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DOI: 10.4230/LIPIcs.SEA.2026.21

An Empirical Analysis of Approximation Algorithms for the Unweighted Tree Augmentation Problem

Authors: Luke Hawranick, Matthew Williamson, Jacob Restanio, K. Subramani, and Cody Klingler

Published in: LIPIcs, Volume 371, 24th International Symposium on Experimental Algorithms (SEA 2026)

Abstract

In this paper, we perform an experimental study of approximation algorithms for the unweighted tree augmentation problem (UTAP). Our goal is to establish a baseline performance for several existing approximation algorithms on actual instances rather than worst-case instances. In particular, we are interested in whether the algorithms' performance in practical instances is consistent with their worst-case guarantee rankings. We are also interested in whether preprocessing times, implementation difficulties, and running times justify the use of an algorithm in practice. We profile and analyze three approximation algorithms from the literature against a simple randomized algorithm. The performance of each algorithm was evaluated using metrics for space usage, running time, and solution quality. We found that the simple randomized algorithm is very competitive with the approximation algorithms and that the algorithms do not necessarily rank according to their theoretical guarantees. The randomized algorithm is easier to implement and understand, using less space than any of the more sophisticated approximation algorithms.

Cite as

Luke Hawranick, Matthew Williamson, Jacob Restanio, K. Subramani, and Cody Klingler. An Empirical Analysis of Approximation Algorithms for the Unweighted Tree Augmentation Problem. In 24th International Symposium on Experimental Algorithms (SEA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 371, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{hawranick_et_al:LIPIcs.SEA.2026.21,
  author =	{Hawranick, Luke and Williamson, Matthew and Restanio, Jacob and Subramani, K. and Klingler, Cody},
  title =	{{An Empirical Analysis of Approximation Algorithms for the Unweighted Tree Augmentation Problem}},
  booktitle =	{24th International Symposium on Experimental Algorithms (SEA 2026)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-422-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{371},
  editor =	{Aum\"{u}ller, Martin and Finocchi, Irene},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2026.21},
  URN =		{urn:nbn:de:0030-drops-260259},
  doi =		{10.4230/LIPIcs.SEA.2026.21},
  annote =	{Keywords: Graphs, Networks, Tree Augmentation, Approximation Algorithms, Empirical}
}

Document

DOI: 10.4230/LIPIcs.SWAT.2026.9

QPTAS for MWIS and Finding Large Sparse Induced Subgraphs in Graphs with Few Independent Long Holes

Authors: Édouard Bonnet, Jadwiga Czyżewska, Tomáš Masařík, Marcin Pilipczuk, and Paweł Rzążewski

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract

We present a quasipolynomial-time approximation scheme (QPTAS) for the Maximum Independent Set (MWIS) in graphs with a bounded number of pairwise vertex-disjoint and non-adjacent long induced cycles. More formally, for every fixed s and t, we show a QPTAS for MWIS in graphs that exclude sC_t as an induced minor. Combining this with known results, we obtain a QPTAS for the problem of finding a largest induced subgraph of bounded treewidth with given hereditary property definable in Counting Monadic Second Order Logic, in the same classes of graphs. This is a step towards a conjecture of Gartland and Lokshtanov which asserts that for any planar graph H, graphs that exclude H as an induced minor admit a polynomial-time algorithm for the latter problem. This conjecture is notoriously open and even its weaker variants are confirmed only for very restricted graphs H.

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Édouard Bonnet, Jadwiga Czyżewska, Tomáš Masařík, Marcin Pilipczuk, and Paweł Rzążewski. QPTAS for MWIS and Finding Large Sparse Induced Subgraphs in Graphs with Few Independent Long Holes. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bonnet_et_al:LIPIcs.SWAT.2026.9,
  author =	{Bonnet, \'{E}douard and Czy\.{z}ewska, Jadwiga and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Pilipczuk, Marcin and Rz\k{a}\.{z}ewski, Pawe{\l}},
  title =	{{QPTAS for MWIS and Finding Large Sparse Induced Subgraphs in Graphs with Few Independent Long Holes}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{9:1--9:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.9},
  URN =		{urn:nbn:de:0030-drops-260454},
  doi =		{10.4230/LIPIcs.SWAT.2026.9},
  annote =	{Keywords: independent set, long holes, QPTAS, induced subgraphs}
}

