19 Search Results for "Schwartz, Roy"

Document
DOI: 10.4230/LIPIcs.MFCS.2024.25
Multiway Cuts with a Choice of Representatives

Authors: Kristóf Bérczi, Tamás Király, and Daniel P. Szabo

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
In the Multiway Cut problem, we are given an undirected graph with nonnegative edge weights and a subset of k terminals, and the goal is to determine a set of edges of minimum total weight whose removal disconnects each terminal from the rest. The problem is APX-hard for k ≥ 3, and an extensive line of research has concentrated on closing the gap between the best upper and lower bounds for approximability and inapproximability, respectively. In this paper, we study several generalizations of Multiway Cut where the terminals can be chosen as representatives from sets of candidates T₁,…,T_q. In this setting, one is allowed to choose these representatives so that the minimum-weight cut separating these sets via their representatives is as small as possible. We distinguish different cases depending on (A) whether the representative of a candidate set has to be separated from the other candidate sets completely or only from the representatives, and (B) whether there is a single representative for each candidate set or the choice of representative is independent for each pair of candidate sets. For fixed q, we give approximation algorithms for each of these problems that match the best known approximation guarantee for Multiway Cut. Our technical contribution is a new extension of the CKR relaxation that preserves approximation guarantees. For general q, we show o(log q)-inapproximability for all cases where the choice of representatives may depend on the pair of candidate sets, as well as for the case where the goal is to separate a fixed node from a single representative from each candidate set. As a positive result, we give a 2-approximation algorithm for the case where we need to choose a single representative from each candidate set. This is a generalization of the (2-2/k)-approximation for k-Cut, and we can solve it by relating the tree case to optimization over a gammoid.
Cite as

Kristóf Bérczi, Tamás Király, and Daniel P. Szabo. Multiway Cuts with a Choice of Representatives. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{berczi_et_al:LIPIcs.MFCS.2024.25,
  author =	{B\'{e}rczi, Krist\'{o}f and Kir\'{a}ly, Tam\'{a}s and Szabo, Daniel P.},
  title =	{{Multiway Cuts with a Choice of Representatives}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.25},
  URN =		{urn:nbn:de:0030-drops-205813},
  doi =		{10.4230/LIPIcs.MFCS.2024.25},
  annote =	{Keywords: Approximation algorithms, Multiway cut, CKR relaxation, Steiner multicut}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.2
Streaming Zero-Knowledge Proofs

Authors: Graham Cormode, Marcel Dall'Agnol, Tom Gur, and Chris Hickey

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
Streaming interactive proofs (SIPs) enable a space-bounded algorithm with one-pass access to a massive stream of data to verify a computation that requires large space, by communicating with a powerful but untrusted prover. This work initiates the study of zero-knowledge proofs for data streams. We define the notion of zero-knowledge in the streaming setting and construct zero-knowledge SIPs for the two main algorithmic building blocks in the streaming interactive proofs literature: the sumcheck and polynomial evaluation protocols. To the best of our knowledge all known streaming interactive proofs are based on either of these tools, and indeed, this allows us to obtain zero-knowledge SIPs for central streaming problems such as index, point and range queries, median, frequency moments, and inner product. Our protocols are efficient in terms of time and space, as well as communication: the verifier algorithm’s space complexity is polylog(n) and, after a non-interactive setup that uses a random string of near-linear length, the remaining parameters are n^o(1). En route, we develop an algorithmic toolkit for designing zero-knowledge data stream protocols, consisting of an algebraic streaming commitment protocol and a temporal commitment protocol. Our analyses rely on delicate algebraic and information-theoretic arguments and reductions from average-case communication complexity.
Cite as

Graham Cormode, Marcel Dall'Agnol, Tom Gur, and Chris Hickey. Streaming Zero-Knowledge Proofs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 2:1-2:66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cormode_et_al:LIPIcs.CCC.2024.2,
  author =	{Cormode, Graham and Dall'Agnol, Marcel and Gur, Tom and Hickey, Chris},
  title =	{{Streaming Zero-Knowledge Proofs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{2:1--2:66},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.2},
  URN =		{urn:nbn:de:0030-drops-203988},
  doi =		{10.4230/LIPIcs.CCC.2024.2},
  annote =	{Keywords: Zero-knowledge proofs, streaming algorithms, computational complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.47
Lower Bounds on 0-Extension with Steiner Nodes

