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Documents authored by Brakensiek, Joshua

Document
DOI: 10.4230/LIPIcs.CP.2026.8
Classification of Non-Redundancy of Boolean Predicates of Arity 4

Authors: Joshua Brakensiek, Venkatesan Guruswami, and Aaron Putterman

Published in: LIPIcs, Volume 379, 32nd International Conference on Principles and Practice of Constraint Programming (CP 2026)

Abstract
Given a constraint satisfaction problem (CSP) predicate P ⊆ D^r, the non-redundancy (NRD) of P is the maximum-sized instance on n variables such that for every clause of the instance, there is an assignment which satisfies all clauses but that one. The study of NRD for various CSPs is an active area of research which combines ideas from extremal combinatorics, logic, lattice theory, and other techniques. Complete classifications are known in the cases r = 2 and (|D| = 2, r = 3). In this paper, we give a near-complete classification of the case (|D| = 2, r = 4). Of the 400 distinct non-trivial Boolean predicates of arity 4, we implement an algorithmic procedure which perfectly classifies 397 of them. Of the remaining three, we solve two by reducing to extremal combinatorics problems - leaving the last one as an open question. Along the way, we identify the first Boolean predicate whose non-redundancy asymptotics are non-polynomial.
Cite as

Joshua Brakensiek, Venkatesan Guruswami, and Aaron Putterman. Classification of Non-Redundancy of Boolean Predicates of Arity 4. In 32nd International Conference on Principles and Practice of Constraint Programming (CP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 379, pp. 8:1-8:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brakensiek_et_al:LIPIcs.CP.2026.8,
  author =	{Brakensiek, Joshua and Guruswami, Venkatesan and Putterman, Aaron},
  title =	{{Classification of Non-Redundancy of Boolean Predicates of Arity 4}},
  booktitle =	{32nd International Conference on Principles and Practice of Constraint Programming (CP 2026)},
  pages =	{8:1--8:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-432-1},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{379},
  editor =	{Beldiceanu, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2026.8},
  URN =		{urn:nbn:de:0030-drops-266412},
  doi =		{10.4230/LIPIcs.CP.2026.8},
  annote =	{Keywords: constraint satisfaction problem, redundancy}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2026.43
Unique Decoding of Reed–Solomon and Related Codes for Semi-Adversarial Errors

Authors: Joshua Brakensiek, Yeyuan Chen, Manik Dhar, and Zihan Zhang

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract
Motivated by recent developments in coding theory, particular in list-decoding, we introduce a new error model which we call semi-adversarial errors. This error model bridges between fully random errors and fully adversarial errors by allowing some symbols of a message to be corrupted by an adversary while others are replaced with uniformly random symbols. As our main quest, we seek to understand optimal efficient unique decoding algorithms in the semi-adversarial model. For interleaved Reed-Solomon (IRS), folded Reed-Solomon (FRS) and univariate multiplicity codes, we design decoding algorithms running in near-linear time for most mixtures of random and adversarial errors. Our analysis matches the information-theoretic optimum for semi-adversarial errors. Our algorithm for interleaved Reed-Solomon codes is an improved implementation of the decoding algorithm by Bleichenbacher-Kiayias-Yung (BKY) for fully random errors. We use a novel monomial-tracking technique to analyze its performance in this new semi-adversarial errors. Inspired by the BKY algorithm, we use novel interpolations to extend our approach to the settings of folded Reed-Solomon and multiplicity codes, resulting in fast algorithms for unique decoding against semi-adversarial errors. Our new decoders for FRS and multiplicity codes replace the sophisticated root-finding step in traditional algorithms, such as the Guruswami-Wang algorithm, with a straightforward polynomial long division. Analysis of these algorithms requires more robust monomial-tracking arguments than IRS codes.
Cite as

Joshua Brakensiek, Yeyuan Chen, Manik Dhar, and Zihan Zhang. Unique Decoding of Reed–Solomon and Related Codes for Semi-Adversarial Errors. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 43:1-43:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brakensiek_et_al:LIPIcs.ICALP.2026.43,
  author =	{Brakensiek, Joshua and Chen, Yeyuan and Dhar, Manik and Zhang, Zihan},
  title =	{{Unique Decoding of Reed–Solomon and Related Codes for Semi-Adversarial Errors}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{43:1--43:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.43},
  URN =		{urn:nbn:de:0030-drops-264326},
  doi =		{10.4230/LIPIcs.ICALP.2026.43},
  annote =	{Keywords: coding theory, interleaved codes, Reed-Solomon codes, semi-random models, unique decoding}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2026.44
Multiplicative Error Set System Sparsification: A Simpler Proof via Chain Length Contraction

