69 Search Results for "Sanit�, Laura"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.79
Finding Almost Tight Witness Trees

Authors: Dylan Hyatt-Denesik, Afrouz Jabal Ameli, and Laura Sanità

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
This paper addresses a graph optimization problem, called the Witness Tree problem, which seeks a spanning tree of a graph minimizing a certain non-linear objective function. This problem is of interest because it plays a crucial role in the analysis of the best approximation algorithms for two fundamental network design problems: Steiner Tree and Node-Tree Augmentation. We will show how a wiser choice of witness trees leads to an improved approximation for Node-Tree Augmentation, and for Steiner Tree in special classes of graphs.
Cite as

Dylan Hyatt-Denesik, Afrouz Jabal Ameli, and Laura Sanità. Finding Almost Tight Witness Trees. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 79:1-79:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{hyattdenesik_et_al:LIPIcs.ICALP.2023.79,
  author =	{Hyatt-Denesik, Dylan and Jabal Ameli, Afrouz and Sanit\`{a}, Laura},
  title =	{{Finding Almost Tight Witness Trees}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{79:1--79:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.79},
  URN =		{urn:nbn:de:0030-drops-181314},
  doi =		{10.4230/LIPIcs.ICALP.2023.79},
  annote =	{Keywords: Algorithms, Network Design, Approximation}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021
LIPIcs, Volume 207, APPROX/RANDOM 2021, Complete Volume

Authors: Mary Wootters and Laura Sanità

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
LIPIcs, Volume 207, APPROX/RANDOM 2021, Complete Volume
Cite as

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 1-1240, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Proceedings{wootters_et_al:LIPIcs.APPROX/RANDOM.2021,
  title =	{{LIPIcs, Volume 207, APPROX/RANDOM 2021, Complete Volume}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{1--1240},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021},
  URN =		{urn:nbn:de:0030-drops-146929},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021},
  annote =	{Keywords: LIPIcs, Volume 207, APPROX/RANDOM 2021, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Mary Wootters and Laura Sanità

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 0:i-0:x, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{wootters_et_al:LIPIcs.APPROX/RANDOM.2021.0,
  author =	{Wootters, Mary and Sanit\`{a}, Laura},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{0:i--0:x},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.0},
  URN =		{urn:nbn:de:0030-drops-146933},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.1
On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources

Authors: Umang Bhaskar, A. R. Sricharan, and Rohit Vaish

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We study the fair allocation of undesirable indivisible items, or chores. While the case of desirable indivisible items (or goods) is extensively studied, with many results known for different notions of fairness, less is known about the fair division of chores. We study envy-free allocation of chores and make three contributions. First, we show that determining the existence of an envy-free allocation is NP-complete even in the simple case when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up to one chore (EF1), correcting a claim in the existing literature. A modification of our algorithm can be used to compute an EF1 allocation for doubly monotone instances (where each agent can partition the set of items into objective goods and objective chores). Our third result applies to a mixed resources model consisting of indivisible items and a divisible, undesirable heterogeneous resource (i.e., a bad cake). We show that there always exists an allocation that satisfies envy-freeness for mixed resources (EFM) in this setting, complementing a recent result of Bei et al. [Bei et al., 2021] for indivisible goods and divisible cake.
Cite as

Umang Bhaskar, A. R. Sricharan, and Rohit Vaish. On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 1:1-1:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhaskar_et_al:LIPIcs.APPROX/RANDOM.2021.1,
  author =	{Bhaskar, Umang and Sricharan, A. R. and Vaish, Rohit},
  title =	{{On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{1:1--1:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.1},
  URN =		{urn:nbn:de:0030-drops-146944},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.1},
  annote =	{Keywords: Fair Division, Indivisible Chores, Approximate Envy-Freeness}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.2
Optimal Algorithms for Online b-Matching with Variable Vertex Capacities

