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URN: urn:nbn:de:0030-drops-155991
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Improved Merlin-Arthur Protocols for Central Problems in Fine-Grained Complexity

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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.

BibTeX - Entry

@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/opus/volltexte/2022/15599},
  URN =		{urn:nbn:de:0030-drops-155991},
  doi =		{10.4230/LIPIcs.ITCS.2022.3},
  annote =	{Keywords: Fine-grained complexity, Merlin-Arthur proofs}
}

Keywords: Fine-grained complexity, Merlin-Arthur proofs
Seminar: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue date: 2022
Date of publication: 25.01.2022


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