Document

DOI: 10.4230/LIPIcs.SWAT.2026.19

Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means

Authors: Nicole Funk, Annika Hennes, Johanna Hillebrand, and Sarah Sturm

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract

We study discrete k-clustering problems in general metric spaces that are constrained by a combination of two different fairness conditions within the demographic fairness model. Given a metric space (P,d), where every point in P is equipped with a protected attribute, and a number k, the goal is to partition P into k clusters with a designated center each, such that a center-based objective function is minimized and the attributes are fairly distributed with respect to the following two fairness concepts: 1) group fairness: We aim for clusters with balanced numbers of attributes by specifying lower and upper bounds for the desired attribute proportions. 2) diverse center selection: Clusters have natural representatives, i.e., their centers. We ask for a balanced set of representatives by specifying the desired number of centers to choose from each attribute. Dickerson, Esmaeili, Morgenstern, and Pena [John P. Dickerson et al., 2023] denote the combination of these two constraints as doubly constrained fair clustering. They present algorithms whose guarantees depend on the best known approximation factors for either of these problems. Currently, this implies an 8-approximation with a small additive violation on the group fairness constraint. For k-center, we improve this approximation factor to 4 with a small additive violation. This guarantee also depends on the currently best algorithm for DS-fair k-center given by Jones, Nguyen and Nguyen [Matthew Jones et al., 2020]. For k-median and k-means, we propose the first constant-factor approximation algorithms. Our algorithms transform a solution that satisfies diverse center selection into a doubly constrained fair clustering using an LP-based approach. Furthermore, our results are generalizable to other center-selection constraints, such as matroid k-clustering and knapsack constraints.

Cite as

Nicole Funk, Annika Hennes, Johanna Hillebrand, and Sarah Sturm. Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{funk_et_al:LIPIcs.SWAT.2026.19,
  author =	{Funk, Nicole and Hennes, Annika and Hillebrand, Johanna and Sturm, Sarah},
  title =	{{Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.19},
  URN =		{urn:nbn:de:0030-drops-260551},
  doi =		{10.4230/LIPIcs.SWAT.2026.19},
  annote =	{Keywords: Clustering, Fairness, Approximation Algorithms, k-center, k-median, k-means}
}

Document

DOI: 10.4230/LIPIcs.FORC.2026.9

Serving Clients Fairly: On Facility Location and k-Median with Fair Outliers

Authors: Rajni Dabas, Samir Khuller, and Emilie Rivkin

Published in: LIPIcs, Volume 368, 7th Symposium on Foundations of Responsible Computing (FORC 2026)

Abstract

Classical clustering problems such as Facility Location and k-Median aim to efficiently serve a set of clients from a subset of facilities - minimizing the total cost of facility openings and client assignments in Facility Location, and minimizing assignment (service) cost under a facility count constraint in k-Median. These problems are highly sensitive to outliers, and therefore researchers have studied variants that allow excluding a small number of clients as outliers to reduce cost. However, in many real-world settings, clients belong to different demographic or functional groups, and unconstrained outlier removal can disproportionately exclude certain groups, raising fairness concerns, especially when the facilities correspond to critically needed facilities for emergencies such as fire stations, hospitals and other emergency services. We study Facility Location with Fair Outliers, where each group is allowed a specified number of outliers, and the objective is to minimize total cost while respecting group-wise fairness constraints. We present a bicriteria approximation with a O(1/ε) approximation factor and (1+ 2ε) factor violation in outliers per group. For k-Median with Fair Outliers, we design a bicriteria approximation with a 4(1+ω/ε) approximation factor and (ω + ε) violation in outliers per group improving on prior work by avoiding dependence on k in outlier violations. We also prove that the problems are W[1]-hard parameterized by ω. We complement our algorithmic contributions with a detailed empirical analysis, demonstrating that fairness can be achieved with negligible increase in cost and that the integrality gap of the standard LP is small in practice.