Authors: Joshua Brakensiek, Venkatesan Guruswami, and Aaron Putterman

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract
The chain length of a set family 𝒮 ⊆ 2^[m] is the largest ascending sequence of sets in containment order in the union-closure of S. In this work, we provide a significantly simpler and more optimal characterization of the sparsifiability of set systems in terms of their chain length, improving on the work of Brakensiek and Guruswami [STOC 2025]. Our proof relies on a generalization of Karger’s [SODA 1993] famous contraction algorithm and its recent linear algebraic extensions [Khanna-Putterman-Sudan SODA 2024], and our resulting bounds show that, just as VC dimension characterizes the additive sparsifiability of a set system, chain length governs the multiplicative sparsifiability. As a corollary, we obtain improved bounds for weighted CSP sparsification.
Cite as

Joshua Brakensiek, Venkatesan Guruswami, and Aaron Putterman. Multiplicative Error Set System Sparsification: A Simpler Proof via Chain Length Contraction. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 44:1-44:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brakensiek_et_al:LIPIcs.ICALP.2026.44,
  author =	{Brakensiek, Joshua and Guruswami, Venkatesan and Putterman, Aaron},
  title =	{{Multiplicative Error Set System Sparsification: A Simpler Proof via Chain Length Contraction}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{44:1--44:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.44},
  URN =		{urn:nbn:de:0030-drops-264331},
  doi =		{10.4230/LIPIcs.ICALP.2026.44},
  annote =	{Keywords: constraint satisfaction problem, chain length, sparsification, VC dimension}
}
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
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.4
The Quest for Strong Inapproximability Results with Perfect Completeness

Authors: Joshua Brakensiek and Venkatesan Guruswami

Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

Abstract
The Unique Games Conjecture (UGC) has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the UGC. For the important case when the input CSP instance admits a satisfying assignment, it therefore remains wide open to understand how well it can be approximated. This work is motivated by the pursuit of a better understanding of the inapproximability of perfectly satisfiable instances of CSPs. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover which we call "V label cover." Assuming a conjecture concerning the inapproximability of V label cover on perfectly satisfiable instances, we prove the following implications: * There is an absolute constant c0 such that for k >= 3, given a satisfiable instance of Boolean k-CSP, it is hard to find an assignment satisfying more than c0 k^2/2^k fraction of the constraints. * Given a k-uniform hypergraph, k >= 2, for all epsilon > 0, it is hard to tell if it is q-strongly colorable or has no independent set with an epsilon fraction of vertices, where q = ceiling[k + sqrt(k) - 0.5]. * Given a k-uniform hypergraph, k >= 3, for all epsilon > 0, it is hard to tell if it is (k-1)-rainbow colorable or has no independent set with an epsilon fraction of vertices. We further supplement the above results with a proof that an ``almost Unique'' version of Label Cover can be approximated within a constant factor on satisfiable instances.
Cite as

Joshua Brakensiek and Venkatesan Guruswami. The Quest for Strong Inapproximability Results with Perfect Completeness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{brakensiek_et_al:LIPIcs.APPROX-RANDOM.2017.4,
  author =	{Brakensiek, Joshua and Guruswami, Venkatesan},
  title =	{{The Quest for Strong Inapproximability Results with Perfect Completeness}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.4},
  URN =		{urn:nbn:de:0030-drops-75537},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.4},
  annote =	{Keywords: inapproximability, hardness of approximation, dictatorship testing, constraint satisfaction, hypergraph coloring}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.33
Vertex Isoperimetry and Independent Set Stability for Tensor Powers of Cliques

