10 Search Results for "Harvey, Nicholas J. A."


Document
Scalable Hard Instances for Independent Set Reconfiguration

Authors: Takehide Soh, Takumu Watanabe, Jun Kawahara, Akira Suzuki, and Takehiro Ito

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)


Abstract
The Token Jumping problem, also known as the independent set reconfiguration problem under the token jumping model, is defined as follows: Given a graph and two same-sized independent sets, determine whether one can be transformed into the other via a sequence of independent sets. Token Jumping has been extensively studied, mainly from the viewpoint of algorithmic theory, but its practical study has just begun. To develop a practically good solver, it is important to construct benchmark datasets that are scalable and hard. Here, "scalable" means the ability to change the scale of the instance while maintaining its characteristics by adjusting the given parameters; and "hard" means that the instance can become so difficult that it cannot be solved within a practical time frame by a solver. In this paper, we propose four types of instance series for Token Jumping. Our instance series is scalable in the sense that instance scales are controlled by the number of vertices. To establish their hardness, we focus on the numbers of transformation steps; our instance series requires exponential numbers of steps with respect to the number of vertices. Interestingly, three types of instance series are constructed by importing theories developed by algorithmic research. We experimentally evaluate the scalability and hardness of the proposed instance series, using the SAT solver and award-winning solvers of the international competition for Token Jumping.

Cite as

Takehide Soh, Takumu Watanabe, Jun Kawahara, Akira Suzuki, and Takehiro Ito. Scalable Hard Instances for Independent Set Reconfiguration. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{soh_et_al:LIPIcs.SEA.2024.26,
  author =	{Soh, Takehide and Watanabe, Takumu and Kawahara, Jun and Suzuki, Akira and Ito, Takehiro},
  title =	{{Scalable Hard Instances for Independent Set Reconfiguration}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.26},
  URN =		{urn:nbn:de:0030-drops-203913},
  doi =		{10.4230/LIPIcs.SEA.2024.26},
  annote =	{Keywords: Combinatorial reconfiguration, Benckmark dataset, Graph Algorithm, PSPACE-complete}
}
Document
Track A: Algorithms, Complexity and Games
On the Streaming Complexity of Expander Decomposition

Authors: Yu Chen, Michael Kapralov, Mikhail Makarov, and Davide Mazzali

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


Abstract
In this paper we study the problem of finding (ε, ϕ)-expander decompositions of a graph in the streaming model, in particular for dynamic streams of edge insertions and deletions. The goal is to partition the vertex set so that every component induces a ϕ-expander, while the number of inter-cluster edges is only an ε fraction of the total volume. It was recently shown that there exists a simple algorithm to construct a (O(ϕ log n), ϕ)-expander decomposition of an n-vertex graph using Õ(n/ϕ²) bits of space [Filtser, Kapralov, Makarov, ITCS'23]. This result calls for understanding the extent to which a dependence in space on the sparsity parameter ϕ is inherent. We move towards answering this question on two fronts. We prove that a (O(ϕ log n), ϕ)-expander decomposition can be found using Õ(n) space, for every ϕ. At the core of our result is the first streaming algorithm for computing boundary-linked expander decompositions, a recently introduced strengthening of the classical notion [Goranci et al., SODA'21]. The key advantage is that a classical sparsifier [Fung et al., STOC'11], with size independent of ϕ, preserves the cuts inside the clusters of a boundary-linked expander decomposition within a multiplicative error. Notable algorithmic applications use sequences of expander decompositions, in particular one often repeatedly computes a decomposition of the subgraph induced by the inter-cluster edges (e.g., the seminal work of Spielman and Teng on spectral sparsifiers [Spielman, Teng, SIAM Journal of Computing 40(4)], or the recent maximum flow breakthrough [Chen et al., FOCS'22], among others). We prove that any streaming algorithm that computes a sequence of (O(ϕ log n), ϕ)-expander decompositions requires Ω̃(n/ϕ) bits of space, even in insertion only streams.