Authors: Susanne Albers and Sebastian Schubert

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We study the b-matching problem, which generalizes classical online matching introduced by Karp, Vazirani and Vazirani (STOC 1990). Consider a bipartite graph G = (S ̇∪ R,E). Every vertex s ∈ S is a server with a capacity b_s, indicating the number of possible matching partners. The vertices r ∈ R are requests that arrive online and must be matched immediately to an eligible server. The goal is to maximize the cardinality of the constructed matching. In contrast to earlier work, we study the general setting where servers may have arbitrary, individual capacities. We prove that the most natural and simple online algorithms achieve optimal competitive ratios. As for deterministic algorithms, we give a greedy algorithm RelativeBalance and analyze it by extending the primal-dual framework of Devanur, Jain and Kleinberg (SODA 2013). In the area of randomized algorithms we study the celebrated Ranking algorithm by Karp, Vazirani and Vazirani. We prove that the original Ranking strategy, simply picking a random permutation of the servers, achieves an optimal competitiveness of 1-1/e, independently of the server capacities. Hence it is not necessary to resort to a reduction, replacing every server s by b_s vertices of unit capacity and to then run Ranking on this graph with ∑_{s ∈ S} b_s vertices on the left-hand side. From a theoretical point of view our result explores the power of randomization and strictly limits the amount of required randomness. From a practical point of view it leads to more efficient allocation algorithms. Technically, we show that the primal-dual framework of Devanur, Jain and Kleinberg cannot establish a competitiveness better than 1/2 for the original Ranking algorithm, choosing a permutation of the servers. Therefore, we formulate a new configuration LP for the b-matching problem and then conduct a primal-dual analysis. We extend this analysis approach to the vertex-weighted b-matching problem. Specifically, we show that the algorithm PerturbedGreedy by Aggarwal, Goel, Karande and Mehta (SODA 2011), again with a sole randomization over the set of servers, is (1-1/e)-competitive. Together with recent work by Huang and Zhang (STOC 2020), our results demonstrate that configuration LPs can be strictly stronger than standard LPs in the analysis of more complex matching problems.
Cite as

Susanne Albers and Sebastian Schubert. Optimal Algorithms for Online b-Matching with Variable Vertex Capacities. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 2:1-2:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{albers_et_al:LIPIcs.APPROX/RANDOM.2021.2,
  author =	{Albers, Susanne and Schubert, Sebastian},
  title =	{{Optimal Algorithms for Online b-Matching with Variable Vertex Capacities}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{2:1--2:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.2},
  URN =		{urn:nbn:de:0030-drops-146957},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.2},
  annote =	{Keywords: Online algorithms, primal-dual analysis, configuration LP, b-matching, variable vertex capacities, unweighted matching, vertex-weighted matching}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.3
Bag-Of-Tasks Scheduling on Related Machines

Authors: Anupam Gupta, Amit Kumar, and Sahil Singla

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We consider online scheduling to minimize weighted completion time on related machines, where each job consists of several tasks that can be concurrently executed. A job gets completed when all its component tasks finish. We obtain an O(K³ log² K)-competitive algorithm in the non-clairvoyant setting, where K denotes the number of distinct machine speeds. The analysis is based on dual-fitting on a precedence-constrained LP relaxation that may be of independent interest.
Cite as

Anupam Gupta, Amit Kumar, and Sahil Singla. Bag-Of-Tasks Scheduling on Related Machines. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gupta_et_al:LIPIcs.APPROX/RANDOM.2021.3,
  author =	{Gupta, Anupam and Kumar, Amit and Singla, Sahil},
  title =	{{Bag-Of-Tasks Scheduling on Related Machines}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.3},
  URN =		{urn:nbn:de:0030-drops-146967},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.3},
  annote =	{Keywords: approximation algorithms, scheduling, bag-of-tasks, related machines}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.4
Hardness of Approximation for Euclidean k-Median