Authors: Yu Chen and Zihan Tan

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
In the 0-Extension problem, we are given an edge-weighted graph G = (V,E,c), a set T ⊆ V of its vertices called terminals, and a semi-metric D over T, and the goal is to find an assignment f of each non-terminal vertex to a terminal, minimizing the sum, over all edges (u,v) ∈ E, the product of the edge weight c(u,v) and the distance D(f(u),f(v)) between the terminals that u,v are mapped to. Current best approximation algorithms on 0-Extension are based on rounding a linear programming relaxation called the semi-metric LP relaxation. The integrality gap of this LP, is upper bounded by O(log|T|/log log|T|) and lower bounded by Ω((log|T|)^{2/3}), has been shown to be closely related to the quality of cut and flow vertex sparsifiers. We study a variant of the 0-Extension problem where Steiner vertices are allowed. Specifically, we focus on the integrality gap of the same semi-metric LP relaxation to this new problem. Following from previous work, this new integrality gap turns out to be closely related to the quality achievable by cut/flow vertex sparsifiers with Steiner nodes, a major open problem in graph compression. We show that the new integrality gap stays superconstant Ω(log log |T|) even if we allow a super-linear O(|T|log^{1-ε}|T|) number of Steiner nodes.
Cite as

Yu Chen and Zihan Tan. Lower Bounds on 0-Extension with Steiner Nodes. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 47:1-47:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2024.47,
  author =	{Chen, Yu and Tan, Zihan},
  title =	{{Lower Bounds on 0-Extension with Steiner Nodes}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{47:1--47:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.47},
  URN =		{urn:nbn:de:0030-drops-201905},
  doi =		{10.4230/LIPIcs.ICALP.2024.47},
  annote =	{Keywords: Graph Algorithms, Zero Extension, Integrality Gap}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.52
Simultaneously Approximating All 𝓁_p-Norms in Correlation Clustering

Authors: Sami Davies, Benjamin Moseley, and Heather Newman

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
This paper considers correlation clustering on unweighted complete graphs. We give a combinatorial algorithm that returns a single clustering solution that is simultaneously O(1)-approximate for all 𝓁_p-norms of the disagreement vector; in other words, a combinatorial O(1)-approximation of the all-norms objective for correlation clustering. This is the first proof that minimal sacrifice is needed in order to optimize different norms of the disagreement vector. In addition, our algorithm is the first combinatorial approximation algorithm for the 𝓁₂-norm objective, and more generally the first combinatorial algorithm for the 𝓁_p-norm objective when 1 < p < ∞. It is also faster than all previous algorithms that minimize the 𝓁_p-norm of the disagreement vector, with run-time O(n^ω), where O(n^ω) is the time for matrix multiplication on n × n matrices. When the maximum positive degree in the graph is at most Δ, this can be improved to a run-time of O(nΔ² log n).
Cite as

Sami Davies, Benjamin Moseley, and Heather Newman. Simultaneously Approximating All 𝓁_p-Norms in Correlation Clustering. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 52:1-52:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{davies_et_al:LIPIcs.ICALP.2024.52,
  author =	{Davies, Sami and Moseley, Benjamin and Newman, Heather},
  title =	{{Simultaneously Approximating All 𝓁\underlinep-Norms in Correlation Clustering}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{52:1--52:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.52},
  URN =		{urn:nbn:de:0030-drops-201950},
  doi =		{10.4230/LIPIcs.ICALP.2024.52},
  annote =	{Keywords: Approximation algorithms, correlation clustering, all-norms, lp-norms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.97
Cut Sparsification and Succinct Representation of Submodular Hypergraphs