Cite as

Yu Chen, Michael Kapralov, Mikhail Makarov, and Davide Mazzali. On the Streaming Complexity of Expander Decomposition. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{chen_et_al:LIPIcs.ICALP.2024.46,
  author =	{Chen, Yu and Kapralov, Michael and Makarov, Mikhail and Mazzali, Davide},
  title =	{{On the Streaming Complexity of Expander Decomposition}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{46:1--46:20},
  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.2024.46},
  URN =		{urn:nbn:de:0030-drops-201890},
  doi =		{10.4230/LIPIcs.ICALP.2024.46},
  annote =	{Keywords: Graph Sketching, Dynamic Streaming, Expander Decomposition}
}
Document
Track A: Algorithms, Complexity and Games
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
Optimal PSPACE-Hardness of Approximating Set Cover Reconfiguration

Authors: Shuichi Hirahara and Naoto Ohsaka

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


Abstract
In the Minmax Set Cover Reconfiguration problem, given a set system ℱ over a universe 𝒰 and its two covers 𝒞^start and 𝒞^goal of size k, we wish to transform 𝒞^start into 𝒞^goal by repeatedly adding or removing a single set of ℱ while covering the universe 𝒰 in any intermediate state. Then, the objective is to minimize the maximum size of any intermediate cover during transformation. We prove that Minmax Set Cover Reconfiguration and Minmax Dominating Set Reconfiguration are PSPACE-hard to approximate within a factor of 2-(1/polyloglog N), where N is the size of the universe and the number of vertices in a graph, respectively, improving upon Ohsaka (SODA 2024) [Ohsaka, 2024] and Karthik C. S. and Manurangsi (2023) [Karthik C. S. and Manurangsi, 2023]. This is the first result that exhibits a sharp threshold for the approximation factor of any reconfiguration problem because both problems admit a 2-factor approximation algorithm as per Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno (Theor. Comput. Sci., 2011) [Takehiro Ito et al., 2011]. Our proof is based on a reconfiguration analogue of the FGLSS reduction [Feige et al., 1996] from Probabilistically Checkable Reconfiguration Proofs of Hirahara and Ohsaka (STOC 2024) [Hirahara and Ohsaka, 2024]. We also prove that for any constant ε ∈ (0,1), Minmax Hypergraph Vertex Cover Reconfiguration on poly(ε^-1)-uniform hypergraphs is PSPACE-hard to approximate within a factor of 2-ε.

Cite as

Shuichi Hirahara and Naoto Ohsaka. Optimal PSPACE-Hardness of Approximating Set Cover Reconfiguration. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 85:1-85:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{hirahara_et_al:LIPIcs.ICALP.2024.85,
  author =	{Hirahara, Shuichi and Ohsaka, Naoto},
  title =	{{Optimal PSPACE-Hardness of Approximating Set Cover Reconfiguration}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{85:1--85:18},
  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.2024.85},
  URN =		{urn:nbn:de:0030-drops-202283},
  doi =		{10.4230/LIPIcs.ICALP.2024.85},
  annote =	{Keywords: reconfiguration problems, hardness of approximation, probabilistic proof systems, FGLSS reduction}
}
Document
Track A: Algorithms, Complexity and Games
Problems on Group-Labeled Matroid Bases

Authors: Florian Hörsch, András Imolay, Ryuhei Mizutani, Taihei Oki, and Tamás Schwarcz

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


Abstract
Consider a matroid equipped with a labeling of its ground set to an abelian group. We define the label of a subset of the ground set as the sum of the labels of its elements. We study a collection of problems on finding bases and common bases of matroids with restrictions on their labels. For zero bases and zero common bases, the results are mostly negative. While finding a non-zero basis of a matroid is not difficult, it turns out that the complexity of finding a non-zero common basis depends on the group. Namely, we show that the problem is hard for a fixed group if it contains an element of order two, otherwise it is polynomially solvable. As a generalization of both zero and non-zero constraints, we further study F-avoiding constraints where we seek a basis or common basis whose label is not in a given set F of forbidden labels. Using algebraic techniques, we give a randomized algorithm for finding an F-avoiding common basis of two matroids represented over the same field for finite groups given as operation tables. The study of F-avoiding bases with groups given as oracles leads to a conjecture stating that whenever an F-avoiding basis exists, an F-avoiding basis can be obtained from an arbitrary basis by exchanging at most |F| elements. We prove the conjecture for the special cases when |F| ≤ 2 or the group is ordered. By relying on structural observations on matroids representable over fixed, finite fields, we verify a relaxed version of the conjecture for these matroids. As a consequence, we obtain a polynomial-time algorithm in these special cases for finding an F-avoiding basis when |F| is fixed.