Cite as

Rajni Dabas, Samir Khuller, and Emilie Rivkin. Serving Clients Fairly: On Facility Location and k-Median with Fair Outliers. In 7th Symposium on Foundations of Responsible Computing (FORC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 368, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dabas_et_al:LIPIcs.FORC.2026.9,
  author =	{Dabas, Rajni and Khuller, Samir and Rivkin, Emilie},
  title =	{{Serving Clients Fairly: On Facility Location and k-Median with Fair Outliers}},
  booktitle =	{7th Symposium on Foundations of Responsible Computing (FORC 2026)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-419-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{368},
  editor =	{Lin, Huijia (Rachel)},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2026.9},
  URN =		{urn:nbn:de:0030-drops-259812},
  doi =		{10.4230/LIPIcs.FORC.2026.9},
  annote =	{Keywords: Approximation algorithms, fairness}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.33

Near-Optimal Bounds for Parameterized Euclidean k-Means

Authors: Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

The k-means problem is a classic objective for modeling clustering in a metric space. Given a set of points in a metric space, the goal is to find k representative points so as to minimize the sum of the squared distances from each point to its closest representative. In this work, we study the approximability of k-means in Euclidean spaces parameterized by the number of clusters, k. In seminal works, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] showed how to obtain a (1+ε)-approximation for high-dimensional Euclidean k-means in time 2^{(k/ε)^O(1)} ⋅ dn^O(1). In this work, we introduce a new fine-grained hypothesis called Exponential Time for Expanders Hypothesis (XXH) which roughly asserts that there are no non-trivial exponential time approximation algorithms for the vertex cover problem on near perfect vertex expanders. Assuming XXH, we close the above long line of work on approximating Euclidean k-means by showing that there is no 2^{(k/ε)^{1-o(1)}} ⋅ n^O(1) time algorithm achieving a (1+ε)-approximation for k-means in Euclidean space. This lower bound is tight as it matches the algorithm given by Feldman, Monemizadeh, and Sohler [SoCG'07] whose runtime is 2^O(k/ε) + O(ndk). Furthermore, assuming XXH, we show that the seminal O(n^{kd+1}) runtime exact algorithm of Inaba, Katoh, and Imai [SoCG'94] for k-means is optimal for small values of k.

Cite as

Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Near-Optimal Bounds for Parameterized Euclidean k-Means. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2026.33,
  author =	{Cohen-Addad, Vincent and C. S., Karthik and Saulpic, David and Schwiegelshohn, Chris},
  title =	{{Near-Optimal Bounds for Parameterized Euclidean k-Means}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{33:1--33:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.33},
  URN =		{urn:nbn:de:0030-drops-258391},
  doi =		{10.4230/LIPIcs.SoCG.2026.33},
  annote =	{Keywords: k-means clustering, Euclidean space, Fine-Grained Complexity}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.34

Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces

Authors: Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

The k-median and k-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the k-median (resp. k-means) problem is to find k representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a (1+ε)-factor approximation in low-dimensional Euclidean metric for both the k-median and k-means problems in near-linear time 2^{(1/ε)^O(d²)} n ⋅ polylog(n) (where d is the dimension and n is the number of input points). We improve this running time to 2^{O(1/ε)^{d-1}} ⋅ n ⋅ polylog(n), and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no 2^o(1/ε^{d-1}) n^O(1) algorithm achieving a (1+ε)-approximation for k-means.

Cite as

Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2026.34,
  author =	{Cohen-Addad, Vincent and Karthik C. S. and Saulpic, David and Schwiegelshohn, Chris},
  title =	{{Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{34:1--34:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.34},
  URN =		{urn:nbn:de:0030-drops-258404},
  doi =		{10.4230/LIPIcs.SoCG.2026.34},
  annote =	{Keywords: k-means clustering, k-median clustering, Euclidean space, Fine-Grained Complexity}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.43

Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-Offs

Authors: Tsuri Farhana, Omrit Filtser, and Shalev Goldshtein

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

We study unlabeled multi-robot motion planning for unit-disk robots in a polygonal environment. Although the problem is hard in general, polynomial-time solutions exist under appropriate separation assumptions on start and target positions. Solovey et al. (RSS'15) provide a near-optimal solution assuming that start/target positions must have pairwise distance at least 4, and at least √5≈2.236 from obstacles. This raises the question of whether polynomial-time algorithms can be obtained in even more densely packed environments. In this paper we present a generalized algorithm that achieve different trade-offs on the robots-separation and obstacles-separation bounds, all significantly improving upon the state of the art. Specifically, we obtain polynomial-time constant-approximation algorithms to minimize the total path length when (i) the robots-separation is 2 2/3 and the obstacles-separation is 1 2/3, or (ii) the robots-separation is ≈3.291 and the obstacles-separation ≈1.354. Additionally, we introduce a different strategy yielding a polynomial-time solution when the robots-separation is only 2, and the obstacles-separation is 3. Finally, we show that without any robots-separation assumption, obstacles-separation of at least 1.5 may be necessary for a solution to exist.

