3 Search Results for "Opršal, Jakub"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.37
Conditional Dichotomy of Boolean Ordered Promise CSPs

Authors: Joshua Brakensiek, Venkatesan Guruswami, and Sai Sandeep

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

Abstract
Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their formal introduction by Austrin, Guruswami, and Håstad [Per Austrin et al., 2017], there has been a flurry of works on PCSPs, including recent breakthroughs in approximate graph coloring [Barto et al., 2018; Andrei A. Krokhin and Jakub Opršal, 2019; Marcin Wrochna and Stanislav Zivný, 2020]. The key tool in studying PCSPs is the algebraic framework developed in the context of CSPs where the closure properties of the satisfying solutions known as polymorphisms are analyzed. The polymorphisms of PCSPs are significantly richer than CSPs - even in the Boolean case, we still do not know if there exists a dichotomy result for PCSPs analogous to Schaefer’s dichotomy result [Thomas J. Schaefer, 1978] for CSPs. In this paper, we study a special case of Boolean PCSPs, namely Boolean Ordered PCSPs where the Boolean PCSPs have the predicate x ≤ y. In the algebraic framework, this is the special case of Boolean PCSPs when the polymorphisms are monotone functions. We prove that Boolean Ordered PCSPs exhibit a computational dichotomy assuming the Rich 2-to-1 Conjecture [Mark Braverman et al., 2021] which is a perfect completeness surrogate of the Unique Games Conjecture. In particular, assuming the Rich 2-to-1 Conjecture, we prove that a Boolean Ordered PCSP can be solved in polynomial time if for every ε > 0, it has polymorphisms where each coordinate has Shapley value at most ε, else it is NP-hard. The algorithmic part of our dichotomy result is based on a structural lemma showing that Boolean monotone functions with each coordinate having low Shapley value have arbitrarily large threshold functions as minors. The hardness part proceeds by showing that the Shapley value is consistent under a uniformly random 2-to-1 minor. As a structural result of independent interest, we construct an example to show that the Shapley value can be inconsistent under an adversarial 2-to-1 minor.
Cite as

Joshua Brakensiek, Venkatesan Guruswami, and Sai Sandeep. Conditional Dichotomy of Boolean Ordered Promise CSPs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 37:1-37:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brakensiek_et_al:LIPIcs.ICALP.2021.37,
  author =	{Brakensiek, Joshua and Guruswami, Venkatesan and Sandeep, Sai},
  title =	{{Conditional Dichotomy of Boolean Ordered Promise CSPs}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{37:1--37:15},
  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.2021.37},
  URN =		{urn:nbn:de:0030-drops-141060},
  doi =		{10.4230/LIPIcs.ICALP.2021.37},
  annote =	{Keywords: promise constraint satisfaction, Boolean ordered PCSP, Shapley value, rich 2-to-1 conjecture, random minor}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.34
Revisiting Alphabet Reduction in Dinur’s PCP

Authors: Venkatesan Guruswami, Jakub Opršal, and Sai Sandeep

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

Abstract
Dinur’s celebrated proof of the PCP theorem alternates two main steps in several iterations: gap amplification to increase the soundness gap by a large constant factor (at the expense of much larger alphabet size), and a composition step that brings back the alphabet size to an absolute constant (at the expense of a fixed constant factor loss in the soundness gap). We note that the gap amplification can produce a Label Cover CSP. This allows us to reduce the alphabet size via a direct long-code based reduction from Label Cover to a Boolean CSP. Our composition step thus bypasses the concept of Assignment Testers from Dinur’s proof, and we believe it is more intuitive - it is just a gadget reduction. The analysis also uses only elementary facts (Parseval’s identity) about Fourier Transforms over the hypercube.
Cite as

Venkatesan Guruswami, Jakub Opršal, and Sai Sandeep. Revisiting Alphabet Reduction in Dinur’s PCP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 34:1-34:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{guruswami_et_al:LIPIcs.APPROX/RANDOM.2020.34,
  author =	{Guruswami, Venkatesan and Opr\v{s}al, Jakub and Sandeep, Sai},
  title =	{{Revisiting Alphabet Reduction in Dinur’s PCP}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{34:1--34:14},
  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.2020.34},
  URN =		{urn:nbn:de:0030-drops-126372},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.34},
  annote =	{Keywords: PCP theorem, CSP, discrete Fourier analysis, label cover, long code}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.15
Rainbow Coloring Hardness via Low Sensitivity Polymorphisms

Authors: Venkatesan Guruswami and Sai Sandeep

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
A k-uniform hypergraph is said to be r-rainbow colorable if there is an r-coloring of its vertices such that every hyperedge intersects all r color classes. Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k >= 3 and r >= 2. Therefore, one settles for finding a rainbow coloring with fewer colors (which is an easier task). When r=k (the maximum possible value), i.e., the hypergraph is k-partite, one can efficiently 2-rainbow color the hypergraph, i.e., 2-color its vertices so that there are no monochromatic edges. In this work we consider the next smaller value of r=k-1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := ceil[(k-2)/2] colors. In particular, for k <=6, it is NP-hard to 2-color (k-1)-rainbow colorable k-uniform hypergraphs. Our proof follows the algebraic approach to promise constraint satisfaction problems. It proceeds by characterizing the polymorphisms associated with the approximate rainbow coloring problem, which are rainbow colorings of some product hypergraphs on vertex set [r]^n. We prove that any such polymorphism f: [r]^n -> [q] must be C-fixing, i.e., there is a small subset S of C coordinates and a setting a in [q]^S such that fixing x_{|S} = a determines the value of f(x). The key step in our proof is bounding the sensitivity of certain rainbow colorings, thereby arguing that they must be juntas. Armed with the C-fixing characterization, our NP-hardness is obtained via a reduction from smooth Label Cover.
Cite as

Venkatesan Guruswami and Sai Sandeep. Rainbow Coloring Hardness via Low Sensitivity Polymorphisms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 15:1-15:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{guruswami_et_al:LIPIcs.APPROX-RANDOM.2019.15,
  author =	{Guruswami, Venkatesan and Sandeep, Sai},
  title =	{{Rainbow Coloring Hardness via Low Sensitivity Polymorphisms}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.15},
  URN =		{urn:nbn:de:0030-drops-112303},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.15},
  annote =	{Keywords: inapproximability, hardness of approximation, constraint satisfaction, hypergraph coloring, polymorphisms}
}
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