DROPS

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DOI: 10.4230/LIPIcs.ITCS.2026.60

Total Search Problems in ZPP

Authors: Noah Fleming, Stefan Grosser, Siddhartha Jain, Jiawei Li, Hanlin Ren, Morgan Shirley, and Weiqiang Yuan

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We initiate a systematic study of TFZPP, the class of total NP search problems solvable by polynomial time randomized algorithms. TFZPP contains a variety of important search problems such as Bertrand-Chebyshev (finding a prime between N and 2N), refuter problems for many circuit lower bounds, and Lossy-Code. The Lossy-Code problem has found prominence due to its fundamental connections to derandomization, catalytic computing, and the metamathematics of complexity theory, among other areas. While TFZPP collapses to FP under standard derandomization assumptions in the white-box setting, we are able to separate TFZPP from the major TFNP subclasses in the black-box setting. In fact, we are able to separate it from every uniform TFNP class assuming that NP is not in quasi-polynomial time. To do so, we extend the connection between proof complexity and black-box TFNP to randomized proof systems and randomized reductions. Next, we turn to developing a taxonomy of TFZPP problems. We highlight a problem called Nephew, originating from an infinity axiom in set theory. We show that Nephew is in PWPP∩ TFZPP and conjecture that it is not reducible to Lossy-Code. Intriguingly, except for some artificial examples, most other black-box TFZPP problems that we are aware of reduce to Lossy-Code: - We define a problem called Empty-Child capturing finding a leaf in a rooted (binary) tree, and show that this problem is equivalent to Lossy-Code. We also show that a variant of Empty-Child with "heights" is complete for the intersection of SOPL and Lossy-Code. - We strengthen Lossy-Code with several combinatorial inequalities such as the AM-GM inequality. Somewhat surprisingly, we show the resulting new problems are still reducible to Lossy-Code. A technical highlight of this result is that they are proved by formalizations in bounded arithmetic, specifically in Jeřábek’s theory APC₁ (JSL 2007). - Finally, we show that the Dense-Linear-Ordering problem reduces to Lossy-Code.

Cite as

Noah Fleming, Stefan Grosser, Siddhartha Jain, Jiawei Li, Hanlin Ren, Morgan Shirley, and Weiqiang Yuan. Total Search Problems in ZPP. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 60:1-60:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fleming_et_al:LIPIcs.ITCS.2026.60,
  author =	{Fleming, Noah and Grosser, Stefan and Jain, Siddhartha and Li, Jiawei and Ren, Hanlin and Shirley, Morgan and Yuan, Weiqiang},
  title =	{{Total Search Problems in ZPP}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{60:1--60:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.60},
  URN =		{urn:nbn:de:0030-drops-253473},
  doi =		{10.4230/LIPIcs.ITCS.2026.60},
  annote =	{Keywords: TFNP, lossy code, randomized proof systems, query complexity}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.57

List-Recovery of Random Linear Codes over Small Fields

Authors: Dean Doron, Jonathan Mosheiff, Nicolas Resch, and João Ribeiro

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract

We study list-recoverability of random linear codes over small fields, both from errors and from erasures. We consider codes of rate ε-close to capacity, and aim to bound the dependence of the output list size L on ε, the input list size 𝓁, and the alphabet size q. Prior to our work, the best upper bound was L = q^O(𝓁/ε) (Zyablov and Pinsker, Prob. Per. Inf. 1981). Previous work has identified cases in which linear codes provably perform worse than non-linear codes with respect to list-recovery. While there exist non-linear codes that achieve L = O(𝓁/ε), we know that L ≥ 𝓁^Ω(1/ε) is necessary for list recovery from erasures over fields of small characteristic, and for list recovery from errors over large alphabets. We show that in other relevant regimes there is no significant price to pay for linearity, in the sense that we get the correct dependence on the gap-to-capacity ε and go beyond the Zyablov-Pinsker bound for the first time. Specifically, when q is constant and ε approaches zero, - For list-recovery from erasures over prime fields, we show that L ≤ C₁/ε. By prior work, such a result cannot be obtained for low-characteristic fields. - For list-recovery from errors over arbitrary fields, we prove that L ≤ C₂/ε. Above, C₁ and C₂ depend on the decoding radius, input list size, and field size. We provide concrete bounds on the constants above, and the upper bounds on L improve upon the Zyablov-Pinsker bound whenever q ≤ 2^{(1/ε)^c} for some small universal constant c > 0.

