15 Search Results for "Ahmed, Reyan"

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.ISAAC.2025.19
Traffic-Oblivious Multi-Commodity Flow Network Design

Authors: Markus Chimani and Max Ilsen

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

Abstract
We consider the Minimum Multi-Commodity Flow Subgraph (MMCFS) problem: given a directed graph G with edge capacities cap and a retention ratio α ∈ (0,1), find an edge-wise minimum subgraph G' ⊆ G such that for all traffic matrices T routable in G using a multi-commodity flow, α⋅ T is routable in G'. This natural yet novel problem is motivated by recent research that investigates how the power consumption in backbone computer networks can be reduced by turning off connections during times of low demand without compromising the quality of service. Since the actual traffic demands are generally not known beforehand, our approach must be traffic-oblivious, i.e., work for all possible sets of simultaneously routable traffic demands in the original network. In this paper we present the problem, relate it to other known problems in literature, and show several structural results, including a reformulation, maximum possible deviations from the optimum, and NP-hardness (as well as a certain inapproximability) already on very restricted instances. The most significant contribution is a max(1/α, 2)-approximation based on a surprisingly simple LP-rounding scheme. We also give instances where this worst-case approximation ratio is met and thus prove that our analysis is tight.
Cite as

Markus Chimani and Max Ilsen. Traffic-Oblivious Multi-Commodity Flow Network Design. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chimani_et_al:LIPIcs.ISAAC.2025.19,
  author =	{Chimani, Markus and Ilsen, Max},
  title =	{{Traffic-Oblivious Multi-Commodity Flow Network Design}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{19:1--19:17},
  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.2025.19},
  URN =		{urn:nbn:de:0030-drops-249273},
  doi =		{10.4230/LIPIcs.ISAAC.2025.19},
  annote =	{Keywords: Multi-commodity flow, Digraphs, LP-rounding, Approximation algorithm}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.22
NNP-NET: Accelerating t-SNE Graph Drawing for Very Large Graphs by Neural Networks

Authors: Ilan Hartskeerl, Tamara Mchedlidze, Simon van Wageningen, Peter Vangorp, and Alexandru Telea

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
tsNET is a recent graph drawing (GD) method that creates high quality layouts but suffers from a very high runtime. We present a new GD method, NNP-NET, which reduces tsNET’s time complexity to generate layouts for very large graphs in seconds. Additionally, we extend tsNET to support drawing graphs with edge weights. We accomplish this by replacing tsNET’s t-SNE projection with Neural Network Projection (NNP), a fast dimensionality reduction (DR) method that can imitate any given DR method. Our experiments show that NNP-NET gets good quality results when compared to other state-of-the art GD methods while yielding a better computational scalability.
Cite as

Ilan Hartskeerl, Tamara Mchedlidze, Simon van Wageningen, Peter Vangorp, and Alexandru Telea. NNP-NET: Accelerating t-SNE Graph Drawing for Very Large Graphs by Neural Networks. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 22:1-22:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hartskeerl_et_al:LIPIcs.GD.2025.22,
  author =	{Hartskeerl, Ilan and Mchedlidze, Tamara and van Wageningen, Simon and Vangorp, Peter and Telea, Alexandru},
  title =	{{NNP-NET: Accelerating t-SNE Graph Drawing for Very Large Graphs by Neural Networks}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{22:1--22:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.22},
  URN =		{urn:nbn:de:0030-drops-250087},
  doi =		{10.4230/LIPIcs.GD.2025.22},
  annote =	{Keywords: supervised graph drawing, dimensionality reduction, t-SNE}
}
Document
Poster Abstract
DOI: 10.4230/LIPIcs.GD.2025.56
Using Reinforcement Learning to Optimize the Global and Local Crossing Number (Poster Abstract)

Authors: Timo Brand, Henry Förster, Stephen Kobourov, Robin Schukrafft, Markus Wallinger, and Johannes Zink

