2 Search Results for "Raj, Malvika"

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
DOI: 10.4230/LIPIcs.ITCS.2022.3
Improved Merlin-Arthur Protocols for Central Problems in Fine-Grained Complexity

Authors: Shyan Akmal, Lijie Chen, Ce Jin, Malvika Raj, and Ryan Williams

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
In a Merlin-Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin-Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include: - Certifying that a list of n integers has no 3-SUM solution can be done in Merlin-Arthur time Õ(n). Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in Õ(n^{1.5}) time (that is, there is a proof system with proofs of length Õ(n^{1.5}) and a deterministic verifier running in Õ(n^{1.5}) time). - Counting the number of k-cliques with total edge weight equal to zero in an n-node graph can be done in Merlin-Arthur time Õ(n^{⌈ k/2⌉}) (where k ≥ 3). For odd k, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in an m-edge graph can be done in Merlin-Arthur time Õ(m). Previous Merlin-Arthur protocols by Williams [CCC'16] and Björklund and Kaski [PODC'16] could only count k-cliques in unweighted graphs, and had worse running times for small k. - Computing the All-Pairs Shortest Distances matrix for an n-node graph can be done in Merlin-Arthur time Õ(n²). Note this is optimal, as the matrix can have Ω(n²) nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an Õ(n^{2.94}) nondeterministic time algorithm. - Certifying that an n-variable k-CNF is unsatisfiable can be done in Merlin-Arthur time 2^{n/2 - n/O(k)}. We also observe an algebrization barrier for the previous 2^{n/2}⋅ poly(n)-time Merlin-Arthur protocol of R. Williams [CCC'16] for #SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol for k-UNSAT running in 2^{n/2}/n^{ω(1)} time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol. - Certifying a Quantified Boolean Formula is true can be done in Merlin-Arthur time 2^{4n/5}⋅ poly(n). Previously, the only nontrivial result known along these lines was an Arthur-Merlin-Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in 2^{2n/3}⋅poly(n) time. Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution to n integers can be done in Merlin-Arthur time 2^{n/3}⋅poly(n), improving on the previous best protocol by Nederlof [IPL 2017] which took 2^{0.49991n}⋅poly(n) time.
Cite as

Shyan Akmal, Lijie Chen, Ce Jin, Malvika Raj, and Ryan Williams. Improved Merlin-Arthur Protocols for Central Problems in Fine-Grained Complexity. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 3:1-3:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{akmal_et_al:LIPIcs.ITCS.2022.3,
  author =	{Akmal, Shyan and Chen, Lijie and Jin, Ce and Raj, Malvika and Williams, Ryan},
  title =	{{Improved Merlin-Arthur Protocols for Central Problems in Fine-Grained Complexity}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{3:1--3:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.3},
  URN =		{urn:nbn:de:0030-drops-155991},
  doi =		{10.4230/LIPIcs.ITCS.2022.3},
  annote =	{Keywords: Fine-grained complexity, Merlin-Arthur proofs}
}
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