Cite as

Dean Doron, Jonathan Mosheiff, Nicolas Resch, and João Ribeiro. List-Recovery of Random Linear Codes over Small Fields. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 57:1-57:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2025.57,
  author =	{Doron, Dean and Mosheiff, Jonathan and Resch, Nicolas and Ribeiro, Jo\~{a}o},
  title =	{{List-Recovery of Random Linear Codes over Small Fields}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{57:1--57:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.57},
  URN =		{urn:nbn:de:0030-drops-244239},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.57},
  annote =	{Keywords: List recovery, random linear codes}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.53

Near-Optimal List-Recovery of Linear Code Families

Authors: Ray Li and Nikhil Shagrithaya

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract

We prove several results on linear codes achieving list-recovery capacity. We show that random linear codes achieve list-recovery capacity with constant output list size (independent of the alphabet size and length). That is, over alphabets of size at least 𝓁^Ω(1/ε), random linear codes of rate R are (1-R-ε, 𝓁, (𝓁/ε)^O(𝓁/ε))-list-recoverable for all R ∈ (0,1) and 𝓁. Together with a result of Levi, Mosheiff, and Shagrithaya, this implies that randomly punctured Reed-Solomon codes also achieve list-recovery capacity. We also prove that our output list size is near-optimal among all linear codes: all (1-R-ε, 𝓁, L)-list-recoverable linear codes must have L ≥ 𝓁^{Ω(R/ε)}. Our simple upper bound combines the Zyablov-Pinsker argument with recent bounds from Kopparty, Ron-Zewi, Saraf, Wootters, and Tamo on the maximum intersection of a "list-recovery ball" and a low-dimensional subspace with large distance. Our lower bound is inspired by a recent lower bound of Chen and Zhang.

Cite as

Ray Li and Nikhil Shagrithaya. Near-Optimal List-Recovery of Linear Code Families. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{li_et_al:LIPIcs.APPROX/RANDOM.2025.53,
  author =	{Li, Ray and Shagrithaya, Nikhil},
  title =	{{Near-Optimal List-Recovery of Linear Code Families}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{53:1--53:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.53},
  URN =		{urn:nbn:de:0030-drops-244199},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.53},
  annote =	{Keywords: Error-Correcting Codes, Randomness, List-Recovery, Reed-Solomon Codes, Random Linear Codes}
}

Document

DOI: 10.4230/LIPIcs.CCC.2025.25

Quantum LDPC Codes of Almost Linear Distance via Iterated Homological Products

Authors: Louis Golowich and Venkatesan Guruswami

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract

The first linear-distance quantum LDPC codes were recently constructed by a line of breakthrough works (culminating in the result of Panteleev & Kalachev, 2021). All such constructions, even when allowing for almost-linear distance, are based on an operation called a balanced (or lifted) product, which is used in a one-shot manner to combine a pair of large classical codes possessing a group symmetry. We present a new construction of almost-linear distance quantum LDPC codes that is iterative in nature. Our construction is based on a more basic and widely used product, namely the homological product (i.e. the tensor product of chain complexes). Specifically, for every ε > 0, we obtain a family of [[N,N^{1-ε},N^{1-ε}]] (subsystem) quantum LDPC codes via repeated homological products of a constant-sized quantum locally testable code. Our key idea is to remove certain low-weight codewords using subsystem codes (while still maintaining constant stabilizer weight), in order to circumvent a particular obstruction that limited the distance of many prior homological product code constructions to at most Õ(√N).

Cite as

Louis Golowich and Venkatesan Guruswami. Quantum LDPC Codes of Almost Linear Distance via Iterated Homological Products. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 25:1-25:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{golowich_et_al:LIPIcs.CCC.2025.25,
  author =	{Golowich, Louis and Guruswami, Venkatesan},
  title =	{{Quantum LDPC Codes of Almost Linear Distance via Iterated Homological Products}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{25:1--25:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.25},
  URN =		{urn:nbn:de:0030-drops-237196},
  doi =		{10.4230/LIPIcs.CCC.2025.25},
  annote =	{Keywords: Quantum Error Correction, Quantum LDPC Code, Homological Product, Iterative Construction}
}

