- Award ID(s):
- 1741137
- NSF-PAR ID:
- 10078625
- Date Published:
- Journal Name:
- Computation Complexity Conference (CCC 2017)
- ISSN:
- 1016-3328
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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null (Ed.)The complexity class ZPP NP[1] (corresponding to zero-error randomized algorithms with access to one NP oracle query) is known to have a number of curious properties. We further explore this class in the settings of time complexity, query complexity, and communication complexity. • For starters, we provide a new characterization: ZPP NP[1] equals the restriction of BPP NP[1] where the algorithm is only allowed to err when it forgoes the opportunity to make an NP oracle query. • Using the above characterization, we prove a query-to-communication lifting theorem , which translates any ZPP NP[1] decision tree lower bound for a function f into a ZPP NP[1] communication lower bound for a two-party version of f . • As an application, we use the above lifting theorem to prove that the ZPP NP[1] communication lower bound technique introduced by Göös, Pitassi, and Watson (ICALP 2016) is not tight. We also provide a “primal” characterization of this lower bound technique as a complexity class.more » « less
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Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in
and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ , by showing that$\mathcal { C}$ admits non-trivial satisfiability and/or$\mathcal { C}$ # SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial# SAT algorithm for a circuit class . Say that a symmetric Boolean function${\mathcal C}$ f (x 1,…,x n ) issparse if it outputs 1 onO (1) values of . We show that for every sparse${\sum }_{i} x_{i}$ f , and for all “typical” , faster$\mathcal { C}$ # SAT algorithms for circuits imply lower bounds against the circuit class$\mathcal { C}$ , which may be$f \circ \mathcal { C}$ stronger than itself. In particular:$\mathcal { C}$ # SAT algorithms forn k -size -circuits running in 2$\mathcal { C}$ n /n k time (for allk ) implyN E X P does not have -circuits of polynomial size.$(f \circ \mathcal { C})$ # SAT algorithms for -size$2^{n^{{\varepsilon }}}$ -circuits running in$\mathcal { C}$ time (for some$2^{n-n^{{\varepsilon }}}$ ε > 0) implyQ u a s i -N P does not have -circuits of polynomial size.$(f \circ \mathcal { C})$ Applying
# SAT algorithms from the literature, one immediate corollary of our results is thatQ u a s i -N P does not haveE M A J ∘A C C 0∘T H R circuits of polynomial size, whereE M A J is the “exact majority” function, improving previous lower bounds againstA C C 0[Williams JACM’14] andA C C 0∘T H R [Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class. -
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 . Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in$$\tilde{O}(n)$$ time (that is, there is a proof system with proofs of length$$\tilde{O}(n^{1.5})$$ and a deterministic verifier running in$$\tilde{O}(n^{1.5})$$ time).$$\tilde{O}(n^{1.5})$$ Counting the number of
k -cliques with total edge weight equal to zero in ann -node graph can be done in Merlin–Arthur time (where$${\tilde{O}}(n^{\lceil k/2\rceil })$$ ). For odd$$k\ge 3$$ k , this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in anm -edge graph can be done in Merlin–Arthur time . Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only count$${\tilde{O}}(m)$$ k -cliques in unweighted graphs, and had worse running times for smallk .Computing the All-Pairs Shortest Distances matrix for an
n -node graph can be done in Merlin–Arthur time . Note this is optimal, as the matrix can have$$\tilde{O}(n^2)$$ nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an$$\Omega (n^2)$$ nondeterministic time algorithm.$$\tilde{O}(n^{2.94})$$ Certifying that an
n -variablek -CNF is unsatisfiable can be done in Merlin–Arthur time . We also observe an algebrization barrier for the previous$$2^{n/2 - n/O(k)}$$ -time Merlin–Arthur protocol of R. Williams [CCC’16] for$$2^{n/2}\cdot \textrm{poly}(n)$$ SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol for$$\#$$ k -UNSAT running in time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol.$$2^{n/2}/n^{\omega (1)}$$ Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time
. 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$$2^{4n/5}\cdot \textrm{poly}(n)$$ time.$$2^{2n/3}\cdot \textrm{poly}(n)$$ n integers can be done in Merlin–Arthur time , improving on the previous best protocol by Nederlof [IPL 2017] which took$$2^{n/3}\cdot \textrm{poly}(n)$$ time.$$2^{0.49991n}\cdot \textrm{poly}(n)$$ -
We investigate the power of randomness in two-party communication complexity. In particular, we study the model where the parties can make a constant number of queries to a function with an efficient one-sided-error randomized protocol. The complexity classes defined by this model comprise the Randomized Boolean Hierarchy, which is analogous to the Boolean Hierarchy but defined with one-sided-error randomness instead of nondeterminism. Our techniques connect the Nondeterministic and Randomized Boolean Hierarchies, and we provide a complete picture of the relationships among complexity classes within and across these two hierarchies. In particular, we prove that the Randomized Boolean Hierarchy does not collapse, and we prove a query-to-communication lifting theorem for all levels of the Nondeterministic Boolean Hierarchy and use it to resolve an open problem stated in the paper by Halstenberg and Reischuk (CCC 1988) which initiated the study of this hierarchy.more » « less
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