Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained Complexity

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.