Document

DOI: 10.4230/LIPIcs.CCC.2025.14

On the Automatability of Tree-Like k-DNF Resolution

Authors: Gaia Carenini and Susanna F. de Rezende

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract

A proof system 𝒫 is said to be automatable in time f(N) if there exists an algorithm that given as input an unsatisfiable formula F outputs a refutation of F in the proof system 𝒫 in time f(N), where N is the size of the smallest 𝒫-refutation of F plus the size of F. Atserias and Bonet (ECCC 2002), observed that tree-like k-DNF resolution is automatable in time N^{c⋅klog N} for a universal constant c. We show that, under the randomized exponential-time hypothesis (rETH), this is tight up to a O(log k)-factor in the exponent, i.e., we prove that tree-like k-DNF resolution, for k at most logarithmic in the number of variables of F, is not automatable in time N^o((k/log k)⋅log N) unless rETH is false. Our proof builds on the non-automatability results for resolution by Atserias and Müller (FOCS 2019), for algebraic proof systems by de Rezende, Göös, Nordström, Pitassi, Robere and Sokolov (STOC 2021), and for tree-like resolution by de Rezende (LAGOS 2021).

Cite as

Gaia Carenini and Susanna F. de Rezende. On the Automatability of Tree-Like k-DNF Resolution. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 14:1-14:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{carenini_et_al:LIPIcs.CCC.2025.14,
  author =	{Carenini, Gaia and de Rezende, Susanna F.},
  title =	{{On the Automatability of Tree-Like k-DNF Resolution}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{14:1--14:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.14},
  URN =		{urn:nbn:de:0030-drops-237081},
  doi =		{10.4230/LIPIcs.CCC.2025.14},
  annote =	{Keywords: Proof Complexity, Tree-like k-DNF Resolution, Automatability}
}

Document

DOI: 10.4230/LIPIcs.CCC.2025.10

Biased Linearity Testing in the 1% Regime

Authors: Subhash Khot and Kunal Mittal

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract

We study linearity testing over the p-biased hypercube ({0,1}ⁿ, μ_p^{⊗n}) in the 1% regime. For a distribution ν supported over {x ∈ {0,1}^k:∑_{i=1}^k x_i = 0 (mod 2)}, with marginal distribution μ_p in each coordinate, the corresponding k-query linearity test Lin(ν) proceeds as follows: Given query access to a function f:{0,1}ⁿ → {-1,1}, sample (x_1,… ,x_k)∼ ν^{⊗n}, query f on x_1,… ,x_k, and accept if and only if ∏_{i ∈ [k]} f(x_i) = 1. Building on the work of Bhangale, Khot, and Minzer (STOC '23), we show, for 0 < p ≤ 1/2, that if k ≥ 1+1/p, then there exists a distribution ν such that the test Lin(ν) works in the 1% regime; that is, any function f:{0,1}ⁿ → {-1,1} passing the test Lin(ν) with probability ≥ 1/2+ε, for some constant ε > 0, satisfies Pr_{x∼μ_p^{⊗n}}[f(x) = g(x)] ≥ 1/2+δ, for some linear function g, and a constant δ = δ(ε) > 0. Conversely, we show that if k < 1+1/p, then no such test Lin(ν) works in the 1% regime. Our key observation is that the linearity test Lin(ν) works if and only if the distribution ν satisfies a certain pairwise independence property.

Cite as

Subhash Khot and Kunal Mittal. Biased Linearity Testing in the 1% Regime. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 10:1-10:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{khot_et_al:LIPIcs.CCC.2025.10,
  author =	{Khot, Subhash and Mittal, Kunal},
  title =	{{Biased Linearity Testing in the 1\% Regime}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{10:1--10:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.10},
  URN =		{urn:nbn:de:0030-drops-237046},
  doi =		{10.4230/LIPIcs.CCC.2025.10},
  annote =	{Keywords: Linearity test, 1\% regime, p-biased}
}

Document

DOI: 10.4230/LIPIcs.CCC.2025.36

Lifting with Colourful Sunflowers

Authors: Susanna F. de Rezende and Marc Vinyals

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract

We show that a generalization of the DAG-like query-to-communication lifting theorem, when proven using sunflowers over non-binary alphabets, yields lower bounds on the monotone circuit complexity and proof complexity of natural functions and formulas that are better than previously known results obtained using the approximation method. These include an n^Ω(k) lower bound for the clique function up to k ≤ n^{1/2-ε}, and an exp(Ω(n^{1/3-ε})) lower bound for a function in P.