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
We present a novel approach to graph drawing based on reinforcement learning for minimizing the global and the local crossing number, that is, the total number of edge crossings and the maximum number of crossings on any edge, respectively. An agent learns how to move a vertex based on a given observation vector. The agent receives feedback in the form of local reward signals tied to crossing reduction. To generate an initial layout, we use a stress-based graph-drawing algorithm. We compare our method against force- and stress-based baseline algorithms as well as three established algorithms for global crossing minimization on a suite of benchmark graphs. The experiments show mixed results: our current algorithm is mainly competitive for the local crossing number.
Cite as

Timo Brand, Henry Förster, Stephen Kobourov, Robin Schukrafft, Markus Wallinger, and Johannes Zink. Using Reinforcement Learning to Optimize the Global and Local Crossing Number (Poster Abstract). In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 56:1-56:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{brand_et_al:LIPIcs.GD.2025.56,
  author =	{Brand, Timo and F\"{o}rster, Henry and Kobourov, Stephen and Schukrafft, Robin and Wallinger, Markus and Zink, Johannes},
  title =	{{Using Reinforcement Learning to Optimize the Global and Local Crossing Number}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{56:1--56:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.56},
  URN =		{urn:nbn:de:0030-drops-250420},
  doi =		{10.4230/LIPIcs.GD.2025.56},
  annote =	{Keywords: Reinforcement Learning, Crossing Minimization, Local Crossing Number}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.7
Same Quality Metrics, Different Graph Drawings

Authors: Simon van Wageningen, Tamara Mchedlidze, and Alexandru C. Telea

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
Graph drawings are commonly used to visualize relational data. User understanding and performance are linked to the quality of such drawings, which is measured by quality metrics. The tacit knowledge in the graph drawing community about these quality metrics is that they are not always able to accurately capture the quality of graph drawings. In particular, such metrics may rate drawings with very poor quality as very good. In this work we make this tacit knowledge explicit by showing that we can modify existing graph drawings into arbitrary target shapes while keeping one or more quality metrics almost identical. This supports the claim that more advanced quality metrics are needed to capture the "goodness" of a graph drawing and that we cannot confidently rely on the value of a single (or several) certain quality metrics.
Cite as

Simon van Wageningen, Tamara Mchedlidze, and Alexandru C. Telea. Same Quality Metrics, Different Graph Drawings. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanwageningen_et_al:LIPIcs.GD.2025.7,
  author =	{van Wageningen, Simon and Mchedlidze, Tamara and Telea, Alexandru C.},
  title =	{{Same Quality Metrics, Different Graph Drawings}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{7:1--7:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.7},
  URN =		{urn:nbn:de:0030-drops-249935},
  doi =		{10.4230/LIPIcs.GD.2025.7},
  annote =	{Keywords: graph drawing, quality metrics, assumptions, fooling}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.38
Stress in Graph Drawings: Perception, Preference, and Performance

Authors: Gavin J. Mooney, Jacob Miller, Michael Wybrow, Stephen Kobourov, and Helen C. Purchase

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
Stress in a graph drawing has been a popular layout principle for more than two decades. Low stress drawings exhibit the property that the geometric distances between all pairs of nodes correlate with the shortest paths between them. The assumption has always been that low stress drawings are "nicer" and better support human perception and comprehension than high stress drawings. In this paper, we put these assumptions to the test. We use a normalised scale-independent and rotation-independent metric for stress; this is necessary to ensure strict controls on our experimental stimuli. We report on three experiments, exploring human perception of stress, preference for stress, and the effect of stress on a graph performance task. We conclude that people can see stress in a graph drawing, that they prefer low stress drawings, and that their performance in a shortest path task improves as stress decreases - thus empirically confirming long-standing assumptions.
Cite as