Authors: Yotam Kenneth and Robert Krauthgamer

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
In cut sparsification, all cuts of a hypergraph H = (V,E,w) are approximated within 1±ε factor by a small hypergraph H'. This widely applied method was generalized recently to a setting where the cost of cutting each hyperedge e is provided by a splitting function g_e: 2^e → ℝ_+. This generalization is called a submodular hypergraph when the functions {g_e}_{e ∈ E} are submodular, and it arises in machine learning, combinatorial optimization, and algorithmic game theory. Previous work studied the setting where H' is a reweighted sub-hypergraph of H, and measured the size of H' by the number of hyperedges in it. In this setting, we present two results: (i) all submodular hypergraphs admit sparsifiers of size polynomial in n = |V| and ε^{-1}; (ii) we propose a new parameter, called spread, and use it to obtain smaller sparsifiers in some cases. We also show that for a natural family of splitting functions, relaxing the requirement that H' be a reweighted sub-hypergraph of H yields a substantially smaller encoding of the cuts of H (almost a factor n in the number of bits). This is in contrast to graphs, where the most succinct representation is attained by reweighted subgraphs. A new tool in our construction of succinct representation is the notion of deformation, where a splitting function g_e is decomposed into a sum of functions of small description, and we provide upper and lower bounds for deformation of common splitting functions.
Cite as

Yotam Kenneth and Robert Krauthgamer. Cut Sparsification and Succinct Representation of Submodular Hypergraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 97:1-97:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kenneth_et_al:LIPIcs.ICALP.2024.97,
  author =	{Kenneth, Yotam and Krauthgamer, Robert},
  title =	{{Cut Sparsification and Succinct Representation of Submodular Hypergraphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{97:1--97:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.97},
  URN =		{urn:nbn:de:0030-drops-202406},
  doi =		{10.4230/LIPIcs.ICALP.2024.97},
  annote =	{Keywords: Cut Sparsification, Submodular Hypergraphs, Succinct Representation}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.100
Subquadratic Submodular Maximization with a General Matroid Constraint

Authors: Yusuke Kobayashi and Tatsuya Terao

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We consider fast algorithms for monotone submodular maximization with a general matroid constraint. We present a randomized (1 - 1/e - ε)-approximation algorithm that requires Õ_{ε}(√r n) independence oracle and value oracle queries, where n is the number of elements in the matroid and r ≤ n is the rank of the matroid. This improves upon the previously best algorithm by Buchbinder-Feldman-Schwartz [Mathematics of Operations Research 2017] that requires Õ_{ε}(r² + √rn) queries. Our algorithm is based on continuous relaxation, as with other submodular maximization algorithms in the literature. To achieve subquadratic query complexity, we develop a new rounding algorithm, which is our main technical contribution. The rounding algorithm takes as input a point represented as a convex combination of t bases of a matroid and rounds it to an integral solution. Our rounding algorithm requires Õ(r^{3/2} t) independence oracle queries, while the previously best rounding algorithm by Chekuri-Vondrák-Zenklusen [FOCS 2010] requires O(r² t) independence oracle queries. A key idea in our rounding algorithm is to use a directed cycle of arbitrary length in an auxiliary graph, while the algorithm of Chekuri-Vondrák-Zenklusen focused on directed cycles of length two.
Cite as

Yusuke Kobayashi and Tatsuya Terao. Subquadratic Submodular Maximization with a General Matroid Constraint. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 100:1-100:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kobayashi_et_al:LIPIcs.ICALP.2024.100,
  author =	{Kobayashi, Yusuke and Terao, Tatsuya},
  title =	{{Subquadratic Submodular Maximization with a General Matroid Constraint}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{100:1--100:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.100},
  URN =		{urn:nbn:de:0030-drops-202437},
  doi =		{10.4230/LIPIcs.ICALP.2024.100},
  annote =	{Keywords: submodular maximization, matroid constraint, approximation algorithm, rounding algorithm, query complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.115
On the Cut-Query Complexity of Approximating Max-Cut