Cite as

Tsuri Farhana, Omrit Filtser, and Shalev Goldshtein. Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-Offs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{farhana_et_al:LIPIcs.SoCG.2026.43,
  author =	{Farhana, Tsuri and Filtser, Omrit and Goldshtein, Shalev},
  title =	{{Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-Offs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{43:1--43:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.43},
  URN =		{urn:nbn:de:0030-drops-258495},
  doi =		{10.4230/LIPIcs.SoCG.2026.43},
  annote =	{Keywords: multi-robot motion planning}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.48

FPT Approximations for Capacitated Sum of Radii and Diameters

Authors: Arnold Filtser and Ameet Gadekar

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

The Capacitated Sum of Radii problem involves partitioning a set of points P, where each point p ∈ P has capacity U_p, into k clusters that minimize the sum of cluster radii, such that the number of points in the cluster centered at point p is at most U_p. We begin by showing that the problem is APX-hard, and that under gap-ETH there is no parameterized approximation scheme (FPT-AS). We then construct a ≈5.83-approximation algorithm in FPT time (improving a previous ≈7.61 approximation in FPT time). Our results also hold when the objective is a general monotone symmetric norm of radii. We also improve the approximation factors for the uniform capacity case, and for the closely related problem of Capacitated Sum of Diameters.

Cite as

Arnold Filtser and Ameet Gadekar. FPT Approximations for Capacitated Sum of Radii and Diameters. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{filtser_et_al:LIPIcs.SoCG.2026.48,
  author =	{Filtser, Arnold and Gadekar, Ameet},
  title =	{{FPT Approximations for Capacitated Sum of Radii and Diameters}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{48:1--48:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.48},
  URN =		{urn:nbn:de:0030-drops-258545},
  doi =		{10.4230/LIPIcs.SoCG.2026.48},
  annote =	{Keywords: clustering, sum of radii, sum of diameter, capacitated clustering, fpt}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.71

Approximating Euclidean Shallow-Light Trees

Authors: Hung Le, Shay Solomon, Cuong Than, Csaba D. Tóth, and Tianyi Zhang

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

For a weighted graph G = (V, E, w) and a designated source vertex s ∈ V, a spanning tree that simultaneously approximates a shortest-path tree w.r.t. source s and a minimum spanning tree is called a shallow-light tree (SLT). Specifically, an (α, β)-SLT of G w.r.t. s ∈ V is a spanning tree of G with root-stretch α (preserving all distances between s and all other vertices up to a factor of α) and lightness β (its weight is at most β times the weight of a minimum spanning tree of G). It was shown in the early 1990s that (1) for any graph, any source, and any ε > 0, there is a (1 + ε, O(1/ε))-SLT, and (2) there exist graphs for which β = Ω(1/ε) for any (1+ε,β)-SLT. The focus of this work is on SLTs in low-dimensional Euclidean spaces, which are of special interest for some applications of SLTs, in geometric network optimization problems. The aforementioned existential lower bound applies to Euclidean plane, as well. It was shown more than a decade ago that (1) by using Steiner points, one can reduce the lightness bound from O(1/ε) to O(√{1/ε}), and (2) there exist point sets in the plane for which β = Ω(√{1/ε}) for any Steiner (1+ε,β)-SLT. These tight existential bounds for the Euclidean case yield approximation factors of O(1/ε) and O(√{1/ε}) on the minimum weight of any non-Steiner and Steiner tree with root-stretch 1+ε, respectively. Despite the large body of work on SLTs, the basic question of whether a better approximation algorithm exists was left untouched to date, and this holds in any graph family. This paper makes a first nontrivial step towards resolving this question by presenting two bicriteria approximation algorithms. For any ε > 0, a set P of n points in constant-dimensional Euclidean space and a source s ∈ P, our first (respectively, second) algorithm returns, in O(n log n ⋅ polylog(ε^{-1})) time, a non-Steiner (resp., Steiner) tree with root-stretch 1+O(ε log ε^{-1}) and weight at most O(opt_ε ⋅ log² ε^{-1}) (resp., O(opt_ε ⋅ log ε^{-1})), where opt_ε denotes the minimum weight of a non-Steiner (resp., Steiner) tree with root-stretch 1+ε.