Gavin J. Mooney, Jacob Miller, Michael Wybrow, Stephen Kobourov, and Helen C. Purchase. Stress in Graph Drawings: Perception, Preference, and Performance. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 38:1-38:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mooney_et_al:LIPIcs.GD.2025.38,
  author =	{Mooney, Gavin J. and Miller, Jacob and Wybrow, Michael and Kobourov, Stephen and Purchase, Helen C.},
  title =	{{Stress in Graph Drawings: Perception, Preference, and Performance}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{38:1--38:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.38},
  URN =		{urn:nbn:de:0030-drops-250240},
  doi =		{10.4230/LIPIcs.GD.2025.38},
  annote =	{Keywords: Graph Drawing, Graph Drawing Metrics, Stress, Visual Perception, User Study}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.30
Universal Quality Metrics for Graph Drawings: Which Graphs Excite Us Most?

Authors: Gavin J. Mooney, Tim Hegemann, Alexander Wolff, Michael Wybrow, and Helen C. Purchase

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
Graphs are drawn for various purposes, and drawings are meant to display various features of a graph (such as planarity, Hamiltonicity). Still, there is a long history in measuring the quality of a graph drawing. Most of the metrics that have been implemented and used in large studies assume that graphs are drawn straight-line. Most of the studies use randomly generated graphs or one of very few existing benchmark sets that consist of graphs with a specific technical background (e.g., telecommunication networks). In this paper, we extend ten commonly used metrics to node-link diagrams where edges can be curves or polygonal chains. We implement these measures and use them to evaluate a new collection of graph drawings that we have extracted from 27 proceedings of the Graph Drawing conference using an automated pipeline. We compare the "metrics landscape" of our new benchmark set, the GD-collection-v1, which seems to mostly contain manually drawn graphs, to the metric landscape of a benchmark set with randomly generated graphs and computer-generated straight-line drawings that has been used in a recent study [Mooney et al.; PacificVis 2024]. Comparing the GD-collection-v1 with the Mooney at al. dataset reveals a distinct metrics landscape: GD drawings come from much smaller graphs (median vertex number 11 vs. 48) and therefore attain higher medians on most readability metrics. For example, Neighbourhood Preservation (0.5 vs. 0.239) is markedly higher in the GD-collection-v1. We also find that a large proportion of extracted drawings contain curved and/or polygonal edges (57%), motivating the extended metric definitions.
Cite as

Gavin J. Mooney, Tim Hegemann, Alexander Wolff, Michael Wybrow, and Helen C. Purchase. Universal Quality Metrics for Graph Drawings: Which Graphs Excite Us Most?. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mooney_et_al:LIPIcs.GD.2025.30,
  author =	{Mooney, Gavin J. and Hegemann, Tim and Wolff, Alexander and Wybrow, Michael and Purchase, Helen C.},
  title =	{{Universal Quality Metrics for Graph Drawings: Which Graphs Excite Us Most?}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{30:1--30:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.30},
  URN =		{urn:nbn:de:0030-drops-250162},
  doi =		{10.4230/LIPIcs.GD.2025.30},
  annote =	{Keywords: Graph drawing metrics, metric landscape, straight-line drawings, polyline drawings, curved drawings, automated extraction of graph drawings}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.22
Directed Buy-At-Bulk Spanners