Authors: Orestis Plevrakis, Seyoon Ragavan, and S. Matthew Weinberg

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We consider the problem of query-efficient global max-cut on a weighted undirected graph in the value oracle model examined by [Rubinstein et al., 2018]. Graph algorithms in this cut query model and other query models have recently been studied for various other problems such as min-cut, connectivity, bipartiteness, and triangle detection. Max-cut in the cut query model can also be viewed as a natural special case of submodular function maximization: on query S ⊆ V, the oracle returns the total weight of the cut between S and V\S. Our first main technical result is a lower bound stating that a deterministic algorithm achieving a c-approximation for any c > 1/2 requires Ω(n) queries. This uses an extension of the cut dimension to rule out approximation (prior work of [Graur et al., 2020] introducing the cut dimension only rules out exact solutions). Secondly, we provide a randomized algorithm with Õ(n) queries that finds a c-approximation for any c < 1. We achieve this using a query-efficient sparsifier for undirected weighted graphs (prior work of [Rubinstein et al., 2018] holds only for unweighted graphs). To complement these results, for most constants c ∈ (0,1], we nail down the query complexity of achieving a c-approximation, for both deterministic and randomized algorithms (up to logarithmic factors). Analogously to general submodular function maximization in the same model, we observe a phase transition at c = 1/2: we design a deterministic algorithm for global c-approximate max-cut in O(log n) queries for any c < 1/2, and show that any randomized algorithm requires Ω(n/log n) queries to find a c-approximate max-cut for any c > 1/2. Additionally, we show that any deterministic algorithm requires Ω(n²) queries to find an exact max-cut (enough to learn the entire graph).
Cite as

Orestis Plevrakis, Seyoon Ragavan, and S. Matthew Weinberg. On the Cut-Query Complexity of Approximating Max-Cut. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 115:1-115:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{plevrakis_et_al:LIPIcs.ICALP.2024.115,
  author =	{Plevrakis, Orestis and Ragavan, Seyoon and Weinberg, S. Matthew},
  title =	{{On the Cut-Query Complexity of Approximating Max-Cut}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{115:1--115:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.115},
  URN =		{urn:nbn:de:0030-drops-202587},
  doi =		{10.4230/LIPIcs.ICALP.2024.115},
  annote =	{Keywords: query complexity, maximum cut, approximation algorithms, graph sparsification}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.52
A Tight Competitive Ratio for Online Submodular Welfare Maximization

Authors: Amit Ganz, Pranav Nuti, and Roy Schwartz

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
In this paper we consider the online Submodular Welfare (SW) problem. In this problem we are given n bidders each equipped with a general non-negative (not necessarily monotone) submodular utility and m items that arrive online. The goal is to assign each item, once it arrives, to a bidder or discard it, while maximizing the sum of utilities. When an adversary determines the items' arrival order we present a simple randomized algorithm that achieves a tight competitive ratio of 1/4. The algorithm is a specialization of an algorithm due to [Harshaw-Kazemi-Feldman-Karbasi MOR`22], who presented the previously best known competitive ratio of 3-2√2≈ 0.171573 to the problem. When the items' arrival order is uniformly random, we present a competitive ratio of ≈ 0.27493, improving the previously known 1/4 guarantee. Our approach for the latter result is based on a better analysis of the (offline) Residual Random Greedy (RRG) algorithm of [Buchbinder-Feldman-Naor-Schwartz SODA`14], which we believe might be of independent interest.
Cite as

Amit Ganz, Pranav Nuti, and Roy Schwartz. A Tight Competitive Ratio for Online Submodular Welfare Maximization. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 52:1-52:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ganz_et_al:LIPIcs.ESA.2023.52,
  author =	{Ganz, Amit and Nuti, Pranav and Schwartz, Roy},
  title =	{{A Tight Competitive Ratio for Online Submodular Welfare Maximization}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{52:1--52:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.52},
  URN =		{urn:nbn:de:0030-drops-187052},
  doi =		{10.4230/LIPIcs.ESA.2023.52},
  annote =	{Keywords: Online Algorithms, Submodular Maximization, Welfare Maximization, Approximation Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.69
An Improved Approximation Algorithm for the Max-3-Section Problem

Authors: Dor Katzelnick, Aditya Pillai, Roy Schwartz, and Mohit Singh

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
We consider the Max--Section problem, where we are given an undirected graph G=(V,E)equipped with non-negative edge weights w: E → R_+ and the goal is to find a partition of V into three equisized parts while maximizing the total weight of edges crossing between different parts. Max-3-Section is closely related to other well-studied graph partitioning problems, e.g., Max-Cut, Max-3-Cut, and Max-Bisection. We present a polynomial time algorithm achieving an approximation of 0.795, that improves upon the previous best known approximation of 0.673. The requirement of multiple parts that have equal sizes renders Max-3-Section much harder to cope with compared to, e.g., Max-Bisection. We show a new algorithm that combines the existing approach of Lassere hierarchy along with a random cut strategy that suffices to give our result.
Cite as

