Arghya and Harsha, Prahladh},
  title =	{{Optimal Online Bipartite Matching in Degree-2 Graphs}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{13:1--13:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.13},
  URN =		{urn:nbn:de:0030-drops-249216},
  doi =		{10.4230/LIPIcs.ISAAC.2025.13},
  annote =	{Keywords: Online Algorithm, Bipartite matching}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.53

A Parameterized Study of Secluded Structures in Directed Graphs

Authors: Jonas Schmidt, Shaily Verma, and Nadym Mallek

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

Given an undirected graph G and an integer k, the Secluded Π-Subgraph problem asks you to find a maximum size induced subgraph that satisfies a property Π and has at most k neighbors in the rest of the graph. This problem has been extensively studied; however, there is no prior study of the problem in directed graphs. This question has been mentioned by Jansen et al. [ISAAC'23]. In this paper, we initiate the study of Secluded Subgraph problems in directed graphs by incorporating different notions of neighborhoods: in-neighborhood, out-neighborhood, and their union. Formally, we call these problems {In, Out, Total}-Secluded Π-Subgraph, where given a directed graph G and an integer k, we want to find an induced subgraph satisfying Π of maximum size that has at most k in/out/total-neighbors in the rest of the graph, respectively. We investigate the parameterized complexity of these problems for different properties Π. In particular, we prove the following parameterized results: - We design an FPT algorithm for the Total-Secluded Strongly Connected Subgraph problem when parameterized by k. - We show that the Out-Secluded ℱ-Free Subgraph problem with parameter k is W[1]-hard, where ℱ is a family of directed graphs except any subgraph of a star graph whose edges are directed towards the center. This result also implies that In/Out-Secluded DAG is W[1]-hard, unlike the undirected variants of the two problems, which are FPT. - We design an FPT-algorithm for In/Out/Total-Secluded α-Bounded Subgraph when parameterized by k, where α-bounded graphs are a superclass of tournaments. - For undirected graphs, we improve the best-known FPT algorithm for Secluded Clique by providing a faster FPT algorithm that runs in time 1.6181^k n^𝒪(1).

Cite as

Jonas Schmidt, Shaily Verma, and Nadym Mallek. A Parameterized Study of Secluded Structures in Directed Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{schmidt_et_al:LIPIcs.ISAAC.2025.53,
  author =	{Schmidt, Jonas and Verma, Shaily and Mallek, Nadym},
  title =	{{A Parameterized Study of Secluded Structures in Directed Graphs}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{53:1--53:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.53},
  URN =		{urn:nbn:de:0030-drops-249616},
  doi =		{10.4230/LIPIcs.ISAAC.2025.53},
  annote =	{Keywords: Secluded Subgraph, Parametrized Complexity, Directed Graphs, Strong Connectivity}
}

Document

DOI: 10.4230/LIPIcs.DISC.2025.40

On the Randomized Locality of Matching Problems in Regular Graphs

Authors: Seri Khoury, Manish Purohit, Aaron Schild, and Joshua R. Wang

Published in: LIPIcs, Volume 356, 39th International Symposium on Distributed Computing (DISC 2025)

Abstract

The main goal in distributed symmetry-breaking is to understand the locality of problems: the radius of the neighborhood that a node must explore to determine its part of a global solution. In this work, we study the locality of matching problems in the family of regular graphs, which is one of the main benchmarks for establishing lower bounds on the locality of symmetry-breaking problems, as well as for obtaining classification results. Our main results are summarized as follows: 1) Approximate matching: We develop randomized algorithms to show that (1 + ε)-approximate matching in regular graphs is truly local, i.e., the locality depends only on ε and is independent of all other graph parameters. Furthermore, as long as the degree Δ is not very small (namely, as long as Δ ≥ poly(1/ε)), this dependence is only logarithmic in 1/ε. This stands in sharp contrast to maximal matching in regular graphs which requires some dependence on the number of nodes n or the degree Δ. 2) Maximal matching: Our techniques further allow us to establish a strong separation between the node-averaged complexity and worst-case complexity of maximal matching in regular graphs, by showing that the former is only O(1). Central to our main technical contribution is a novel martingale-based analysis for the ≈ 40-year-old algorithm by Luby. In particular, our analysis shows that applying one round of Luby’s algorithm on the line graph of a Δ-regular graph results in an almost Δ/2-regular graph.

Cite as

Seri Khoury, Manish Purohit, Aaron Schild, and Joshua R. Wang. On the Randomized Locality of Matching Problems in Regular Graphs. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{khoury_et_al:LIPIcs.DISC.2025.40,
  author =	{Khoury, Seri and Purohit, Manish and Schild, Aaron and Wang, Joshua R.},
  title =	{{On the Randomized Locality of Matching Problems in Regular Graphs}},
  booktitle =	{39th International Symposium on Distributed Computing (DISC 2025)},
  pages =	{40:1--40:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-402-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{356},
  editor =	{Kowalski, Dariusz R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2025.40},
  URN =		{urn:nbn:de:0030-drops-248570},
  doi =		{10.4230/LIPIcs.DISC.2025.40},
  annote =	{Keywords: regular graphs, maximum matching, augmenting paths, distributed algorithms, Luby’s algorithm, martingales}
}

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