Authors: Elena Grigorescu, Nithish Kumar, and Young-San Lin

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We present a framework that unifies directed buy-at-bulk network design and directed spanner problems, namely, buy-at-bulk spanners. The goal is to find a minimum-cost routing solution for network design problems that captures economies at scale, while satisfying demands and distance constraints for terminal pairs. A more restricted version of this problem was shown to be O(2^{log^{1-ε} n})-hard to approximate, where n is the number of vertices, under a standard complexity assumption, by Elkin and Peleg (Theory of Computing Systems, 2007). Our results for buy-at-bulk spanners are the following. - When the edge lengths are integral with magnitude polynomial in n we present: 1) An Õ(n^{4/5 + ε})-approximation polynomial-time randomized algorithm for uniform demands. 2) An Õ(k^{1/2 + ε})-approximation polynomial-time randomized algorithm for general demands, where k is the number of terminal pairs. This can be improved to an Õ(k^{ε})-approximation algorithm for the single-source problem. The same approximation ratios hold in the online setting. - When the edge lengths are rational and well-conditioned, we present an Õ(k^{1/2 + ε})-approximation polynomial-time randomized algorithm that may slightly violate the distance constraints. The result can be improved to an Õ(k^ε)-approximation algorithm for the single-source problem. The same approximation ratios hold for the online setting when the condition number is given in advance. To the best of our knowledge, these are the first sublinear factor approximation algorithms for directed buy-at-bulk spanners. We allow the edge lengths to be negative and the demands to be non-unit, unlike the previous literature. Our approximation ratios match the state-of-the-art ratios in special cases, namely, buy-at-bulk network design by Antonakopoulos (WAOA, 2010) and (online) weighted spanners by Grigorescu, Kumar, and Lin (APPROX 2023). Furthermore, we improve the competitive ratio for online buy-at-bulk by Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018) by a factor of log R, where R is the ratio between the maximum demand and the minimum demand.
Cite as

Elena Grigorescu, Nithish Kumar, and Young-San Lin. Directed Buy-At-Bulk Spanners. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 22:1-22:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{grigorescu_et_al:LIPIcs.APPROX/RANDOM.2025.22,
  author =	{Grigorescu, Elena and Kumar, Nithish and Lin, Young-San},
  title =	{{Directed Buy-At-Bulk Spanners}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{22:1--22:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.22},
  URN =		{urn:nbn:de:0030-drops-243885},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.22},
  annote =	{Keywords: buy-at-bulk spanners, minimum density junction tree, resource constrained shortest path}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.94
A Nearly Optimal Deterministic Algorithm for Online Transportation Problem

Authors: Tsubasa Harada and Toshiya Itoh

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
For the online transportation problem with m server sites, it has long been known that the competitive ratio of any deterministic algorithm is at least 2m-1. Kalyanasundaram and Pruhs conjectured in 1998 that a deterministic (2m-1)-competitive algorithm exists for this problem, a conjecture that has remained open for over two decades. In this paper, we propose a new deterministic algorithm for the online transportation problem and show that it achieves a competitive ratio of at most 8m-5. This is the first O(m)-competitive deterministic algorithm, coming close to the lower bound of 2m-1 within a constant factor.
Cite as

Tsubasa Harada and Toshiya Itoh. A Nearly Optimal Deterministic Algorithm for Online Transportation Problem. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 94:1-94:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{harada_et_al:LIPIcs.ICALP.2025.94,
  author =	{Harada, Tsubasa and Itoh, Toshiya},
  title =	{{A Nearly Optimal Deterministic Algorithm for Online Transportation Problem}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{94:1--94:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.94},
  URN =		{urn:nbn:de:0030-drops-234712},
  doi =		{10.4230/LIPIcs.ICALP.2025.94},
  annote =	{Keywords: Online algorithms, Competitive analysis, Online metric matching, Online weighted matching, Online minimum weight perfect matching, Online transportation problem, Online facility assignment, Greedy algorithm}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.66
Lipschitz Decompositions of Finite 𝓁_{p} Metrics

Authors: Robert Krauthgamer and Nir Petruschka

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for n-point subsets of 𝓁_p, for p > 2, remained open, see e.g. [Naor, SODA 2017]. We make significant progress on this question and establish the bound β = O(log^{1-1/p} n). Building on prior work, we demonstrate applications of this result to two problems, high-dimensional geometric spanners and distance labeling schemes. In addition, we sharpen a related decomposition bound for 1 < p < 2, due to Filtser and Neiman [Algorithmica 2022].
Cite as

