Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained Complexity
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Author(s)
Akmal, Shyan
Chen, Lijie
Jin, Ce
Raj, Malvika
Williams, Ryan
Date Issued
February 17, 2023
Publisher
Springer US
Citation
Akmal, Shyan, Chen, Lijie, Jin, Ce, Raj, Malvika and Williams, Ryan. 2023. "Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained Complexity."
Version
Final published version
Abstract
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
$$\tilde{O}(n)$$
O
~
(
n
)
. Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in
$$\tilde{O}(n^{1.5})$$
O
~
(
n
1.5
)
time (that is, there is a proof system with proofs of length
$$\tilde{O}(n^{1.5})$$
O
~
(
n
1.5
)
and a deterministic verifier running in
$$\tilde{O}(n^{1.5})$$
O
~
(
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
$${\tilde{O}}(n^{\lceil k/2\rceil })$$
O
~
(
n
⌈
k
/
2
⌉
)
(where
$$k\ge 3$$
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
$${\tilde{O}}(m)$$
O
~
(
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
$$\tilde{O}(n^2)$$
O
~
(
n
2
)
. Note this is optimal, as the matrix can have
$$\Omega (n^2)$$
Ω
(
n
2
)
nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an
$$\tilde{O}(n^{2.94})$$
O
~
(
n
2.94
)
nondeterministic time algorithm.
Certifying that an n-variable k-CNF is unsatisfiable can be done in Merlin–Arthur time
$$2^{n/2 - n/O(k)}$$
2
n
/
2
-
n
/
O
(
k
)
. We also observe an algebrization barrier for the previous
$$2^{n/2}\cdot \textrm{poly}(n)$$
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
$$2^{n/2}/n^{\omega (1)}$$
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
$$2^{4n/5}\cdot \textrm{poly}(n)$$
2
4
n
/
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
$$2^{2n/3}\cdot \textrm{poly}(n)$$
2
2
n
/
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
$$2^{n/3}\cdot \textrm{poly}(n)$$
2
n
/
3
·
poly
(
n
)
, improving on the previous best protocol by Nederlof [IPL 2017] which took
$$2^{0.49991n}\cdot \textrm{poly}(n)$$
2
0.49991
n
·
poly
(
n
)
time.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00453-023-01102-6