Cite as

Hung Le, Shay Solomon, Cuong Than, Csaba D. Tóth, and Tianyi Zhang. Approximating Euclidean Shallow-Light Trees. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 71:1-71:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{le_et_al:LIPIcs.SoCG.2026.71,
  author =	{Le, Hung and Solomon, Shay and Than, Cuong and T\'{o}th, Csaba D. and Zhang, Tianyi},
  title =	{{Approximating Euclidean Shallow-Light Trees}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{71:1--71:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.71},
  URN =		{urn:nbn:de:0030-drops-258789},
  doi =		{10.4230/LIPIcs.SoCG.2026.71},
  annote =	{Keywords: geometric network design, optimization, shallow-light tree, Steiner point}
}

Document

DOI: 10.4230/LIPIcs.ICDT.2026.11

Factorised Representations of Join Queries: Tight Bounds and a New Dichotomy

Authors: Christoph Berkholz and Harry Vinall-Smeeth

Published in: LIPIcs, Volume 365, 29th International Conference on Database Theory (ICDT 2026)

Abstract

A common theme in factorised databases and knowledge compilation is the representation of solution sets in a useful yet succinct data structure. In this paper, we study the representation of the result of join queries (or, equivalently, the set of homomorphisms between two relational structures). We focus on the very general format of {∪,×}-circuits - also known as d-representations or DNNF circuits - and aim to find the limits of this approach. In prior work, it has been shown that there always exists a {∪,×}-circuit of size N^O(subw) representing the query result, where N is the size of the database and subw the submodular width of the query. If the arity of all relations is bounded by a constant, then subw is linear in the treewidth tw of the query. In this setting, the authors of this paper proved a lower bound of N^Ω(tw^ε) on the circuit size (ICALP 2023), where ε > 0 depends on the excluded grid theorem. Our first main contribution is to improve this lower bound to N^Ω(tw), which is tight up to a constant factor in the exponent. Our second contribution is a N^Ω(subw^{1/4}) lower bound on the circuit size for join queries over relations of unbounded arity. Both lower bounds are unconditional lower bounds on the circuit size for well-chosen database instances. Their proofs use a combination of structural (hyper)graph theory with communication complexity in a simple yet novel way. While the second lower bound is asymptotically equivalent to Marx’s conditional bound on the decision complexity (JACM 2013), our N^Θ(tw) bound in the bounded arity setting is tight, while the best conditional bound on the decision complexity is N^Ω(tw/log tw). Note that removing this logarithmic factor in the decision setting is a major open problem.

Cite as

Christoph Berkholz and Harry Vinall-Smeeth. Factorised Representations of Join Queries: Tight Bounds and a New Dichotomy. In 29th International Conference on Database Theory (ICDT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 365, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{berkholz_et_al:LIPIcs.ICDT.2026.11,
  author =	{Berkholz, Christoph and Vinall-Smeeth, Harry},
  title =	{{Factorised Representations of Join Queries: Tight Bounds and a New Dichotomy}},
  booktitle =	{29th International Conference on Database Theory (ICDT 2026)},
  pages =	{11:1--11:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-413-0},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{365},
  editor =	{ten Cate, Balder and Funk, Maurice},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2026.11},
  URN =		{urn:nbn:de:0030-drops-256255},
  doi =		{10.4230/LIPIcs.ICDT.2026.11},
  annote =	{Keywords: join queries, homomorphisms, factorised databases, succinct representation, knowledge compilation, lower bounds}
}

Document

DOI: 10.4230/LIPIcs.STACS.2026.49

2D Minimal Graph Rigidity is in NC for One-Crossing-Minor-Free Graphs

Authors: Rohit Gurjar, Kilian Rothmund, and Thomas Thierauf

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

Abstract

Minimally rigid graphs can be decided and embedded in the plane efficiently, i.e. in polynomial time. There is also an efficient randomized parallel algorithm, i.e. in RNC. We present an NC-algorithm to decide whether one-crossing-minor-free graphs are minimally rigid. In the special case of K_{3,3}-free graphs, we also compute an infinitesimally rigid embedding in NC.