Authors: Anup Bhattacharya, Dishant Goyal, and Ragesh Jaiswal

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
The Euclidean k-median problem is defined in the following manner: given a set 𝒳 of n points in d-dimensional Euclidean space ℝ^d, and an integer k, find a set C ⊂ ℝ^d of k points (called centers) such that the cost function Φ(C,𝒳) ≡ ∑_{x ∈ 𝒳} min_{c ∈ C} ‖x-c‖₂ is minimized. The Euclidean k-means problem is defined similarly by replacing the distance with squared Euclidean distance in the cost function. Various hardness of approximation results are known for the Euclidean k-means problem [Pranjal Awasthi et al., 2015; Euiwoong Lee et al., 2017; Vincent Cohen{-}Addad and {Karthik {C. S.}}, 2019]. However, no hardness of approximation result was known for the Euclidean k-median problem. In this work, assuming the unique games conjecture (UGC), we provide the hardness of approximation result for the Euclidean k-median problem in O(log k) dimensional space. This solves an open question posed explicitly in the work of Awasthi et al. [Pranjal Awasthi et al., 2015]. Furthermore, we study the hardness of approximation for the Euclidean k-means/k-median problems in the bi-criteria setting where an algorithm is allowed to choose more than k centers. That is, bi-criteria approximation algorithms are allowed to output β k centers (for constant β > 1) and the approximation ratio is computed with respect to the optimal k-means/k-median cost. We show the hardness of bi-criteria approximation result for the Euclidean k-median problem for any β < 1.015, assuming UGC. We also show a similar hardness of bi-criteria approximation result for the Euclidean k-means problem with a stronger bound of β < 1.28, again assuming UGC.
Cite as

Anup Bhattacharya, Dishant Goyal, and Ragesh Jaiswal. Hardness of Approximation for Euclidean k-Median. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhattacharya_et_al:LIPIcs.APPROX/RANDOM.2021.4,
  author =	{Bhattacharya, Anup and Goyal, Dishant and Jaiswal, Ragesh},
  title =	{{Hardness of Approximation for Euclidean k-Median}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{4:1--4:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.4},
  URN =		{urn:nbn:de:0030-drops-146979},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.4},
  annote =	{Keywords: Hardness of approximation, bicriteria approximation, approximation algorithms, k-median, k-means}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.5
Online Directed Spanners and Steiner Forests

Authors: Elena Grigorescu, Young-San Lin, and Kent Quanrud

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We present online algorithms for directed spanners and directed Steiner forests. These are well-studied network connectivity problems that fall under the unifying framework of online covering and packing linear programming formulations. This framework was developed in the seminal work of Buchbinder and Naor (Mathematics of Operations Research, 34, 2009) and is based on primal-dual techniques. Specifically, our results include the following: - For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{4/5}) for graphs with general edge lengths, where n is the number of vertices of the given graph. For graphs with uniform edge lengths, we give an efficient randomized algorithm with competitive ratio Õ(n^{2/3+ε}), and an efficient deterministic algorithm with competitive ratio Õ(k^{1/2+ε}), where k is the number of terminal pairs. To the best of our knowledge, these are the first online algorithms for directed spanners. In the offline version, the current best approximation ratio for uniform edge lengths is Õ(n^{3/5 + ε}), due to Chlamt{á}č, Dinitz, Kortsarz, and Laekhanukit (SODA 2017, TALG 2020). - For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{2/3 + ε}). The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is Õ(k^{1/2 + ε})-competitive. In the offline version, the current best approximation ratio with uniform costs is Õ(n^{26/45 + ε}), due to Abboud and Bodwin (SODA 2018). To obtain efficient and competitive online algorithms, we observe that a small modification of the online covering and packing framework by Buchbinder and Naor implies a polynomial-time implementation of the primal-dual approach with separation oracles, which a priori might perform exponentially many calls to the oracle. We convert the online spanner problem into an online covering problem and complete the rounding-step analysis in a problem-specific fashion.
Cite as

Elena Grigorescu, Young-San Lin, and Kent Quanrud. Online Directed Spanners and Steiner Forests. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 5:1-5:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grigorescu_et_al:LIPIcs.APPROX/RANDOM.2021.5,
  author =	{Grigorescu, Elena and Lin, Young-San and Quanrud, Kent},
  title =	{{Online Directed Spanners and Steiner Forests}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{5:1--5:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.5},
  URN =		{urn:nbn:de:0030-drops-146987},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.5},
  annote =	{Keywords: online directed pairwise spanners, online directed Steiner forests, online covering/packing linear programming, primal-dual approach}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.6
Query Complexity of Global Minimum Cut