Cite as

Susanna F. de Rezende and Marc Vinyals. Lifting with Colourful Sunflowers. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 36:1-36:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{derezende_et_al:LIPIcs.CCC.2025.36,
  author =	{de Rezende, Susanna F. and Vinyals, Marc},
  title =	{{Lifting with Colourful Sunflowers}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{36:1--36:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.36},
  URN =		{urn:nbn:de:0030-drops-237303},
  doi =		{10.4230/LIPIcs.CCC.2025.36},
  annote =	{Keywords: lifting, sunflower, clique, colouring, monotone circuit, cutting planes}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.122

Yet Another Simple Proof of the PCRP Theorem

Authors: Naoto Ohsaka

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract

The Probabilistically Checkable Reconfiguration Proof (PCRP) theorem, proven by Hirahara and Ohsaka (STOC 2024) [Hirahara and Ohsaka, 2024] and Karthik C. S. and Manurangsi [{Karthik {C. S.}} and Manurangsi, 2023], provides a new PCP-type characterization of PSPACE: A language L is in PSPACE if and only if there exists a probabilistic verifier 𝒱 and a pair of polynomial-time computable proofs π^ini, π^end such that the following hold for every input x: - If x ∈ L, then π^ini(x) can be transformed into π^end(x) by repeatedly flipping a single bit of the proof at a time, while making 𝒱(x) to accept every intermediate proof with probability 1. - If x ∉ L, then any such transformation induces a proof that is rejected by 𝒱(x) with probability more than 1/2. The PCRP theorem finds many applications in PSPACE-hardness of approximation for reconfiguration problems. In this paper, we present an alternative proof of the PCRP theorem that is "simpler" than those of Hirahara and Ohsaka [Hirahara and Ohsaka, 2024] and Karthik C. S. and Manurangsi [Karthik C. S. and Manurangsi, 2023]. Our PCRP system is obtained by combining simple robustization and composition steps in a modular fashion, which renders its analysis more intuitive. The crux of implementing the robustization step is an error-correcting code that enjoys both list decodability and reconfigurability, the latter of which enables to reconfigure between a pair of codewords, while avoiding getting too close to any other codewords.

Cite as

Naoto Ohsaka. Yet Another Simple Proof of the PCRP Theorem. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 122:1-122:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ohsaka:LIPIcs.ICALP.2025.122,
  author =	{Ohsaka, Naoto},
  title =	{{Yet Another Simple Proof of the PCRP Theorem}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{122:1--122:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.122},
  URN =		{urn:nbn:de:0030-drops-234995},
  doi =		{10.4230/LIPIcs.ICALP.2025.122},
  annote =	{Keywords: reconfiguration problems, hardness of approximation, probabilistic proof systems}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2025.2

Improved Lower Bounds for 3-Query Matching Vector Codes

Authors: Divesh Aggarwal, Pranjal Dutta, Zeyong Li, Maciej Obremski, and Sidhant Saraogi

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

A Matching Vector (MV) family modulo a positive integer m ≥ 2 is a pair of ordered lists U = (u_1, ⋯, u_K) and V = (v_1, ⋯, v_K) where u_i, v_j ∈ ℤ_m^n with the following property: for any i ∈ [K], the inner product ⟨u_i, v_i⟩ = 0 mod m, and for any i ≠ j, ⟨u_i, v_j⟩ ≠ 0 mod m. An MV family is called r-restricted if inner products ⟨u_i, v_j⟩, for all i,j, take at most r different values. The r-restricted MV families are extremely important since the only known construction of constant-query subexponential locally decodable codes (LDCs) are based on them. Such LDCs constructed via matching vector families are called matching vector codes. Let MV(m,n) (respectively MV(m, n, r)) denote the largest K such that there exists an MV family (respectively r-restricted MV family) of size K in ℤ_m^n. Such a MV family can be transformed in a black-box manner to a good r-query locally decodable code taking messages of length K to codewords of length N = m^n. For small prime m, an almost tight bound MV(m,n) ≤ O(m^{n/2}) was first shown by Dvir, Gopalan, Yekhanin (FOCS'10, SICOMP'11), while for general m, the same paper established an upper bound of O(m^{n-1+o_m(1)}), with o_m(1) denoting a function that goes to zero when m grows. For any arbitrary constant r ≥ 3 and composite m, the best upper bound till date on MV(m,n,r) is O(m^{n/2}), is due to Bhowmick, Dvir and Lovett (STOC'13, SICOMP'14).In a breakthrough work, Alrabiah, Guruswami, Kothari and Manohar (STOC'23) implicitly improve this bound for 3-restricted families to MV(m, n, 3) ≤ O(m^{n/3}). In this work, we present an upper bound for r = 3 where MV(m,n,3) ≤ m^{n/6 +O(log n)}, and as a result, any 3-query matching vector code must have codeword length of N ≥ K^{6-o(1)}.

Cite as

Divesh Aggarwal, Pranjal Dutta, Zeyong Li, Maciej Obremski, and Sidhant Saraogi. Improved Lower Bounds for 3-Query Matching Vector Codes. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 2:1-2:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aggarwal_et_al:LIPIcs.ITCS.2025.2,
  author =	{Aggarwal, Divesh and Dutta, Pranjal and Li, Zeyong and Obremski, Maciej and Saraogi, Sidhant},
  title =	{{Improved Lower Bounds for 3-Query Matching Vector Codes}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{2:1--2:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.2},
  URN =		{urn:nbn:de:0030-drops-226308},
  doi =		{10.4230/LIPIcs.ITCS.2025.2},
  annote =	{Keywords: Locally Decodable Codes, Matching Vector Families}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2025.76

A Quantum Unique Games Conjecture

Authors: Hamoon Mousavi and Taro Spirig

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

After the NP-hardness of computational problems such as 3SAT and MaxCut was established, a natural next step was to explore whether these problems remain hard to approximate. While the quantum nonlocal games extensions of some of these problems are known to be hard - indeed undecidable - their inapproximability remains largely unresolved. In this work, we introduce definitions for the quantum extensions of Label-Cover and Unique-Label-Cover. We show that these problems play a similarly crucial role in studying the inapproximability of quantum constraint satisfaction problems as they do in the classical setting.

Cite as

Hamoon Mousavi and Taro Spirig. A Quantum Unique Games Conjecture. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 76:1-76:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mousavi_et_al:LIPIcs.ITCS.2025.76,
  author =	{Mousavi, Hamoon and Spirig, Taro},
  title =	{{A Quantum Unique Games Conjecture}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{76:1--76:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.76},
  URN =		{urn:nbn:de:0030-drops-227047},
  doi =		{10.4230/LIPIcs.ITCS.2025.76},
  annote =	{Keywords: hardness of approximation, quantum computing, noncommutative constraint satisfaction problems, Fourier analysis}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.30

On the List Recoverability of Randomly Punctured Codes

Authors: Ben Lund and Aditya Potukuchi

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

Abstract

We show that a random puncturing of a code with good distance is list recoverable beyond the Johnson bound. In particular, this implies that there are Reed-Solomon codes that are list recoverable beyond the Johnson bound. It was previously known that there are Reed-Solomon codes that do not have this property. As an immediate corollary to our main theorem, we obtain better degree bounds on unbalanced expanders that come from Reed-Solomon codes.

Cite as

Ben Lund and Aditya Potukuchi. On the List Recoverability of Randomly Punctured Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 30:1-30:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lund_et_al:LIPIcs.APPROX/RANDOM.2020.30,
  author =	{Lund, Ben and Potukuchi, Aditya},
  title =	{{On the List Recoverability of Randomly Punctured Codes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{30:1--30:11},
  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.2020.30},
  URN =		{urn:nbn:de:0030-drops-126330},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.30},
  annote =	{Keywords: List recovery, randomly punctured codes, Reed-Solomon codes}
}

Document

DOI: 10.4230/LIPIcs.CCC.2020.28

Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs

Authors: Susanna F. de Rezende, Jakob Nordström, Kilian Risse, and Dmitry Sokolov

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense graphs as in [Raz '04] and [Razborov '03, '04]. We obtain our results by revisiting Razborov’s pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.