Cite as

Rohit Gurjar, Kilian Rothmund, and Thomas Thierauf. 2D Minimal Graph Rigidity is in NC for One-Crossing-Minor-Free Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 49:1-49:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{gurjar_et_al:LIPIcs.STACS.2026.49,
  author =	{Gurjar, Rohit and Rothmund, Kilian and Thierauf, Thomas},
  title =	{{2D Minimal Graph Rigidity is in NC for One-Crossing-Minor-Free Graphs}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{49:1--49:22},
  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.2026.49},
  URN =		{urn:nbn:de:0030-drops-255385},
  doi =		{10.4230/LIPIcs.STACS.2026.49},
  annote =	{Keywords: Graph Rigidity, Parallel Algorithms, Polynomial Identity Testing, Derandomization}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.94

Smoothed Analysis of Online Metric Matching with a Single Sample: Beyond Metric Distortion

Authors: Yingxi Li, Ellen Vitercik, and Mingwei Yang

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

Abstract

In the online metric matching problem, n servers and n requests lie in a metric space. Servers are available upfront, and requests arrive sequentially. An arriving request must be matched immediately and irrevocably to an available server, incurring a cost equal to their distance. The goal is to minimize the total matching cost. We study this problem in [0, 1]^d with the Euclidean metric, when servers are adversarial and requests are independently drawn from distinct distributions that satisfy a mild smoothness condition. Our main result is an O(1)-competitive algorithm for d ≠ 2 that requires no distributional knowledge, relying only on a single sample from each request distribution. To our knowledge, this is the first algorithm to achieve an o(log n) competitive ratio for non-trivial metrics beyond the i.i.d. setting. Our approach bypasses the Ω(log n) barrier introduced by probabilistic metric embeddings: instead of analyzing the embedding distortion and the algorithm separately, we directly bound the cost of the algorithm on the target metric space of a simple deterministic embedding. We then combine this analysis with lower bounds on the offline optimum for Euclidean metrics, derived via majorization arguments, to obtain our guarantees.

Cite as

Yingxi Li, Ellen Vitercik, and Mingwei Yang. Smoothed Analysis of Online Metric Matching with a Single Sample: Beyond Metric Distortion. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 94:1-94:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{li_et_al:LIPIcs.ITCS.2026.94,
  author =	{Li, Yingxi and Vitercik, Ellen and Yang, Mingwei},
  title =	{{Smoothed Analysis of Online Metric Matching with a Single Sample: Beyond Metric Distortion}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{94:1--94:23},
  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.2026.94},
  URN =		{urn:nbn:de:0030-drops-253815},
  doi =		{10.4230/LIPIcs.ITCS.2026.94},
  annote =	{Keywords: Online algorithm, Metric matching, Competitive analysis, Smoothed analysis}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.4

Pseudodeterministic Algorithms for Minimum Cut Problems

Authors: Aryan Agarwala and Nithin Varma

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

Abstract

In this paper we present efficient pseudodeterministic algorithms for both the global minimum cut and minimum s-t cut problems. The running time of our algorithm for the global minimum cut problem is asymptotically better than the fastest sequential deterministic global minimum cut algorithm (Henzinger, Li, Rao, Wang; SODA 2024). Furthermore, we implement our algorithm in streaming, PRAM, and cut-query models, where no efficient deterministic global minimum cut algorithms are known.

Cite as

Aryan Agarwala and Nithin Varma. Pseudodeterministic Algorithms for Minimum Cut Problems. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{agarwala_et_al:LIPIcs.ITCS.2026.4,
  author =	{Agarwala, Aryan and Varma, Nithin},
  title =	{{Pseudodeterministic Algorithms for Minimum Cut Problems}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{4:1--4:15},
  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.2026.4},
  URN =		{urn:nbn:de:0030-drops-252917},
  doi =		{10.4230/LIPIcs.ITCS.2026.4},
  annote =	{Keywords: Minimum Cut, Pseudodeterministic Algorithms}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.72

Forrelation Is Extremally Hard

Authors: Uma Girish and Rocco Servedio

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

Abstract

The Forrelation problem is a central problem that demonstrates an exponential separation between quantum and classical capabilities. In this problem, given query access to n-bit Boolean functions f and g, the goal is to estimate the Forrelation function forr(f,g), which measures the correlation between g and the Fourier transform of f. In this work we provide a new linear algebraic perspective on the Forrelation problem, as opposed to prior analytic approaches. We establish a connection between the Forrelation problem and bent Boolean functions and through this connection, analyze an extremal version of the Forrelation problem where the goal is to distinguish between extremal instances of Forrelation, namely (f,g) with forr(f,g) = 1 and forr(f,g) = -1. We show that this problem can be solved with one quantum query and success probability one, yet requires Ω̃(2^{n/4}) classical randomized queries, even for algorithms with a one-third failure probability, highlighting the remarkable power of one exact quantum query. We also study a restricted variant of this problem where the inputs f,g are computable by small classical circuits and show classical hardness under cryptographic assumptions.