Dor Katzelnick, Aditya Pillai, Roy Schwartz, and Mohit Singh. An Improved Approximation Algorithm for the Max-3-Section Problem. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 69:1-69:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{katzelnick_et_al:LIPIcs.ESA.2023.69,
  author =	{Katzelnick, Dor and Pillai, Aditya and Schwartz, Roy and Singh, Mohit},
  title =	{{An Improved Approximation Algorithm for the Max-3-Section Problem}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{69:1--69:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.69},
  URN =		{urn:nbn:de:0030-drops-187229},
  doi =		{10.4230/LIPIcs.ESA.2023.69},
  annote =	{Keywords: Approximation Algorithms, Semidefinite Programming, Max-Cut, Max-Bisection}
}
Document
DOI: 10.4230/DagRep.12.6.14
Efficient and Equitable Natural Language Processing in the Age of Deep Learning (Dagstuhl Seminar 22232)

Authors: Jesse Dodge, Iryna Gurevych, Roy Schwartz, Emma Strubell, and Betty van Aken

Published in: Dagstuhl Reports, Volume 12, Issue 6 (2023)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 22232 "Efficient and Equitable Natural Language Processing in the Age of Deep Learning". Since 2012, the field of artificial intelligence (AI) has reported remarkable progress on a broad range of capabilities including object recognition, game playing, speech recognition, and machine translation. Much of this progress has been achieved by increasingly large and computationally intensive deep learning models: training costs for state-of-the-art deep learning models have increased 300,000 times between 2012 and 2018 [1]. Perhaps the epitome of this trend is the subfield of natural language processing (NLP) that over the past three years has experienced even sharper growth in model size and corresponding computational requirements in the word embedding approaches (e.g. ELMo, BERT, openGPT-2, Megatron-LM, T5, and GPT-3, one of the largest models ever trained with 175B dense parameters) that are now the basic building blocks of nearly all NLP models. Recent studies indicate that this trend is both environmentally unfriendly and prohibitively expensive, raising barriers to participation in NLP research [2,3]. The goal of this seminar was to mitigate these concerns and promote equity of access in NLP. References. [1] D. Amodei and D. Hernandez. 2018. AI and Compute. https://openai.com/blog/ai-and-compute [2] R. Schwartz, D. Dodge, N. A. Smith, and O. Etzioni. 2020. Green AI. Communications of the ACM (CACM) [3] E. Strubell, A. Ganesh, and A. McCallum. 2019. Energy and Policy Considerations for Deep Learning in NLP. In Proc. of ACL.
Cite as

Jesse Dodge, Iryna Gurevych, Roy Schwartz, Emma Strubell, and Betty van Aken. Efficient and Equitable Natural Language Processing in the Age of Deep Learning (Dagstuhl Seminar 22232). In Dagstuhl Reports, Volume 12, Issue 6, pp. 14-27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{dodge_et_al:DagRep.12.6.14,
  author =	{Dodge, Jesse and Gurevych, Iryna and Schwartz, Roy and Strubell, Emma and van Aken, Betty},
  title =	{{Efficient and Equitable Natural Language Processing in the Age of Deep Learning (Dagstuhl Seminar 22232)}},
  pages =	{14--27},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2023},
  volume =	{12},
  number =	{6},
  editor =	{Dodge, Jesse and Gurevych, Iryna and Schwartz, Roy and Strubell, Emma and van Aken, Betty},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.6.14},
  URN =		{urn:nbn:de:0030-drops-174549},
  doi =		{10.4230/DagRep.12.6.14},
  annote =	{Keywords: deep learning, efficiency, equity, natural language processing (nlp)}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.37
Fair Correlation Clustering in General Graphs