Authors: Arijit Bishnu, Arijit Ghosh, Gopinath Mishra, and Manaswi Paraashar

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
In this work, we resolve the query complexity of global minimum cut problem for a graph by designing a randomized algorithm for approximating the size of minimum cut in a graph, where the graph can be accessed through local queries like Degree, Neighbor, and Adjacency queries. Given ε ∈ (0,1), the algorithm with high probability outputs an estimate t̂ satisfying the following (1-ε) t ≤ t̂ ≤ (1+ε) t, where t is the size of minimum cut in the graph. The expected number of local queries used by our algorithm is min{m+n,m/t}poly(log n,1/(ε)) where n and m are the number of vertices and edges in the graph, respectively. Eden and Rosenbaum showed that Ω(m/t) local queries are required for approximating the size of minimum cut in graphs, {but no local query based algorithm was known. Our algorithmic result coupled with the lower bound of Eden and Rosenbaum [APPROX 2018] resolve the query complexity of the problem of estimating the size of minimum cut in graphs using local queries.} Building on the lower bound of Eden and Rosenbaum, we show that, for all t ∈ ℕ, Ω(m) local queries are required to decide if the size of the minimum cut in the graph is t or t-2. Also, we show that, for any t ∈ ℕ, Ω(m) local queries are required to find all the minimum cut edges even if it is promised that the input graph has a minimum cut of size t. Both of our lower bound results are randomized, and hold even if we can make Random Edge queries in addition to local queries.
Cite as

Arijit Bishnu, Arijit Ghosh, Gopinath Mishra, and Manaswi Paraashar. Query Complexity of Global Minimum Cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bishnu_et_al:LIPIcs.APPROX/RANDOM.2021.6,
  author =	{Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath and Paraashar, Manaswi},
  title =	{{Query Complexity of Global Minimum Cut}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{6:1--6:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.6},
  URN =		{urn:nbn:de:0030-drops-146992},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.6},
  annote =	{Keywords: Query complexity, Global mincut}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.7
A Constant-Factor Approximation for Weighted Bond Cover

Authors: Eun Jung Kim, Euiwoong Lee, and Dimitrios M. Thilikos

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
The Weighted ℱ-Vertex Deletion for a class ℱ of graphs asks, weighted graph G, for a minimum weight vertex set S such that G-S ∈ ℱ. The case when ℱ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted ℱ-Vertex Deletion. Only three cases of minor-closed ℱ are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. We study the problem for the class ℱ of θ_c-minor-free graphs, under the equivalent setting of the Weighted c-Bond Cover problem, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret et al., SIDMA'14] which states the following: any graph G containing a θ_c-minor-model either contains a large two-terminal protrusion, or contains a constant-size θ_c-minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted ℱ-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families.
Cite as

Eun Jung Kim, Euiwoong Lee, and Dimitrios M. Thilikos. A Constant-Factor Approximation for Weighted Bond Cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kim_et_al:LIPIcs.APPROX/RANDOM.2021.7,
  author =	{Kim, Eun Jung and Lee, Euiwoong and Thilikos, Dimitrios M.},
  title =	{{A Constant-Factor Approximation for Weighted Bond Cover}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{7:1--7:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.7},
  URN =		{urn:nbn:de:0030-drops-147002},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.7},
  annote =	{Keywords: Constant-factor approximation algorithms, Primal-dual method, Bonds in graphs, Graph minors, Graph modification problems}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.8
Truly Asymptotic Lower Bounds for Online Vector Bin Packing

Authors: János Balogh, Ilan Reuven Cohen, Leah Epstein, and Asaf Levin

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
In this work, we consider online d-dimensional vector bin packing. It is known that no algorithm can have a competitive ratio of o(d/log² d) in the absolute sense, although upper bounds for this problem have always been presented in the asymptotic sense. Since variants of bin packing are traditionally studied with respect to the asymptotic measure, and since the two measures are different, we focus on the asymptotic measure and prove new lower bounds of the asymptotic competitive ratio. The existing lower bounds prior to this work were known to be smaller than 3, even for very large d. Here, we significantly improved on the best known lower bounds of the asymptotic competitive ratio (and as a byproduct, on the absolute competitive ratio) for online vector packing of vectors with d ≥ 3 dimensions, for every dimension d. To obtain these results, we use several different constructions, one of which is an adaptive construction with a lower bound of Ω(√d). Our main result is that the lower bound of Ω(d/log² d) on the competitive ratio holds also in the asymptotic sense. This result holds also against randomized algorithms, and requires a careful adaptation of constructions for online coloring, rather than simple black-box reductions.
Cite as

