3 Search Results for "Graur, Andrei"

Document
DOI: 10.4230/LIPIcs.CCC.2021.15
On the Cut Dimension of a Graph

Authors: Troy Lee, Tongyang Li, Miklos Santha, and Shengyu Zhang

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
Let G = (V,w) be a weighted undirected graph with m edges. The cut dimension of G is the dimension of the span of the characteristic vectors of the minimum cuts of G, viewed as vectors in {0,1}^m. For every n ≥ 2 we show that the cut dimension of an n-vertex graph is at most 2n-3, and construct graphs realizing this bound. The cut dimension was recently defined by Graur et al. [Andrei Graur et al., 2020], who show that the maximum cut dimension of an n-vertex graph is a lower bound on the number of cut queries needed by a deterministic algorithm to solve the minimum cut problem on n-vertex graphs. For every n ≥ 2, Graur et al. exhibit a graph on n vertices with cut dimension at least 3n/2 -2, giving the first lower bound larger than n on the deterministic cut query complexity of computing mincut. We observe that the cut dimension is even a lower bound on the number of linear queries needed by a deterministic algorithm to solve mincut, where a linear query can ask any vector x ∈ ℝ^{binom(n,2)} and receives the answer w^T x. Our results thus show a lower bound of 2n-3 on the number of linear queries needed by a deterministic algorithm to solve minimum cut on n-vertex graphs, and imply that one cannot show a lower bound larger than this via the cut dimension. We further introduce a generalization of the cut dimension which we call the 𝓁₁-approximate cut dimension. The 𝓁₁-approximate cut dimension is also a lower bound on the number of linear queries needed by a deterministic algorithm to compute minimum cut. It is always at least as large as the cut dimension, and we construct an infinite family of graphs on n = 3k+1 vertices with 𝓁₁-approximate cut dimension 2n-2, showing that it can be strictly larger than the cut dimension.
Cite as

Troy Lee, Tongyang Li, Miklos Santha, and Shengyu Zhang. On the Cut Dimension of a Graph. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 15:1-15:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{lee_et_al:LIPIcs.CCC.2021.15,
  author =	{Lee, Troy and Li, Tongyang and Santha, Miklos and Zhang, Shengyu},
  title =	{{On the Cut Dimension of a Graph}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{15:1--15:35},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.15},
  URN =		{urn:nbn:de:0030-drops-142890},
  doi =		{10.4230/LIPIcs.CCC.2021.15},
  annote =	{Keywords: Query complexity, submodular function minimization, cut dimension}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.14
Efficient Splitting of Necklaces

Authors: Noga Alon and Andrei Graur

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

Abstract
We provide efficient approximation algorithms for the Necklace Splitting problem. The input consists of a sequence of beads of n types and an integer k. The objective is to split the necklace, with a small number of cuts made between consecutive beads, and distribute the resulting intervals into k collections so that the discrepancy between the shares of any two collections, according to each type, is at most 1. We also consider an approximate version where each collection should contain at least a (1-ε)/k and at most a (1+ε)/k fraction of the beads of each type. It is known that there is always a solution making at most n(k-1) cuts, and this number of cuts is optimal in general. The proof is topological and provides no efficient procedure for finding these cuts. It is also known that for k = 2, and some fixed positive ε, finding a solution with n cuts is PPAD-hard. We describe an efficient algorithm that produces an ε-approximate solution for k = 2 making n (2+log (1/ε)) cuts. This is an exponential improvement of a (1/ε)^O(n) bound of Bhatt and Leighton from the 80s. We also present an online algorithm for the problem (in its natural online model), in which the number of cuts made to produce discrepancy at most 1 on each type is Õ(m^{2/3} n), where m is the maximum number of beads of any type. Lastly, we establish a lower bound showing that for the online setup this is tight up to logarithmic factors. Similar results are obtained for k > 2.
Cite as

Noga Alon and Andrei Graur. Efficient Splitting of Necklaces. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{alon_et_al:LIPIcs.ICALP.2021.14,
  author =	{Alon, Noga and Graur, Andrei},
  title =	{{Efficient Splitting of Necklaces}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{14:1--14:17},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.14},
  URN =		{urn:nbn:de:0030-drops-140832},
  doi =		{10.4230/LIPIcs.ICALP.2021.14},
  annote =	{Keywords: necklace splitting, necklace halving, approximation algorithms, online algorithms, discrepancy}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.64
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)

Copy BibTex To Clipboard

@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-dev.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}
}
  • Refine by Author
  • 2 Graur, Andrei
  • 1 Alon, Noga
  • 1 Lee, Troy
  • 1 Li, Tongyang
  • 1 Pollner, Tristan
  • Show More...

  • Refine by Classification
  • 1 Mathematics of computing → Graph theory
  • 1 Theory of computation → Approximation algorithms analysis
  • 1 Theory of computation → Computational complexity and cryptography
  • 1 Theory of computation → Graph algorithms analysis
  • 1 Theory of computation → Submodular optimization and polymatroids

  • Refine by Keyword
  • 1 Query complexity
  • 1 approximation algorithms
  • 1 cut dimension
  • 1 discrepancy
  • 1 min cut
  • Show More...

  • Refine by Type
  • 3 document

  • Refine by Publication Year
  • 2 2021
  • 1 2020

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact