40 Search Results for "Santha, Miklos"

Document
DOI: 10.4230/LIPIcs.STACS.2026.57
Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach

Authors: Antoine Joux and Karol Węgrzycki

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
Lagarias and Odlyzko (J.ACM 1985) proposed a polynomial-time algorithm for solving "almost all" instances of the Subset Sum problem with n integers of size Ω(Γ_LO), where log₂(Γ_LO) > n² log₂(γ) and γ is a parameter of the lattice basis reduction (γ > √{4/3} for LLL). The algorithm of Lagarias and Odlyzko is a cornerstone of cryptography. However, the theoretical guarantee on the density of feasible instances has remained unimproved for almost 40 years. In this paper, we propose an algorithm that solves "almost all" instances of Subset Sum with integers of size Ω(√{Γ_LO}) after a single call to lattice reduction. Additionally, our approach allows solving the Subset Sum problem for multiple targets, whereas the previous method could handle only one target per call to lattice basis reduction. We introduce a modular arithmetic approach to the Subset Sum problem, leveraging lattice reduction to solve a linear system modulo a suitably large prime. By analyzing the lengths of the LLL-reduced basis vectors of both the primal and dual lattices simultaneously, we show that density guarantees can be improved.
Cite as

Antoine Joux and Karol Węgrzycki. Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 57:1-57:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{joux_et_al:LIPIcs.STACS.2026.57,
  author =	{Joux, Antoine and W\k{e}grzycki, Karol},
  title =	{{Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{57:1--57:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.57},
  URN =		{urn:nbn:de:0030-drops-255462},
  doi =		{10.4230/LIPIcs.STACS.2026.57},
  annote =	{Keywords: Average-Case Analysis, Subset Sum, Lattice Reduction, LLL}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.10
On the Complexity of Unique Quantum Witnesses and Quantum Approximate Counting

Authors: Anurag Anshu, Jonas Haferkamp, Yeongwoo Hwang, and Quynh T. Nguyen

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

Abstract
We study the long-standing open question on the power of unique witnesses in quantum protocols, which asks if UniqueQMA, a variant of QMA whose accepting witness space is 1-dimensional, contains QMA under quantum reductions. This work rules out any black-box reduction from QMA to UniqueQMA by showing a quantum oracle separation between BQP^UniqueQMA and QMA. This provides a contrast to the classical case, where the Valiant-Vazirani theorem shows a black-box randomized reduction from UniqueNP to NP, and suggests the need for studying the structure of the ground space of local Hamiltonians in distilling a potential unique witness. Via similar techniques, we show, relative to a quantum oracle, that QMA^QMA cannot decide quantum approximate counting, ruling out a quantum analogue of Stockmeyer’s algorithm in the black-box setting. Our results employ a subspace reflection oracle, previously considered in [Scott Aaronson and Greg Kuperberg, 2007; Scott Aaronson et al., 2020; She and Yuen, 2023], but we introduce new tools which allow us to exploit the unique witness constraint. We also show a strong "polarization" behavior of QMA circuits, which could be of independent interest in studying quantum polynomial hierarchies. We then ask a natural question; what structural properties of the local Hamiltonian problem can we exploit? We introduce a physically motivated candidate by showing that the ground energy of local Hamiltonians that satisfy a computational variant of the eigenstate thermalization hypothesis (ETH) can be estimated through a UniqueQMA protocol. Our protocol can be viewed as a quantum expander test in a low energy subspace of the Hamiltonian and verifies a unique entangled state across two copies of the subspace. This allows us to conclude that if UniqueQMA is not equivalent to QMA, then QMA-hard Hamiltonians must violate ETH under adversarial perturbations (more accurately, further assuming the quantum PCP conjecture if ETH only applies to extensive energy subspaces). Under the same assumption, this also serves as evidence that chaotic local Hamiltonians, such as the SYK model may be computationally simpler than general local Hamiltonians.
Cite as

Anurag Anshu, Jonas Haferkamp, Yeongwoo Hwang, and Quynh T. Nguyen. On the Complexity of Unique Quantum Witnesses and Quantum Approximate Counting. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{anshu_et_al:LIPIcs.ITCS.2026.10,
  author =	{Anshu, Anurag and Haferkamp, Jonas and Hwang, Yeongwoo and Nguyen, Quynh T.},
  title =	{{On the Complexity of Unique Quantum Witnesses and Quantum Approximate Counting}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{10:1--10:20},
  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.2026.10},
  URN =		{urn:nbn:de:0030-drops-252978},
  doi =		{10.4230/LIPIcs.ITCS.2026.10},
  annote =	{Keywords: Quantum complexity, approximate counting, Valiant-Vazirani, eigenstate thermalization hypothesis}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.7
An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem

Authors: Marco Aldi, Sevag Gharibian, and Dorian Rudolph

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

Abstract
The theory of Total Function NP (TFNP) and its subclasses says that, even if one is promised an efficiently verifiable proof exists for a problem, finding this proof can be intractable. Despite the success of the theory at showing intractability of problems such as computing Brouwer fixed points and Nash equilibria, subclasses of TFNP remain arguably few and far between. In this work, we define two new subclasses of TFNP borne of the study of complex polynomial systems: Multi-homogeneous Systems (MHS) and Sparse Fundamental Theorem of Algebra (SFTA). The first of these is based on Bézout’s theorem from algebraic geometry, marking the first TFNP subclass based on an algebraic geometric principle. At the heart of our study is the computational problem known as Quantum SAT (QSAT) with a System of Distinct Representatives (SDR), first studied by [Laumann, Läuchli, Moessner, Scardicchio, and Sondhi 2010]. Among other results, we show that QSAT with SDR is MHS-complete, thus giving not only the first link between quantum complexity theory and TFNP, but also the first TFNP problem whose classical variant (SAT with SDR) is easy but whose quantum variant is hard. We also show how to embed the roots of a sparse, high-degree, univariate polynomial into QSAT with SDR, obtaining that SFTA is contained in a zero-error version of MHS. We conjecture this construction also works in the low-error setting, which would imply SFTA ⊆ MHS.
Cite as

Marco Aldi, Sevag Gharibian, and Dorian Rudolph. An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 7:1-7:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{aldi_et_al:LIPIcs.ITCS.2026.7,
  author =	{Aldi, Marco and Gharibian, Sevag and Rudolph, Dorian},
  title =	{{An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{7:1--7:24},
  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.2026.7},
  URN =		{urn:nbn:de:0030-drops-252946},
  doi =		{10.4230/LIPIcs.ITCS.2026.7},
  annote =	{Keywords: quantum complexity theory, Quantum Merlin Arthur (QMA), Quantum Satisfiability Problem (QSAT), total function NP (TFNP)}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.83
The Pure-State Consistency of Local Density Matrices Problem: In PSPACE and Complete for a Class Between QMA and QMA(2)

Authors: Jonas Kamminga and Dorian Rudolph

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

Abstract
In this work we investigate the computational complexity of the pure consistency of local density matrices (PureCLDM) and pure N-representability (Pure-N-Representability; analog of PureCLDM for bosonic or fermionic systems) problems. In these problems the input is a set of reduced density matrices and the task is to determine whether there exists a global pure state consistent with these reduced density matrices. While mixed CLDM, i.e. where the global state can be mixed, was proven to be QMA-complete by Broadbent and Grilo [JoC 2022], almost nothing was known about the complexity of the pure version. Before our work the best upper and lower bounds were QMA(2) and QMA. Our contribution to the understanding of these problems is twofold. Firstly, we define a pure state analogue of the complexity class QMA^+ of Aharanov and Regev [FOCS 2003], which we call PureSuperQMA. We prove that both pure-N-Representability and PureCLDM are complete for this new class. Along the way we supplement Broadbent and Grilo by proving hardness for 2-qubit reduced density matrices and showing that mixed N-Representability is QMA-complete. Secondly, we improve the upper bound on PureCLDM. Using methods from algebraic geometry, we prove that PureSuperQMA ⊆ PSPACE. Our methods, and the PSPACE upper bound, are also valid for PureCLDM with exponential or even perfect precision, hence precisePureCLDM is not preciseQMA(2) = NEXP-complete, unless PSPACE = NEXP. We view this as evidence for a negative answer to the longstanding open question whether PureCLDM is QMA(2)-complete. The techniques we develop for our PSPACE upper bound are quite general. We are able to use them for various applications: from proving PSPACE upper bounds on other quantum problems to giving an efficient parallel (NC) algorithm for (non-convex) quadratically constrained quadratic programs with few constraints.
Cite as

Jonas Kamminga and Dorian Rudolph. The Pure-State Consistency of Local Density Matrices Problem: In PSPACE and Complete for a Class Between QMA and QMA(2). In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 83:1-83:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kamminga_et_al:LIPIcs.ITCS.2026.83,
  author =	{Kamminga, Jonas and Rudolph, Dorian},
  title =	{{The Pure-State Consistency of Local Density Matrices Problem: In PSPACE and Complete for a Class Between QMA and QMA(2)}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{83:1--83:23},
  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.2026.83},
  URN =		{urn:nbn:de:0030-drops-253701},
  doi =		{10.4230/LIPIcs.ITCS.2026.83},
  annote =	{Keywords: Quantum Complexity Theory, PSPACE, QMA(2), Consistency of Local Density Matrices, Polynomial Optimization}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.2
Oracle Separations for the Quantum-Classical Polynomial Hierarchy

Authors: Avantika Agarwal and Shalev Ben{-}David

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

Abstract
We study the quantum-classical polynomial hierarchy, QCPH, which is the class of languages solvable by a constant number of alternating classical quantifiers followed by a quantum verifier. Our main result is that QCPH is infinite relative to a random oracle (previously, this was not even known relative to any oracle). We further prove that higher levels of PH are not contained in lower levels of QCPH relative to a random oracle; this is a strengthening of the somewhat recent result that PH is infinite relative to a random oracle (Rossman, Servedio, and Tan 2016). The oracle separation requires lower bounding a certain type of low-depth alternating circuit with some quantum gates. To establish this, we give a new switching lemma for quantum algorithms which may be of independent interest. Our lemma says that for any d, if we apply a random restriction to a function f with quantum query complexity Q(f) ≤ n^{1/3}, the restricted function becomes exponentially close (in terms of d) to a depth-d decision tree. Our switching lemma works even in a "worst-case" sense, in that only the indices to be restricted are random; the values they are restricted to are chosen adversarially. Moreover, the switching lemma also works for polynomial degree in place of quantum query complexity.
Cite as

Avantika Agarwal and Shalev Ben-David. Oracle Separations for the Quantum-Classical Polynomial Hierarchy. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 2:1-2:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{agarwal_et_al:LIPIcs.ITCS.2026.2,
  author =	{Agarwal, Avantika and Ben\{-\}David, Shalev},
  title =	{{Oracle Separations for the Quantum-Classical Polynomial Hierarchy}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{2:1--2:22},
  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.2026.2},
  URN =		{urn:nbn:de:0030-drops-252893},
  doi =		{10.4230/LIPIcs.ITCS.2026.2},
  annote =	{Keywords: Switching Lemma, Polynomial Hierarchy, Approximate Degree, Random Oracles, Query Complexity, Quantum Computing}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.72
Forrelation Is Extremally Hard

Authors: Uma Girish and Rocco Servedio

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

Abstract
The Forrelation problem is a central problem that demonstrates an exponential separation between quantum and classical capabilities. In this problem, given query access to n-bit Boolean functions f and g, the goal is to estimate the Forrelation function forr(f,g), which measures the correlation between g and the Fourier transform of f. In this work we provide a new linear algebraic perspective on the Forrelation problem, as opposed to prior analytic approaches. We establish a connection between the Forrelation problem and bent Boolean functions and through this connection, analyze an extremal version of the Forrelation problem where the goal is to distinguish between extremal instances of Forrelation, namely (f,g) with forr(f,g) = 1 and forr(f,g) = -1. We show that this problem can be solved with one quantum query and success probability one, yet requires Ω̃(2^{n/4}) classical randomized queries, even for algorithms with a one-third failure probability, highlighting the remarkable power of one exact quantum query. We also study a restricted variant of this problem where the inputs f,g are computable by small classical circuits and show classical hardness under cryptographic assumptions.
Cite as

Uma Girish and Rocco Servedio. Forrelation Is Extremally Hard. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 72:1-72:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{girish_et_al:LIPIcs.ITCS.2026.72,
  author =	{Girish, Uma and Servedio, Rocco},
  title =	{{Forrelation Is Extremally Hard}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{72:1--72:22},
  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.2026.72},
  URN =		{urn:nbn:de:0030-drops-253594},
  doi =		{10.4230/LIPIcs.ITCS.2026.72},
  annote =	{Keywords: Forrelation, exact quantum, query complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.110
Optimal Two-Round Communication Lower Bound for Graph Connectivity via Pointer Chasing

Authors: Jaikumar Radhakrishnan, Chaitanya Reddy, and Rakesh Venkat

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

Abstract
We consider the communication complexity of the graph connectivity problem, where the edges of an n-vertex undirected graph G are distributed between two parties Alice and Bob, who are then required to communicate to determine if G is connected. We show that in any randomized protocol with two-rounds of communication, Alice and Bob must exchange Ω(nlog n) bits; such a lower bound for one-round protocols was shown by Sun and Woodruff (APPROX/RANDOM 2015). A one-round deterministic protocol, where Alice sends O(n log n) bits and Bob determines the answer, was observed by Hajnal, Maass and Turan (STOC 1988); they also showed a matching lower bound of Ω(n log n) bits for deterministic protocols with unbounded rounds of communication. For randomized protocols, a reduction from the set disjointness problem due to Babai, Frankl and Simon (FOCS 1986) implies a randomized lower bound of Ω(n) even with unbounded rounds of communication. Whether this lower bound can be improved to Ω(n log n) has been an outstanding open question, whose algorithmic implications were recently emphasized by Apers, Efron, Gawrychowski, Lee, Mukopadhyay and Nanongkai (FOCS 2022). Our lower bound for randomized two-round protocols is based on a reduction from a restricted version of the two-player pointer chasing problem originally studied by Papadimitriou and Sipser (JCSS 1984). Using this reduction, we show an ω(n) lower bounds on graph connectivity for any constant number of rounds by extending deterministic lower bounds shown by Ponzio, Radhakrishnan and Venkatesh (JCSS 2001) to the randomized setting.
Cite as

Jaikumar Radhakrishnan, Chaitanya Reddy, and Rakesh Venkat. Optimal Two-Round Communication Lower Bound for Graph Connectivity via Pointer Chasing. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 110:1-110:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{radhakrishnan_et_al:LIPIcs.ITCS.2026.110,
  author =	{Radhakrishnan, Jaikumar and Reddy, Chaitanya and Venkat, Rakesh},
  title =	{{Optimal Two-Round Communication Lower Bound for Graph Connectivity via Pointer Chasing}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{110:1--110:20},
  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.2026.110},
  URN =		{urn:nbn:de:0030-drops-253974},
  doi =		{10.4230/LIPIcs.ITCS.2026.110},
  annote =	{Keywords: Communication complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.112
Lower Bounds Beyond DNF of Parities

Authors: Artur Riazanov, Anastasia Sofronova, and Dmitry Sokolov

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

Abstract
We consider a subclass of AC⁰[2] circuits that simultaneously captures DNF∘Xor and depth-3 AC⁰ circuits. For this class we show a technique for proving lower bounds inspired by the top-down approach. We give lower bounds for the middle slice function, inner product function, and affine dispersers.
Cite as

Artur Riazanov, Anastasia Sofronova, and Dmitry Sokolov. Lower Bounds Beyond DNF of Parities. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 112:1-112:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{riazanov_et_al:LIPIcs.ITCS.2026.112,
  author =	{Riazanov, Artur and Sofronova, Anastasia and Sokolov, Dmitry},
  title =	{{Lower Bounds Beyond DNF of Parities}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{112:1--112:15},
  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.2026.112},
  URN =		{urn:nbn:de:0030-drops-253996},
  doi =		{10.4230/LIPIcs.ITCS.2026.112},
  annote =	{Keywords: boolean circuits, top-down, unpredictability}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.44
Approximation Schemes for k-Subset Sum Ratio and k-Way Number Partitioning Ratio

Authors: Sotiris Kanellopoulos, Giorgos Mitropoulos, Antonis Antonopoulos, Nikos Leonardos, Aris Pagourtzis, Christos Pergaminelis, Stavros Petsalakis, and Kanellos Tsitouras

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
The Subset Sum Ratio problem (SSR) asks, given a multiset A of positive integers, to find two disjoint subsets of A such that the largest-to-smallest ratio of their sums is minimized. In this paper we study the k-version of SSR, namely k-Subset Sum Ratio (k-SSR), which asks to minimize the largest-to-smallest ratio of sums of k disjoint subsets of A. We develop an approximation scheme for k-SSR running in O(n^{2k}/ε^{k-1}) time, where n = |A| and ε is the error parameter. To the best of our knowledge, this is the first FPTAS for k-SSR for fixed k > 2. We also study the k-way Number Partitioning Ratio (k-PART) problem, which differs from k-SSR in that the k subsets must constitute a partition of A; this problem in fact corresponds to the objective of minimizing the largest-to-smallest sum ratio in the family of Multiway Number Partitioning problems. We present a more involved FPTAS for k-PART, also achieving O(n^{2k}/ε^{k-1}) time complexity. Notably, k-PART is also equivalent to the Minimum Envy-Ratio problem with identical valuation functions, which has been studied in the context of fair division of indivisible goods. Thus, for the case of identical valuations, our FPTAS represents a significant improvement over the O(n^{4k²+1}/ε^{2k²}) bound obtained by Nguyen and Rothe’s FPTAS [Trung Thanh Nguyen and Jörg Rothe, 2014] for Minimum Envy-Ratio with general additive valuations. Lastly, we propose a second FPTAS for k-SSR, which employs carefully designed calls to the first one; the new scheme has a time complexity of Õ(n/ε^{3k-1}), thus being much faster when n≫ 1/ ε.
Cite as

Sotiris Kanellopoulos, Giorgos Mitropoulos, Antonis Antonopoulos, Nikos Leonardos, Aris Pagourtzis, Christos Pergaminelis, Stavros Petsalakis, and Kanellos Tsitouras. Approximation Schemes for k-Subset Sum Ratio and k-Way Number Partitioning Ratio. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 44:1-44:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kanellopoulos_et_al:LIPIcs.ISAAC.2025.44,
  author =	{Kanellopoulos, Sotiris and Mitropoulos, Giorgos and Antonopoulos, Antonis and Leonardos, Nikos and Pagourtzis, Aris and Pergaminelis, Christos and Petsalakis, Stavros and Tsitouras, Kanellos},
  title =	{{Approximation Schemes for k-Subset Sum Ratio and k-Way Number Partitioning Ratio}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{44:1--44:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.44},
  URN =		{urn:nbn:de:0030-drops-249521},
  doi =		{10.4230/LIPIcs.ISAAC.2025.44},
  annote =	{Keywords: Fully polynomial-time approximation schemes, Subset Sum Ratio, Number Partitioning, Fair division, Envy minimization, Pseudo-polynomial time algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.100
Cut-Query Algorithms with Few Rounds

Authors: Yotam Kenneth-Mordoch and Robert Krauthgamer

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
In the cut-query model, the algorithm can access the input graph G = (V,E) only via cut queries that report, given a set S ⊆ V, the total weight of edges crossing the cut between S and V⧵ S. This model was introduced by Rubinstein, Schramm and Weinberg [ITCS'18] and its investigation has so far focused on the number of queries needed to solve optimization problems, such as global minimum cut. We turn attention to the round complexity of cut-query algorithms, and show that several classical problems can be solved in this model with only a constant number of rounds. Our main results are algorithms for finding a minimum cut in a graph, that offer different tradeoffs between round complexity and query complexity, where n = |V| and δ(G) denotes the minimum degree of G: (i) Õ(n^{4/3}) cut queries in two rounds in unweighted graphs; (ii) Õ(rn^{1+1/r}/δ(G)^{1/r}) queries in 2r+1 rounds for any integer r ≥ 1 again in unweighted graphs; and (iii) Õ(rn^{1+(1+log_n W)/r}) queries in 4r+3 rounds for any r ≥ 1 in weighted graphs. We also provide algorithms that find a minimum (s,t)-cut and approximate the maximum cut in a few rounds.
Cite as

Yotam Kenneth-Mordoch and Robert Krauthgamer. Cut-Query Algorithms with Few Rounds. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 100:1-100:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kennethmordoch_et_al:LIPIcs.ESA.2025.100,
  author =	{Kenneth-Mordoch, Yotam and Krauthgamer, Robert},
  title =	{{Cut-Query Algorithms with Few Rounds}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{100:1--100:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.100},
  URN =		{urn:nbn:de:0030-drops-245692},
  doi =		{10.4230/LIPIcs.ESA.2025.100},
  annote =	{Keywords: Cut Queries, Round Complexity, Submodular Optimization}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.86
New Algorithms for Pigeonhole Equal Subset Sum

Authors: Ce Jin, Ryan Williams, and Stan Zhang

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We study the Pigeonhole Equal Subset Sum problem, which is a total-search variant of the Subset Sum problem introduced by Papadimitriou (1994): we are given a set of n positive integers {w₁,…,w_n} with the additional restriction that ∑_{i=1}^n w_i < 2ⁿ - 1, and want to find two different subsets A,B ⊆ [n] such that ∑_{i∈A} w_i = ∑_{i∈B} w_i. Very recently, Jin and Wu (ICALP 2024) gave a randomized algorithm solving Pigeonhole Equal Subset Sum in O^*(2^{0.4n}) time, beating the classical meet-in-the-middle algorithm with O^*(2^{n/2}) runtime. In this paper, we refine Jin and Wu’s techniques to improve the runtime even further to O^*(2^{n/3}).
Cite as

Ce Jin, Ryan Williams, and Stan Zhang. New Algorithms for Pigeonhole Equal Subset Sum. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 86:1-86:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jin_et_al:LIPIcs.ESA.2025.86,
  author =	{Jin, Ce and Williams, Ryan and Zhang, Stan},
  title =	{{New Algorithms for Pigeonhole Equal Subset Sum}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{86:1--86:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.86},
  URN =		{urn:nbn:de:0030-drops-245541},
  doi =		{10.4230/LIPIcs.ESA.2025.86},
  annote =	{Keywords: pigeonhole principle, subset sums}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.35
Fooling Near-Maximal Decision Trees

Authors: William M. Hoza and Zelin Lv

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

Abstract
For any constant α > 0, we construct an explicit pseudorandom generator (PRG) that fools n-variate decision trees of size m with error ε and seed length (1 + α) ⋅ log₂ m + O(log(1/ε) + log log n). For context, one can achieve seed length (2 + o(1)) ⋅ log₂ m + O(log(1/ε) + log log n) using well-known constructions and analyses of small-bias distributions, but such a seed length is trivial when m ≥ 2^{n/2}. Our approach is to develop a new variant of the classic concept of almost k-wise independence, which might be of independent interest. We say that a distribution X over {0, 1}ⁿ is k-wise ε-probably uniform if every Boolean function f that depends on only k variables satisfies 𝔼[f(X)] ≥ (1 - ε) ⋅ 𝔼[f]. We show how to sample a k-wise ε-probably uniform distribution using a seed of length (1 + α) ⋅ k + O(log(1/ε) + log log n). Meanwhile, we also show how to construct a set H ⊆ 𝔽₂ⁿ such that every feasible system of k linear equations in n variables over 𝔽₂ has a solution in H. The cardinality of H and the time complexity of enumerating H are at most 2^{k + o(k) + polylog n}, whereas small-bias distributions would give a bound of 2^{2k + O(log(n/k))}. By combining our new constructions with work by Chen and Kabanets (TCS 2016), we obtain nontrivial PRGs and hitting sets for linear-size Boolean circuits. Specifically, we get an explicit PRG with seed length (1 - Ω(1)) ⋅ n that fools circuits of size 2.99 ⋅ n over the U₂ basis, and we get a hitting set with time complexity 2^{(1 - Ω(1)) ⋅ n} for circuits of size 2.49 ⋅ n over the B₂ basis.
Cite as

William M. Hoza and Zelin Lv. Fooling Near-Maximal Decision Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 35:1-35:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hoza_et_al:LIPIcs.APPROX/RANDOM.2025.35,
  author =	{Hoza, William M. and Lv, Zelin},
  title =	{{Fooling Near-Maximal Decision Trees}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{35:1--35:24},
  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.2025.35},
  URN =		{urn:nbn:de:0030-drops-244019},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.35},
  annote =	{Keywords: almost k-wise independence, decision trees, pseudorandom generators}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.55
Lifting to Randomized Parity Decision Trees