Cite as

Uma Girish and Rocco Servedio. Forrelation Is Extremally Hard. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 72:1-72:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{girish_et_al:LIPIcs.ITCS.2026.72,
  author =	{Girish, Uma and Servedio, Rocco},
  title =	{{Forrelation Is Extremally Hard}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{72:1--72:22},
  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.2026.72},
  URN =		{urn:nbn:de:0030-drops-253594},
  doi =		{10.4230/LIPIcs.ITCS.2026.72},
  annote =	{Keywords: Forrelation, exact quantum, query complexity}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.80

FPT Approximations for Connected Maximum Coverage

Authors: Tanmay Inamdar, Satyabrata Jana, Madhumita Kundu, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi

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

Abstract

We revisit connectivity-constrained coverage through a unifying model, Partial Connected Red-Blue Dominating Set (PartialConRBDS). Given a bipartite graph G = (R∪ B,E) with red vertices R and blue vertices B, an auxiliary connectivity graph G_{conn} on R, and integers k,t, the task is to find a set S ⊆ R with |S| ≤ k such that G_{conn}[S] is connected and S dominates at least t blue vertices. This formulation captures connected variants of Maximum Coverage [Hochbaum-Rao, Inf. Proc. Lett., 2020; D'Angelo-Delfaraz, AAMAS 2025], Partial Vertex Cover, and Partial Dominating Set [Khuller et al., SODA 2014; Lamprou et al., TCS 2021] via standard encodings. Limits to parameterized tractability. PartialConRBDS is W[1]-hard parameterized by k even under strong restrictions: it remains hard when G_{conn} is a clique or a star and the incidence graph G is 3-degenerate, or when G is K_{2,2}-free. Inapproximability. For every ε > 0, there is no polynomial-time (1, 1-1/e+ε)-approximation unless 𝖯 = NP. Moreover, under ETH, no algorithm running in f(k)⋅ n^{o(k)} time achieves an g(k)-approximation for k for any computable function g(⋅), or for any ε > 0, a (1-1/e+ε)-approximation for t. Graphical special cases. Partial Connected Dominating Set is W[2]-hard parameterized by k and inherits the same ETH-based f(k)⋅ n^{o(k)} inapproximability bound as above; Partial Connected Vertex Cover is W[1]-hard parameterized by k. These hardness boundaries delineate a natural "sweet spot" for study: within appropriate structural restrictions on the incidence graph, one can still aim for fine-grained (FPT) approximations. Our algorithms. We solve PartialConRBDS exactly by reducing it to Relaxed Directed Steiner Out-Tree in time (2e)^t ⋅ n^{𝒪(1)}. For biclique-free incidences (i.e., when G excludes K_{d,d} as an induced subgraph), we obtain two complementary parameterized schemes: - An Efficient Parameterized Approximation Scheme (EPAS) running in time 2^{𝒪(k² d/ε)}⋅ n^{𝒪(1)} that either returns a connected solution of size at most k covering at least (1-ε)t blue vertices, or correctly reports that no connected size-k solution covers t; and - A Parameterized Approximation Scheme (PAS) running in time 2^{𝒪(kd(k²+log d))}⋅ n^{𝒪(1/ε)} that either returns a connected solution of size at most (1+ε)k covering at least t blue vertices, or correctly reports that no connected size-k solution covers t. Together, these results chart the boundary between hardness and FPT-approximability for connectivity-constrained coverage.

Cite as

Tanmay Inamdar, Satyabrata Jana, Madhumita Kundu, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. FPT Approximations for Connected Maximum Coverage. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 80:1-80:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{inamdar_et_al:LIPIcs.ITCS.2026.80,
  author =	{Inamdar, Tanmay and Jana, Satyabrata and Kundu, Madhumita and Lokshtanov, Daniel and Saurabh, Saket and Zehavi, Meirav},
  title =	{{FPT Approximations for Connected Maximum Coverage}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{80:1--80:24},
  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.2026.80},
  URN =		{urn:nbn:de:0030-drops-253674},
  doi =		{10.4230/LIPIcs.ITCS.2026.80},
  annote =	{Keywords: Partial Dominating Set, Connectivity, Maximum Coverage, FPT Approximation, Fixed-parameter Tractability}
}

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