Authors: Roy Schwartz and Roded Zats

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Abstract
We consider the family of Correlation Clustering optimization problems under fairness constraints. In Correlation Clustering we are given a graph whose every edge is labeled either with a + or a -, and the goal is to find a clustering that agrees the most with the labels: + edges within clusters and - edges across clusters. The notion of fairness implies that there is no over, or under, representation of vertices in the clustering: every vertex has a color and the distribution of colors within each cluster is required to be the same as the distribution of colors in the input graph. Previously, approximation algorithms were known only for fair disagreement minimization in complete unweighted graphs. We prove the following: (1) there is no finite approximation for fair disagreement minimization in general graphs unless P = NP (this hardness holds also for bicriteria algorithms); and (2) fair agreement maximization in general graphs admits a bicriteria approximation of ≈ 0.591 (an improved ≈ 0.609 true approximation is given for the special case of two uniformly distributed colors). Our algorithm is based on proving that the sticky Brownian motion rounding of [Abbasi Zadeh-Bansal-Guruganesh-Nikolov-Schwartz-Singh SODA'20] copes well with uncut edges.
Cite as

Roy Schwartz and Roded Zats. Fair Correlation Clustering in General Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 37:1-37:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{schwartz_et_al:LIPIcs.APPROX/RANDOM.2022.37,
  author =	{Schwartz, Roy and Zats, Roded},
  title =	{{Fair Correlation Clustering in General Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{37:1--37:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.37},
  URN =		{urn:nbn:de:0030-drops-171591},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.37},
  annote =	{Keywords: Correlation Clustering, Approximation Algorithms, Semi-Definite Programming}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.46
Fault Tolerant Max-Cut

Authors: Keren Censor-Hillel, Noa Marelly, Roy Schwartz, and Tigran Tonoyan

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
In this work, we initiate the study of fault tolerant Max-Cut, where given an edge-weighted undirected graph G = (V,E), the goal is to find a cut S ⊆ V that maximizes the total weight of edges that cross S even after an adversary removes k vertices from G. We consider two types of adversaries: an adaptive adversary that sees the outcome of the random coin tosses used by the algorithm, and an oblivious adversary that does not. For any constant number of failures k we present an approximation of (0.878-ε) against an adaptive adversary and of α_{GW}≈ 0.8786 against an oblivious adversary (here α_{GW} is the approximation achieved by the random hyperplane algorithm of [Goemans-Williamson J. ACM `95]). Additionally, we present a hardness of approximation of α_{GW} against both types of adversaries, rendering our results (virtually) tight. The non-linear nature of the fault tolerant objective makes the design and analysis of algorithms harder when compared to the classic Max-Cut. Hence, we employ approaches ranging from multi-objective optimization to LP duality and the ellipsoid algorithm to obtain our results.
Cite as

Keren Censor-Hillel, Noa Marelly, Roy Schwartz, and Tigran Tonoyan. Fault Tolerant Max-Cut. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{censorhillel_et_al:LIPIcs.ICALP.2021.46,
  author =	{Censor-Hillel, Keren and Marelly, Noa and Schwartz, Roy and Tonoyan, Tigran},
  title =	{{Fault Tolerant Max-Cut}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{46:1--46:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.46},
  URN =		{urn:nbn:de:0030-drops-141158},
  doi =		{10.4230/LIPIcs.ICALP.2021.46},
  annote =	{Keywords: fault-tolerance, max-cut, approximation}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.49
Maximizing the Correlation: Extending Grothendieck’s Inequality to Large Domains

Authors: Dor Katzelnick and Roy Schwartz

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
Correlation Clustering is an elegant model where given a graph with edges labeled + or -, the goal is to produce a clustering that agrees the most with the labels: + edges should reside within clusters and - edges should cross between clusters. In this work we study the MaxCorr objective, aiming to find a clustering that maximizes the difference between edges classified correctly and incorrectly. We focus on the case of bipartite graphs and present an improved approximation of 0.254, improving upon the known approximation of 0.219 given by Charikar and Wirth [FOCS`2004] and going beyond the 0.2296 barrier imposed by their technique. Our algorithm is inspired by Krivine’s method for bounding Grothendieck’s constant, and we extend this method to allow for more than two clusters in the output. Moreover, our algorithm leads to two additional results: (1) the first known approximation guarantees for MaxCorr where the output is constrained to have a bounded number of clusters; and (2) a natural extension of Grothendieck’s inequality to large domains.
Cite as