Robert Krauthgamer and Nir Petruschka. Lipschitz Decompositions of Finite 𝓁_{p} Metrics. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{krauthgamer_et_al:LIPIcs.SoCG.2025.66,
  author =	{Krauthgamer, Robert and Petruschka, Nir},
  title =	{{Lipschitz Decompositions of Finite 𝓁\underline\{p\} Metrics}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{66:1--66:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.66},
  URN =		{urn:nbn:de:0030-drops-232182},
  doi =		{10.4230/LIPIcs.SoCG.2025.66},
  annote =	{Keywords: Lipschitz decompositions, metric embeddings, geometric spanners}
}
Document
DOI: 10.4230/LIPIcs.SAND.2025.9
Spanner Enumeration for Temporal Graphs

Authors: Kazuhiro Kurita, Andrea Marino, Jason Schoeters, and Takeaki Uno

Published in: LIPIcs, Volume 330, 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)

Abstract
A spanner of a temporal graph is a subset of edges that preserves connectivity over time between vertices. A minimal spanner is one in which no additional edges can be removed without breaking this connectivity. Our focus is on enumerating minimal spanners for a given temporal graph. We explore several variations of this problem based on the type of connectivity that must be maintained, ranging from one-to-all connectivity to one-to-all-to-one, many-to-all, and finally all-to-all connectivity. We establish that these problems become progressively harder: (i) We present a polynomial-delay enumeration algorithm for one-to-all connectivity; (ii) We prove Dual-hardness for both one-to-all-to-one and many-to-all connectivity, even in the restricted case of two-to-all; (iii) Finally, for all-to-all connectivity, we show that enumeration cannot be performed in output-polynomial time unless P = NP.
Cite as

Kazuhiro Kurita, Andrea Marino, Jason Schoeters, and Takeaki Uno. Spanner Enumeration for Temporal Graphs. In 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 330, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kurita_et_al:LIPIcs.SAND.2025.9,
  author =	{Kurita, Kazuhiro and Marino, Andrea and Schoeters, Jason and Uno, Takeaki},
  title =	{{Spanner Enumeration for Temporal Graphs}},
  booktitle =	{4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)},
  pages =	{9:1--9:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-368-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{330},
  editor =	{Meeks, Kitty and Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2025.9},
  URN =		{urn:nbn:de:0030-drops-230621},
  doi =		{10.4230/LIPIcs.SAND.2025.9},
  annote =	{Keywords: temporal graphs, temporal spanners, one-to-all connectivity, all-to-all connectivity enumeration, NP-completeness, Dual-hardness, binary partition tree, flashlight search, polynomial delay}
}
Document
DOI: 10.4230/LIPIcs.DISC.2021.21
Improved Weighted Additive Spanners

Authors: Michael Elkin, Yuval Gitlitz, and Ofer Neiman

Published in: LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)

Abstract
Graph spanners and emulators are sparse structures that approximately preserve distances of the original graph. While there has been an extensive amount of work on additive spanners, so far little attention was given to weighted graphs. Only very recently [Abu Reyan Ahmed et al., 2020] extended the classical +2 (respectively, +4) spanners for unweighted graphs of size O(n^{3/2}) (resp., O(n^{7/5})) to the weighted setting, where the additive error is +2W (resp., +4W). This means that for every pair u,v, the additive stretch is at most +2W_{u,v}, where W_{u,v} is the maximal edge weight on the shortest u-v path (weights are normalized so that the minimum edge weight is 1). In addition, [Abu Reyan Ahmed et al., 2020] showed a randomized algorithm yielding a +8W_{max} spanner of size O(n^{4/3}), here W_{max} is the maximum edge weight in the entire graph. In this work we improve the latter result by devising a simple deterministic algorithm for a +(6+ε)W spanner for weighted graphs with size O(n^{4/3}) (for any constant ε > 0), thus nearly matching the classical +6 spanner of size O(n^{4/3}) for unweighted graphs. Furthermore, we show a +(2+ε)W subsetwise spanner of size O(n⋅√{|S|}), improving the +4W_{max} result of [Abu Reyan Ahmed et al., 2020] (that had the same size). We also show a simple randomized algorithm for a +4W emulator of size Õ(n^{4/3}). In addition, we show that our technique is applicable for very sparse additive spanners, that have linear size. It is known that such spanners must suffer polynomially large stretch. For weighted graphs, we use a variant of our simple deterministic algorithm that yields a linear size +Õ(√n⋅ W) spanner, and we also obtain a tradeoff between size and stretch. Finally, generalizing the technique of [D. Dor et al., 2000] for unweighted graphs, we devise an efficient randomized algorithm producing a +2W spanner for weighted graphs of size Õ(n^{3/2}) in Õ(n²) time.
Cite as