Authors: Farzan Byramji and Russell Impagliazzo

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

Abstract
We prove a lifting theorem from randomized decision tree depth to randomized parity decision tree (PDT) size. We use the same property of the gadget, stifling, which was introduced by Chattopadhyay, Mande, Sanyal and Sherif [ITCS 23] to prove a lifting theorem for deterministic PDTs. Moreover, even the milder condition that the gadget has minimum parity certificate complexity at least 2 suffices for lifting to randomized PDT size. To improve the dependence on the gadget g in the lower bounds for composed functions, we consider a related problem g_* whose inputs are certificates of g. It is implicit in the work of Chattopadhyay et al. that for any function f, lower bounds for the *-depth of f_* give lower bounds for the PDT size of f. We make this connection explicit in the deterministic case and show that it also holds for randomized PDTs. We then combine this with composition theorems for *-depth, which follow by adapting known composition theorems for decision trees. As a corollary, we get tight lifting theorems when the gadget is Indexing, Inner Product or Disjointness.
Cite as

Farzan Byramji and Russell Impagliazzo. Lifting to Randomized Parity Decision Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 55:1-55:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{byramji_et_al:LIPIcs.APPROX/RANDOM.2025.55,
  author =	{Byramji, Farzan and Impagliazzo, Russell},
  title =	{{Lifting to Randomized Parity Decision Trees}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{55:1--55:22},
  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.2025.55},
  URN =		{urn:nbn:de:0030-drops-244213},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.55},
  annote =	{Keywords: Parity decision trees, composition}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.24
Computational Geometry with Probabilistically Noisy Primitive Operations

Authors: David Eppstein, Michael T. Goodrich, and Vinesh Sridhar

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
Much prior work has been done on designing computational geometry algorithms that handle input degeneracies, data imprecision, and arithmetic round-off errors. We take a new approach, inspired by the noisy sorting literature, and study computational geometry algorithms subject to noisy Boolean primitive operations in which, e.g., the comparison "is point q above line 𝓁?" returns the wrong answer with some fixed probability. We propose a novel technique called path-guided pushdown random walks that generalizes the results of noisy sorting. We apply this technique to solve point-location, plane-sweep, convex hulls in 2D and 3D, and Delaunay triangulations for noisy primitives in optimal time with high probability.
Cite as

David Eppstein, Michael T. Goodrich, and Vinesh Sridhar. Computational Geometry with Probabilistically Noisy Primitive Operations. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{eppstein_et_al:LIPIcs.WADS.2025.24,
  author =	{Eppstein, David and Goodrich, Michael T. and Sridhar, Vinesh},
  title =	{{Computational Geometry with Probabilistically Noisy Primitive Operations}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.24},
  URN =		{urn:nbn:de:0030-drops-242552},
  doi =		{10.4230/LIPIcs.WADS.2025.24},
  annote =	{Keywords: Computational geometry, noisy comparisons, random walks}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.62
The Complexity of Reachability Problems in Strongly Connected Finite Automata

Authors: Stefan Kiefer and Andrew Ryzhikov

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
Several reachability problems in finite automata, such as completeness of NFAs and synchronisation of total DFAs, correspond to fundamental properties of sets of nonnegative matrices. In particular, the two mentioned properties correspond to matrix mortality and ergodicity, which ask whether there exists a product of the input matrices that is equal to, respectively, the zero matrix and a matrix with a column of strictly positive entries only. The case where the input automaton is strongly connected (that is, the corresponding set of nonnegative matrices is irreducible) frequently appears in applications and often admits better properties than the general case. In this paper, we address the existence of such properties from the computational complexity point of view, and develop a versatile technique to show that several NL-complete problems remain NL-complete in the strongly connected case. In particular, we show that deciding if a binary total DFA is synchronising is NL-complete even if it is promised to be strongly connected, and that deciding completeness of a binary unambiguous NFA with very limited nondeterminism is NL-complete under the same promise.
Cite as

Stefan Kiefer and Andrew Ryzhikov. The Complexity of Reachability Problems in Strongly Connected Finite Automata. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 62:1-62:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kiefer_et_al:LIPIcs.MFCS.2025.62,
  author =	{Kiefer, Stefan and Ryzhikov, Andrew},
  title =	{{The Complexity of Reachability Problems in Strongly Connected Finite Automata}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{62:1--62:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.62},
  URN =		{urn:nbn:de:0030-drops-241690},
  doi =		{10.4230/LIPIcs.MFCS.2025.62},
  annote =	{Keywords: unambiguous automata, nonnegative matrices, irreducible matrix sets, strongly connected automata, matrix monoids, mortality, completeness, synchronisation, ergodicity}
}
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