Authors: Joshua Brakensiek

Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

Abstract
The tensor power of the clique on t vertices (denoted by K_t^n) is the graph on vertex set {1, ..., t}^n such that two vertices x, y in {1, ..., t}^n are connected if and only if x_i != y_i for all i in {1, ..., n}. Let the density of a subset S of K_t^n to be mu(S) := |S|/t^n. Also let the vertex boundary of a set S to be the vertices of the graph, including those of S, which are incident to some vertex of S. We investigate two similar problems on such graphs. First, we study the vertex isoperimetry problem. Given a density nu in [0, 1] what is the smallest possible density of the vertex boundary of a subset of K_t^n of density nu? Let Phi_t(nu) be the infimum of these minimum densities as n -> infinity. We find a recursive relation allows one to compute Phi_t(nu) in time polynomial to the number of desired bits of precision. Second, we study given an independent set I of K_t^n of density mu(I) = (1-epsilon)/t, how close it is to a maximum-sized independent set J of density 1/t. We show that this deviation (measured by mu(I\J)) is at most 4 epsilon^{(log t)/(log t - log(t-1))} as long as epsilon < 1 - 3/t + 2/t^2. This substantially improves on results of Alon, Dinur, Friedgut, and Sudakov (2004) and Ghandehari and Hatami (2008) which had an O(epsilon) upper bound. We also show the exponent (log t)/(log t - log(t-1)) is optimal assuming n tending to infinity and epsilon tending to 0. The methods have similarity to recent work by Ellis, Keller, and Lifshitz (2016) in the context of Kneser graphs and other settings. The author hopes that these results have potential applications in hardness of approximation, particularly in approximate graph coloring and independent set problems.
Cite as

Joshua Brakensiek. Vertex Isoperimetry and Independent Set Stability for Tensor Powers of Cliques. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{brakensiek:LIPIcs.APPROX-RANDOM.2017.33,
  author =	{Brakensiek, Joshua},
  title =	{{Vertex Isoperimetry and Independent Set Stability for Tensor Powers of Cliques}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{33:1--33:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.33},
  URN =		{urn:nbn:de:0030-drops-75828},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.33},
  annote =	{Keywords: extremal combinatorics, independent sets, isoperimetry, stability}
}
Document
DOI: 10.4230/LIPIcs.CCC.2016.14
New Hardness Results for Graph and Hypergraph Colorings

Authors: Joshua Brakensiek and Venkatesan Guruswami

Published in: LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

Abstract
Finding a proper coloring of a t-colorable graph G with t colors is a classic NP-hard problem when t >= 3. In this work, we investigate the approximate coloring problem in which the objective is to find a proper c-coloring of G where c >= t. We show that for all t >= 3, it is NP-hard to find a c-coloring when c <= 2t-2. In the regime where t is small, this improves, via a unified approach, the previously best known hardness result of c <= max{2t- 5, t + 2*floor(t/3) - 1} (Garey and Johnson 1976; Khanna, Linial, Safra, 1993; Guruswami, Khanna, 2000). For example, we show that 6-coloring a 4-colorable graph is NP-hard, improving on the NP-hardness of 5-coloring a 4-colorable graph. We also generalize this to related problems on the strong coloring of hypergraphs. A k-uniform hypergraph H is t-strong colorable (where t >= k) if there is a t-coloring of the vertices such that no two vertices in each hyperedge of H have the same color. We show that if t = ceiling(3k/2), then it is NP-hard to find a 2-coloring of the vertices of H such that no hyperedge is monochromatic. We conjecture that a similar hardness holds for t=k+1. We establish the NP-hardness of these problems by reducing from the hardness of the Label Cover problem, via a "dictatorship test" gadget graph. By combinatorially classifying all possible colorings of this graph, we can infer labels to provide to the label cover problem. This approach generalizes the "weak polymorphism" framework of (Austrin, Guruswami, Hastad, 2014), though interestingly our results are "PCP-free" in that they do not require any approximation gap in the starting Label Cover instance.
Cite as

Joshua Brakensiek and Venkatesan Guruswami. New Hardness Results for Graph and Hypergraph Colorings. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 14:1-14:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{brakensiek_et_al:LIPIcs.CCC.2016.14,
  author =	{Brakensiek, Joshua and Guruswami, Venkatesan},
  title =	{{New Hardness Results for Graph and Hypergraph Colorings}},
  booktitle =	{31st Conference on Computational Complexity (CCC 2016)},
  pages =	{14:1--14:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-008-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{50},
  editor =	{Raz, Ran},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.14},
  URN =		{urn:nbn:de:0030-drops-58291},
  doi =		{10.4230/LIPIcs.CCC.2016.14},
  annote =	{Keywords: hardness of approximation, graph coloring, hypergraph coloring, polymor- phisms, combinatorics}
}
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