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Susanna F. de Rezende, Jakob Nordström, Kilian Risse, and Dmitry Sokolov. Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 28:1-28:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{derezende_et_al:LIPIcs.CCC.2020.28,
  author =	{de Rezende, Susanna F. and Nordstr\"{o}m, Jakob and Risse, Kilian and Sokolov, Dmitry},
  title =	{{Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{28:1--28:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.28},
  URN =		{urn:nbn:de:0030-drops-125804},
  doi =		{10.4230/LIPIcs.CCC.2020.28},
  annote =	{Keywords: proof complexity, resolution, weak pigeonhole principle, perfect matching, sparse graphs}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2019.24

Approximate Counting of k-Paths: Deterministic and in Polynomial Space

Authors: Andreas Björklund, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract

A few years ago, Alon et al. [ISMB 2008] gave a simple randomized O((2e)^km epsilon^{-2})-time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 +/- epsilon. Shortly afterwards, Alon and Gutner [IWPEC 2009, TALG 2010] gave a deterministic exponential-space algorithm with running time (2e)^{k+O(log^3k)}m log n whenever epsilon^{-1}=k^{O(1)}. Recently, Brand et al. [STOC 2018] provided a speed-up at the cost of reintroducing randomization. Specifically, they gave a randomized O(4^km epsilon^{-2})-time exponential-space algorithm. In this article, we revisit the algorithm by Alon and Gutner. We modify the foundation of their work, and with a novel twist, obtain the following results. - We present a deterministic 4^{k+O(sqrt{k}(log^2k+log^2 epsilon^{-1}))}m log n-time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. - Additionally, we present a randomized 4^{k+O(log k(log k + log epsilon^{-1}))}m log n-time polynomial-space algorithm. While Brand et al. make non-trivial use of exterior algebra, our algorithm is very simple; we only make elementary use of the probabilistic method. Thus, the algorithm by Brand et al. runs in time 4^{k+o(k)}m whenever epsilon^{-1}=2^{o(k)}, while our deterministic and randomized algorithms run in time 4^{k+o(k)}m log n whenever epsilon^{-1}=2^{o(k^{1/4})} and epsilon^{-1}=2^{o(k/(log k))}, respectively. Prior to our work, no 2^{O(k)}n^{O(1)}-time polynomial-space algorithm was known. Additionally, our approach is embeddable in the classic framework of divide-and-color, hence it immediately extends to approximate counting of graphs of bounded treewidth; in comparison, Brand et al. note that their approach is limited to graphs of bounded pathwidth.

Cite as

Andreas Björklund, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Approximate Counting of k-Paths: Deterministic and in Polynomial Space. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bjorklund_et_al:LIPIcs.ICALP.2019.24,
  author =	{Bj\"{o}rklund, Andreas and Lokshtanov, Daniel and Saurabh, Saket and Zehavi, Meirav},
  title =	{{Approximate Counting of k-Paths: Deterministic and in Polynomial Space}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{24:1--24:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.24},
  URN =		{urn:nbn:de:0030-drops-106001},
  doi =		{10.4230/LIPIcs.ICALP.2019.24},
  annote =	{Keywords: parameterized complexity, approximate counting, \{ k\}-Path}
}

Document

DOI: 10.4230/LIPIcs.SOCG.2015.537

Bisector Energy and Few Distinct Distances

Authors: Ben Lund, Adam Sheffer, and Frank de Zeeuw

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract

We introduce the bisector energy of an n-point set P in the real plane, defined as the number of quadruples (a,b,c,d) from P such that a and b determine the same perpendicular bisector as c and d. If no line or circle contains M(n) points of P, then we prove that the bisector energy is O(M(n)^{2/5}n^{12/5} + M(n)n^2). We also prove the lower bound M(n)n^2, which matches our upper bound when M(n) is large. We use our upper bound on the bisector energy to obtain two rather different results: (i) If P determines O(n / sqrt(log n)) distinct distances, then for any 0 < a < 1/4, either there exists a line or circle that contains n^a points of P, or there exist n^{8/5 - 12a/5} distinct lines that contain sqrt(log n) points of P. This result provides new information on a conjecture of Erdös regarding the structure of point sets with few distinct distances. (ii) If no line or circle contains M(n) points of P, then the number of distinct perpendicular bisectors determined by P is min{M(n)^{-2/5}n^{8/5}, M(n)^{-1}n^2}). This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over the real numbers, initiated by Elekes and Ronyai.

Cite as

Ben Lund, Adam Sheffer, and Frank de Zeeuw. Bisector Energy and Few Distinct Distances. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 537-552, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{lund_et_al:LIPIcs.SOCG.2015.537,
  author =	{Lund, Ben and Sheffer, Adam and de Zeeuw, Frank},
  title =	{{Bisector Energy and Few Distinct Distances}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{537--552},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.537},
  URN =		{urn:nbn:de:0030-drops-51086},
  doi =		{10.4230/LIPIcs.SOCG.2015.537},
  annote =	{Keywords: Combinatorial geometry, distinct distances, incidence geometry}
}

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