János Balogh, Ilan Reuven Cohen, Leah Epstein, and Asaf Levin. Truly Asymptotic Lower Bounds for Online Vector Bin Packing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{balogh_et_al:LIPIcs.APPROX/RANDOM.2021.8,
  author =	{Balogh, J\'{a}nos and Cohen, Ilan Reuven and Epstein, Leah and Levin, Asaf},
  title =	{{Truly Asymptotic Lower Bounds for Online Vector Bin Packing}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{8:1--8:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.8},
  URN =		{urn:nbn:de:0030-drops-147013},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.8},
  annote =	{Keywords: Bin packing, online algorithms, approximation algorithms, vector packing}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.9
Fine-Grained Completeness for Optimization in P

Authors: Karl Bringmann, Alejandro Cassis, Nick Fischer, and Marvin Künnemann

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We initiate the study of fine-grained completeness theorems for exact and approximate optimization in the polynomial-time regime. Inspired by the first completeness results for decision problems in P (Gao, Impagliazzo, Kolokolova, Williams, TALG 2019) as well as the classic class MaxSNP and MaxSNP-completeness for NP optimization problems (Papadimitriou, Yannakakis, JCSS 1991), we define polynomial-time analogues MaxSP and MinSP, which contain a number of natural optimization problems in P, including Maximum Inner Product, general forms of nearest neighbor search and optimization variants of the k-XOR problem. Specifically, we define MaxSP as the class of problems definable as max_{x₁,… ,x_k} #{(y₁,… ,y_𝓁) : ϕ(x₁,… ,x_k, y₁,… ,y_𝓁)}, where ϕ is a quantifier-free first-order property over a given relational structure (with MinSP defined analogously). On m-sized structures, we can solve each such problem in time O(m^{k+𝓁-1}). Our results are: - We determine (a sparse variant of) the Maximum/Minimum Inner Product problem as complete under deterministic fine-grained reductions: A strongly subquadratic algorithm for Maximum/Minimum Inner Product would beat the baseline running time of O(m^{k+𝓁-1}) for all problems in MaxSP/MinSP by a polynomial factor. - This completeness transfers to approximation: Maximum/Minimum Inner Product is also complete in the sense that a strongly subquadratic c-approximation would give a (c+ε)-approximation for all MaxSP/MinSP problems in time O(m^{k+𝓁-1-δ}), where ε > 0 can be chosen arbitrarily small. Combining our completeness with (Chen, Williams, SODA 2019), we obtain the perhaps surprising consequence that refuting the OV Hypothesis is equivalent to giving a O(1)-approximation for all MinSP problems in faster-than-O(m^{k+𝓁-1}) time. - By fine-tuning our reductions, we obtain mild algorithmic improvements for solving and approximating all problems in MaxSP and MinSP, using the fastest known algorithms for Maximum/Minimum Inner Product.
Cite as

Karl Bringmann, Alejandro Cassis, Nick Fischer, and Marvin Künnemann. Fine-Grained Completeness for Optimization in P. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 9:1-9:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bringmann_et_al:LIPIcs.APPROX/RANDOM.2021.9,
  author =	{Bringmann, Karl and Cassis, Alejandro and Fischer, Nick and K\"{u}nnemann, Marvin},
  title =	{{Fine-Grained Completeness for Optimization in P}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{9:1--9:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.9},
  URN =		{urn:nbn:de:0030-drops-147024},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.9},
  annote =	{Keywords: Fine-grained Complexity \& Algorithm Design, Completeness, Hardness of Approximation in P, Dimensionality Reductions}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.10
An Estimator for Matching Size in Low Arboricity Graphs with Two Applications