Michael Elkin, Yuval Gitlitz, and Ofer Neiman. Improved Weighted Additive Spanners. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{elkin_et_al:LIPIcs.DISC.2021.21,
  author =	{Elkin, Michael and Gitlitz, Yuval and Neiman, Ofer},
  title =	{{Improved Weighted Additive Spanners}},
  booktitle =	{35th International Symposium on Distributed Computing (DISC 2021)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-210-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{209},
  editor =	{Gilbert, Seth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.21},
  URN =		{urn:nbn:de:0030-drops-148232},
  doi =		{10.4230/LIPIcs.DISC.2021.21},
  annote =	{Keywords: Graph theory, Pure additive spanners}
}
Document
DOI: 10.4230/LIPIcs.SEA.2021.16
Multi-Level Weighted Additive Spanners

Authors: Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence

Published in: LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

Abstract
Given a graph G = (V,E), a subgraph H is an additive +β spanner if dist_H(u,v) ≤ dist_G(u,v) + β for all u, v ∈ V. A pairwise spanner is a spanner for which the above inequality is only required to hold for specific pairs P ⊆ V × V given on input; when the pairs have the structure P = S × S for some S ⊆ V, it is called a subsetwise spanner. Additive spanners in unweighted graphs have been studied extensively in the literature, but have only recently been generalized to weighted graphs. In this paper, we consider a multi-level version of the subsetwise additive spanner in weighted graphs motivated by multi-level network design and visualization, where the vertices in S possess varying level, priority, or quality of service (QoS) requirements. The goal is to compute a nested sequence of spanners with the minimum total number of edges. We first generalize the +2 subsetwise spanner of [Pettie 2008, Cygan et al., 2013] to the weighted setting. We experimentally measure the performance of this and several existing algorithms by [Ahmed et al., 2020] for weighted additive spanners, both in terms of runtime and sparsity of the output spanner, when applied as a subroutine to multi-level problem. We provide an experimental evaluation on graphs using several different random graph generators and show that these spanner algorithms typically achieve much better guarantees in terms of sparsity and additive error compared with the theoretical maximum. By analyzing our experimental results, we additionally developed a new technique of changing a certain initialization parameter which provides better spanners in practice at the expense of a small increase in running time.
Cite as

Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence. Multi-Level Weighted Additive Spanners. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 16:1-16:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ahmed_et_al:LIPIcs.SEA.2021.16,
  author =	{Ahmed, Reyan and Bodwin, Greg and Sahneh, Faryad Darabi and Hamm, Keaton and Kobourov, Stephen and Spence, Richard},
  title =	{{Multi-Level Weighted Additive Spanners}},
  booktitle =	{19th International Symposium on Experimental Algorithms (SEA 2021)},
  pages =	{16:1--16:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-185-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{190},
  editor =	{Coudert, David and Natale, Emanuele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.16},
  URN =		{urn:nbn:de:0030-drops-137885},
  doi =		{10.4230/LIPIcs.SEA.2021.16},
  annote =	{Keywords: multi-level, graph spanner, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.4
Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem

Authors: Reyan Ahmed, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
We study the multi-level Steiner tree problem: a generalization of the Steiner tree problem in graphs where terminals T require varying priority, level, or quality of service. In this problem, we seek to find a minimum cost tree containing edges of varying rates such that any two terminals u, v with priorities P(u), P(v) are connected using edges of rate min{P(u),P(v)} or better. The case where edge costs are proportional to their rate is approximable to within a constant factor of the optimal solution. For the more general case of non-proportional costs, this problem is hard to approximate with ratio c log log n, where n is the number of vertices in the graph. A simple greedy algorithm by Charikar et al., however, provides a min{2(ln |T|+1), 𝓁 ρ}-approximation in this setting, where ρ is an approximation ratio for a heuristic solver for the Steiner tree problem and 𝓁 is the number of priorities or levels (Byrka et al. give a Steiner tree algorithm with ρ≈1.39, for example). In this paper, we describe a natural generalization to the multi-level case of the classical (single-level) Steiner tree approximation algorithm based on Kruskal’s minimum spanning tree algorithm. We prove that this algorithm achieves an approximation ratio at least as good as Charikar et al., and experimentally performs better with respect to the optimum solution. We develop an integer linear programming formulation to compute an exact solution for the multi-level Steiner tree problem with non-proportional edge costs and use it to evaluate the performance of our algorithm on both random graphs and multi-level instances derived from SteinLib.
Cite as

Reyan Ahmed, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence. Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 4:1-4:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ahmed_et_al:LIPIcs.ESA.2020.4,
  author =	{Ahmed, Reyan and Sahneh, Faryad Darabi and Hamm, Keaton and Kobourov, Stephen and Spence, Richard},
  title =	{{Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{4:1--4:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.4},
  URN =		{urn:nbn:de:0030-drops-128709},
  doi =		{10.4230/LIPIcs.ESA.2020.4},
  annote =	{Keywords: multi-level, Steiner tree, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.SEA.2018.15
Multi-Level Steiner Trees

Authors: Reyan Ahmed, Patrizio Angelini, Faryad Darabi Sahneh, Alon Efrat, David Glickenstein, Martin Gronemann, Niklas Heinsohn, Stephen G. Kobourov, Richard Spence, Joseph Watkins, and Alexander Wolff

Published in: LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

Abstract
In the classical Steiner tree problem, one is given an undirected, connected graph G=(V,E) with non-negative edge costs and a set of terminals T subseteq V. The objective is to find a minimum-cost edge set E' subseteq E that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of rho = ln(4)+epsilon < 1.39. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals T_1 subset ... subset T_k subseteq V, compute nested edge sets E_1 subseteq ... subseteq E_k subseteq E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under names such as Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two natural heuristics with approximation factor O(k). Based on these, we introduce a composite algorithm that requires 2^k Steiner tree computations. We determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio and needs at most 2k Steiner tree computations. We compare five algorithms experimentally on several classes of graphs using four types of graph generators. We also implemented an integer linear program for MLST to provide ground truth. Our combined algorithm outperforms the others both in theory and in practice when the number of levels is small (k <= 22), which works well for applications such as designing multi-level infrastructure or network visualization.
Cite as

Reyan Ahmed, Patrizio Angelini, Faryad Darabi Sahneh, Alon Efrat, David Glickenstein, Martin Gronemann, Niklas Heinsohn, Stephen G. Kobourov, Richard Spence, Joseph Watkins, and Alexander Wolff. Multi-Level Steiner Trees. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ahmed_et_al:LIPIcs.SEA.2018.15,
  author =	{Ahmed, Reyan and Angelini, Patrizio and Sahneh, Faryad Darabi and Efrat, Alon and Glickenstein, David and Gronemann, Martin and Heinsohn, Niklas and Kobourov, Stephen G. and Spence, Richard and Watkins, Joseph and Wolff, Alexander},
  title =	{{Multi-Level Steiner Trees}},
  booktitle =	{17th International Symposium on Experimental Algorithms (SEA 2018)},
  pages =	{15:1--15:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-070-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{103},
  editor =	{D'Angelo, Gianlorenzo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.15},
  URN =		{urn:nbn:de:0030-drops-89506},
  doi =		{10.4230/LIPIcs.SEA.2018.15},
  annote =	{Keywords: Approximation algorithm, Steiner tree, multi-level graph representation}
}
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