Cite as

Florian Hörsch, András Imolay, Ryuhei Mizutani, Taihei Oki, and Tamás Schwarcz. Problems on Group-Labeled Matroid Bases. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 86:1-86:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{horsch_et_al:LIPIcs.ICALP.2024.86,
  author =	{H\"{o}rsch, Florian and Imolay, Andr\'{a}s and Mizutani, Ryuhei and Oki, Taihei and Schwarcz, Tam\'{a}s},
  title =	{{Problems on Group-Labeled Matroid Bases}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{86:1--86:20},
  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.2024.86},
  URN =		{urn:nbn:de:0030-drops-202299},
  doi =		{10.4230/LIPIcs.ICALP.2024.86},
  annote =	{Keywords: matroids, matroid intersection, congruency constraint, exact-weight constraint, additive combinatorics, algebraic algorithm, strongly base orderability}
}
Document
Track A: Algorithms, Complexity and Games
Alphabet Reduction for Reconfiguration Problems

Authors: Naoto Ohsaka

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


Abstract
We present a reconfiguration analogue of alphabet reduction à la Dinur (J. ACM, 2007) and its applications. Given a binary constraint graph G and its two satisfying assignments ψ^ini and ψ^tar, the Maxmin 2-CSP Reconfiguration problem requests to transform ψ^ini into ψ^tar by repeatedly changing the value of a single vertex so that the minimum fraction of satisfied edges is maximized. We demonstrate a polynomial-time reduction from Maxmin 2-CSP Reconfiguration with arbitrarily large alphabet size W ∈ ℕ to itself with universal alphabet size W₀ ∈ ℕ such that 1) the perfect completeness is preserved, and 2) if any reconfiguration for the former violates ε-fraction of edges, then Ω(ε)-fraction of edges must be unsatisfied during any reconfiguration for the latter. The crux of its construction is the reconfigurability of Hadamard codes, which enables to reconfigure between a pair of codewords, while avoiding getting too close to the other codewords. Combining this alphabet reduction with gap amplification due to Ohsaka (SODA 2024), we are able to amplify the 1 vs. 1-ε gap for arbitrarily small ε ∈ (0,1) up to the 1 vs. 1-ε₀ for some universal ε₀ ∈ (0,1) without blowing up the alphabet size. In particular, a 1 vs. 1-ε₀ gap version of Maxmin 2-CSP Reconfiguration with alphabet size W₀ is PSPACE-hard given a probabilistically checkable reconfiguration proof system having any soundness error 1-ε due to Hirahara and Ohsaka (STOC 2024) and Karthik C. S. and Manurangsi (2023). As an immediate corollary, we show that there exists a universal constant ε₀ ∈ (0,1) such that many popular reconfiguration problems are PSPACE-hard to approximate within a factor of 1-ε₀, including those of 3-SAT, Independent Set, Vertex Cover, Clique, Dominating Set, and Set Cover. This may not be achieved only by gap amplification of Ohsaka, which makes the alphabet size gigantic depending on ε^-1.

Cite as

Naoto Ohsaka. Alphabet Reduction for Reconfiguration Problems. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 113:1-113:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{ohsaka:LIPIcs.ICALP.2024.113,
  author =	{Ohsaka, Naoto},
  title =	{{Alphabet Reduction for Reconfiguration Problems}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{113:1--113:17},
  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.2024.113},
  URN =		{urn:nbn:de:0030-drops-202560},
  doi =		{10.4230/LIPIcs.ICALP.2024.113},
  annote =	{Keywords: reconfiguration problems, hardness of approximation, Hadamard codes, alphabet reduction}
}
Document
Track A: Algorithms, Complexity and Games
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
New Query Lower Bounds for Submodular Function Minimization

Authors: Andrei Graur, Tristan Pollner, Vidhya Ramaswamy, and S. Matthew Weinberg

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)


Abstract
We consider submodular function minimization in the oracle model: given black-box access to a submodular set function f:2^[n] → ℝ, find an element of arg min_S {f(S)} using as few queries to f(⋅) as possible. State-of-the-art algorithms succeed with Õ(n²) queries [Yin Tat Lee et al., 2015], yet the best-known lower bound has never been improved beyond n [Nicholas J. A. Harvey, 2008]. We provide a query lower bound of 2n for submodular function minimization, a 3n/2-2 query lower bound for the non-trivial minimizer of a symmetric submodular function, and a binom{n}{2} query lower bound for the non-trivial minimizer of an asymmetric submodular function. Our 3n/2-2 lower bound results from a connection between SFM lower bounds and a novel concept we term the cut dimension of a graph. Interestingly, this yields a 3n/2-2 cut-query lower bound for finding the global mincut in an undirected, weighted graph, but we also prove it cannot yield a lower bound better than n+1 for s-t mincut, even in a directed, weighted graph.