Authors: Hossein Jowhari

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
In this paper, we present a new degree-based estimator for the size of maximum matching in bounded arboricity graphs. When the arboricity of the graph is bounded by α, the estimator gives a α+2 factor approximation of the matching size. For planar graphs, we show the estimator does better and returns a 3.5 approximation of the matching size. Using this estimator, we get new results for approximating the matching size of planar graphs in the streaming and distributed models of computation. In particular, in the vertex-arrival streams, we get a randomized O((√n)/(ε²)log n) space algorithm for approximating the matching size of a planar graph on n vertices within (3.5+ε) factor. Similarly, we get a simultaneous protocol in the vertex-partition model for approximating the matching size within (3.5+ε) factor using O((n^{2/3})/(ε²)log n) communication from each player. In comparison with the previous estimators, the estimator in this paper does not need to know the arboricity of the input graph and improves the approximation factor in the case of planar graphs.
Cite as

Hossein Jowhari. An Estimator for Matching Size in Low Arboricity Graphs with Two Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 10:1-10:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jowhari:LIPIcs.APPROX/RANDOM.2021.10,
  author =	{Jowhari, Hossein},
  title =	{{An Estimator for Matching Size in Low Arboricity Graphs with Two Applications}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{10:1--10:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.10},
  URN =		{urn:nbn:de:0030-drops-147039},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.10},
  annote =	{Keywords: Data Stream Algorithms, Maximum Matching, Planar Graphs}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.11
An Optimal Algorithm for Triangle Counting in the Stream

Authors: Rajesh Jayaram and John Kallaugher

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We present a new algorithm for approximating the number of triangles in a graph G whose edges arrive as an arbitrary order stream. If m is the number of edges in G, T the number of triangles, Δ_E the maximum number of triangles which share a single edge, and Δ_V the maximum number of triangles which share a single vertex, then our algorithm requires space: Õ(m/T⋅(Δ_E + √{Δ_V})) Taken with the Ω((m Δ_E)/T) lower bound of Braverman, Ostrovsky, and Vilenchik (ICALP 2013), and the Ω((m √{Δ_V})/T) lower bound of Kallaugher and Price (SODA 2017), our algorithm is optimal up to log factors, resolving the complexity of a classic problem in graph streaming.
Cite as

Rajesh Jayaram and John Kallaugher. An Optimal Algorithm for Triangle Counting in the Stream. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 11:1-11:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jayaram_et_al:LIPIcs.APPROX/RANDOM.2021.11,
  author =	{Jayaram, Rajesh and Kallaugher, John},
  title =	{{An Optimal Algorithm for Triangle Counting in the Stream}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{11:1--11:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.11},
  URN =		{urn:nbn:de:0030-drops-147046},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.11},
  annote =	{Keywords: Triangle Counting, Streaming, Graph Algorithms, Sampling, Sketching}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.12
Matching Drivers to Riders: A Two-Stage Robust Approach

Authors: Omar El Housni, Vineet Goyal, Oussama Hanguir, and Clifford Stein

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
Matching demand (riders) to supply (drivers) efficiently is a fundamental problem for ride-hailing platforms who need to match the riders (almost) as soon as the request arrives with only partial knowledge about future ride requests. A myopic approach that computes an optimal matching for current requests ignoring future uncertainty can be highly sub-optimal. In this paper, we consider a two-stage robust optimization framework for this matching problem where future demand uncertainty is modeled using a set of demand scenarios (specified explicitly or implicitly). The goal is to match the current request to drivers (in the first stage) so that the cost of first stage matching and the worst-case cost over all scenarios for the second stage matching is minimized. We show that this two-stage robust matching is NP-hard under both explicit and implicit models of uncertainty. We present constant approximation algorithms for both models of uncertainty under different settings and show they improve significantly over standard greedy approaches.
Cite as

Omar El Housni, Vineet Goyal, Oussama Hanguir, and Clifford Stein. Matching Drivers to Riders: A Two-Stage Robust Approach. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{housni_et_al:LIPIcs.APPROX/RANDOM.2021.12,
  author =	{Housni, Omar El and Goyal, Vineet and Hanguir, Oussama and Stein, Clifford},
  title =	{{Matching Drivers to Riders: A Two-Stage Robust Approach}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{12:1--12:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.12},
  URN =		{urn:nbn:de:0030-drops-147054},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.12},
  annote =	{Keywords: matching, robust optimization, approximation algorithms}
}
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