Dor Katzelnick and Roy Schwartz. Maximizing the Correlation: Extending Grothendieck’s Inequality to Large Domains. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 49:1-49:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{katzelnick_et_al:LIPIcs.APPROX/RANDOM.2020.49,
  author =	{Katzelnick, Dor and Schwartz, Roy},
  title =	{{Maximizing the Correlation: Extending Grothendieck’s Inequality to Large Domains}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{49:1--49:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.49},
  URN =		{urn:nbn:de:0030-drops-126525},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.49},
  annote =	{Keywords: Correlation Clustering, Grothendieck’s Inequality, Approximation}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.53
Approximating Requirement Cut via a Configuration LP

Authors: Roy Schwartz and Yotam Sharoni

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
We consider the {Requirement Cut} problem, where given an undirected graph G = (V,E) equipped with non-negative edge weights c:E → R_{+}, and g groups of vertices X₁,…,X_{g} ⊆ V each equipped with a requirement r_i, the goal is to find a collection of edges F ⊆ E, with total minimum weight, such that once F is removed from G in the resulting graph every X_{i} is broken into at least r_{i} connected components. {Requirement Cut} captures multiple classic cut problems in graphs, e.g., {Multicut}, {Multiway Cut}, {Min k-Cut}, {Steiner k-Cut}, {Steiner Multicut}, and {Multi-Multiway Cut}. Nagarajan and Ravi [Algoritmica`10] presented an approximation of O(log{n}log{R}) for the problem, which was subsequently improved to O(log{g} log{k}) by Gupta, Nagarajan and Ravi [Operations Research Letters`10] (here R = ∑ _{i = 1}^g r_i and k = |∪ _{i = 1}^g X_i |). We present an approximation of O(Xlog{R} √{log{k}}log{log{k}}) for {Requirement Cut} (here X = max _{i = 1,…,g} {|X_i|}). Our approximation in general is incomparable to the above mentioned previous results, however when all groups are not too large, i.e., X = o((√{log{k}}log{g})/(log{R}log{log{k}})), it is better. Our algorithm is based on a new configuration linear programming relaxation for the problem, which is accompanied by a remarkably simple randomized rounding procedure.
Cite as

Roy Schwartz and Yotam Sharoni. Approximating Requirement Cut via a Configuration LP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{schwartz_et_al:LIPIcs.APPROX/RANDOM.2020.53,
  author =	{Schwartz, Roy and Sharoni, Yotam},
  title =	{{Approximating Requirement Cut via a Configuration LP}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{53:1--53:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.53},
  URN =		{urn:nbn:de:0030-drops-126560},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.53},
  annote =	{Keywords: Approximation, Requirement Cut, Sparsest Cut, Metric Embedding}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2019.27
Online and Offline Algorithms for Circuit Switch Scheduling

Authors: Roy Schwartz, Mohit Singh, and Sina Yazdanbod

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Abstract
Motivated by the use of high speed circuit switches in large scale data centers, we consider the problem of circuit switch scheduling. In this problem we are given demands between pairs of servers and the goal is to schedule at every time step a matching between the servers while maximizing the total satisfied demand over time. The crux of this scheduling problem is that once one shifts from one matching to a different one a fixed delay delta is incurred during which no data can be transmitted. For the offline version of the problem we present a (1-(1/e)-epsilon) approximation ratio (for any constant epsilon >0). Since the natural linear programming relaxation for the problem has an unbounded integrality gap, we adopt a hybrid approach that combines the combinatorial greedy with randomized rounding of a different suitable linear program. For the online version of the problem we present a (bi-criteria) ((e-1)/(2e-1)-epsilon)-competitive ratio (for any constant epsilon >0 ) that exceeds time by an additive factor of O(delta/epsilon). We note that no uni-criteria online algorithm is possible. Surprisingly, we obtain the result by reducing the online version to the offline one.
Cite as

Roy Schwartz, Mohit Singh, and Sina Yazdanbod. Online and Offline Algorithms for Circuit Switch Scheduling. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{schwartz_et_al:LIPIcs.FSTTCS.2019.27,
  author =	{Schwartz, Roy and Singh, Mohit and Yazdanbod, Sina},
  title =	{{Online and Offline Algorithms for Circuit Switch Scheduling}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{27:1--27:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.27},
  URN =		{urn:nbn:de:0030-drops-115893},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.27},
  annote =	{Keywords: approximation algorithm, online, matching, scheduling}
}
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