Cite as

Andrei Graur, Tristan Pollner, Vidhya Ramaswamy, and S. Matthew Weinberg. New Query Lower Bounds for Submodular Function Minimization. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{graur_et_al:LIPIcs.ITCS.2020.64,
  author =	{Graur, Andrei and Pollner, Tristan and Ramaswamy, Vidhya and Weinberg, S. Matthew},
  title =	{{New Query Lower Bounds for Submodular Function Minimization}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{64:1--64:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.64},
  URN =		{urn:nbn:de:0030-drops-117493},
  doi =		{10.4230/LIPIcs.ITCS.2020.64},
  annote =	{Keywords: submodular functions, query lower bounds, min cut}
}
Document
Approximating Hit Rate Curves using Streaming Algorithms

Authors: Zachary Drudi, Nicholas J. A. Harvey, Stephen Ingram, Andrew Warfield, and Jake Wires

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
A hit rate curve is a function that maps cache size to the proportion of requests that can be served from the cache. (The caching policy and sequence of requests are assumed to be fixed.) Hit rate curves have been studied for decades in the operating system, database and computer architecture communities. They are useful tools for designing appropriate cache sizes, dynamically allocating memory between competing caches, and for summarizing locality properties of the request sequence. In this paper we focus on the widely-used LRU caching policy. Computing hit rate curves is very efficient from a runtime standpoint, but existing algorithms are not efficient in their space usage. For a stream of m requests for n cacheable objects, all existing algorithms that provably compute the hit rate curve use space linear in n. In the context of modern storage systems, n can easily be in the billions or trillions, so the space usage of these algorithms makes them impractical. We present the first algorithm for provably approximating hit rate curves for the LRU policy with sublinear space. Our algorithm uses O( p^2 * log(n) * log^2(m) / epsilon^2 ) bits of space and approximates the hit rate curve at p uniformly-spaced points to within additive error epsilon. This is not far from optimal. Any single-pass algorithm with the same guarantees must use Omega(p^2 + epsilon^{-2} + log(n)) bits of space. Furthermore, our use of additive error is necessary. Any single-pass algorithm achieving multiplicative error requires Omega(n) bits of space.

Cite as

Zachary Drudi, Nicholas J. A. Harvey, Stephen Ingram, Andrew Warfield, and Jake Wires. Approximating Hit Rate Curves using Streaming Algorithms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 225-241, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{drudi_et_al:LIPIcs.APPROX-RANDOM.2015.225,
  author =	{Drudi, Zachary and Harvey, Nicholas J. A. and Ingram, Stephen and Warfield, Andrew and Wires, Jake},
  title =	{{Approximating Hit Rate Curves using Streaming Algorithms}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{225--241},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.225},
  URN =		{urn:nbn:de:0030-drops-53056},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.225},
  annote =	{Keywords: Cache analysis, hit rate curves, miss rate curves, streaming algorithms}
}
Document
Discrepancy Without Partial Colorings

Authors: Nicholas J. A. Harvey, Roy Schwartz, and Mohit Singh

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)


Abstract
Spencer's theorem asserts that, for any family of n subsets of ground set of size n, the elements of the ground set can be "colored" by the values +1 or -1 such that the sum of every set is O(sqrt(n)) in absolute value. All existing proofs of this result recursively construct "partial colorings", which assign +1 or -1 values to half of the ground set. We devise the first algorithm for Spencer's theorem that directly computes a coloring, without recursively computing partial colorings.

Cite as

Nicholas J. A. Harvey, Roy Schwartz, and Mohit Singh. Discrepancy Without Partial Colorings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 258-273, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


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@InProceedings{harvey_et_al:LIPIcs.APPROX-RANDOM.2014.258,
  author =	{Harvey, Nicholas J. A. and Schwartz, Roy and Singh, Mohit},
  title =	{{Discrepancy Without Partial Colorings}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{258--273},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.258},
  URN =		{urn:nbn:de:0030-drops-47014},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.258},
  annote =	{Keywords: Combinatorial Discrepancy, Brownian Motion, Semi-Definite Programming, Randomized Algorithm}
}
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