46 research outputs found
Average-Case Hardness of NP and PH from Worst-Case Fine-Grained Assumptions
What is a minimal worst-case complexity assumption that implies non-trivial average-case hardness of NP or PH? This question is well motivated by the theory of fine-grained average-case complexity and fine-grained cryptography. In this paper, we show that several standard worst-case complexity assumptions are sufficient to imply non-trivial average-case hardness of NP or PH:
- NTIME[n] cannot be solved in quasi-linear time on average if UP ? ? DTIME[2^{O?(?n)}].
- ??TIME[n] cannot be solved in quasi-linear time on average if ?_kSAT cannot be solved in time 2^{O?(?n)} for some constant k. Previously, it was not known if even average-case hardness of ??SAT implies the average-case hardness of ??TIME[n].
- Under the Exponential-Time Hypothesis (ETH), there is no average-case n^{1+?}-time algorithm for NTIME[n] whose running time can be estimated in time n^{1+?} for some constant ? > 0.
Our results are given by generalizing the non-black-box worst-case-to-average-case connections presented by Hirahara (STOC 2021) to the settings of fine-grained complexity. To do so, we construct quite efficient complexity-theoretic pseudorandom generators under the assumption that the nondeterministic linear time is easy on average, which may be of independent interest
On the Power of Regular and Permutation Branching Programs
We give new upper and lower bounds on the power of several restricted classes of arbitrary-order read-once branching programs (ROBPs) and standard-order ROBPs (SOBPs) that have received significant attention in the literature on pseudorandomness for space-bounded computation.
- Regular SOBPs of length n and width ?w(n+1)/2? can exactly simulate general SOBPs of length n and width w, and moreover an n/2-o(n) blow-up in width is necessary for such a simulation. Our result extends and simplifies prior average-case simulations (Reingold, Trevisan, and Vadhan (STOC 2006), Bogdanov, Hoza, Prakriya, and Pyne (CCC 2022)), in particular implying that weighted pseudorandom generators (Braverman, Cohen, and Garg (SICOMP 2020)) for regular SOBPs of width poly(n) or larger automatically extend to general SOBPs. Furthermore, our simulation also extends to general (even read-many) oblivious branching programs.
- There exist natural functions computable by regular SOBPs of constant width that are average-case hard for permutation SOBPs of exponential width. Indeed, we show that Inner-Product mod 2 is average-case hard for arbitrary-order permutation ROBPs of exponential width.
- There exist functions computable by constant-width arbitrary-order permutation ROBPs that are worst-case hard for exponential-width SOBPs.
- Read-twice permutation branching programs of subexponential width can simulate polynomial-width arbitrary-order ROBPs
Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four
Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists
an explicit -variate and degree polynomial such
that if any depth four circuit of bounded formal degree which computes
a polynomial of bounded individual degree , that is functionally
equivalent to , then must have size .
The motivation for their work comes from Boolean Circuit Complexity. Based on
a characterization for circuits by Yao [FOCS, 1985] and Beigel and
Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that
functions in can also be computed by algebraic
circuits (i.e., circuits of the form -- sums
of powers of polynomials) of size. Thus they argued that a
"functional" lower bound for an explicit
polynomial against circuits would imply a
lower bound for the "corresponding Boolean function" of against non-uniform
. In their work, they ask if their lower bound be extended to
circuits.
In this paper, for large integers and such that , we show that any circuit of
bounded individual degree at most that
functionally computes Iterated Matrix Multiplication polynomial
() over must have size . Since Iterated
Matrix Multiplication over is functionally in
, improvement of the afore mentioned lower bound to hold for
quasipolynomially large values of individual degree would imply a fine-grained
separation of from
Improved Learning from Kolmogorov Complexity
Carmosino, Impagliazzo, Kabanets, and Kolokolova (CCC, 2016) showed that the existence of natural properties in the sense of Razborov and Rudich (JCSS, 1997) implies PAC learning algorithms in the sense of Valiant (Comm. ACM, 1984), for boolean functions in P/poly, under the uniform distribution and with membership queries. It is still an open problem to get from natural properties learning algorithms that do not rely on membership queries but rather use randomly drawn labeled examples.
Natural properties may be understood as an average-case version of MCSP, the problem of deciding the minimum size of a circuit computing a given truth-table. Problems related to MCSP include those concerning time-bounded Kolmogorov complexity. MKTP, for example, asks for the KT-complexity of a given string. KT-complexity is a relaxation of circuit size, as it does away with the requirement that a short description of a string be interpreted as a boolean circuit. In this work, under assumptions of MKTP and the related problem MK^tP being easy on average, we get learning algorithms for boolean functions in P/poly that
- work over any distribution D samplable by a family of polynomial-size circuits (given explicitly in the case of MKTP),
- only use randomly drawn labeled examples from D, and
- are agnostic (do not require the target function to belong to the hypothesis class). Our results build upon the recent work of Hirahara and Nanashima (FOCS, 2021) who showed similar learning consequences but under a stronger assumption that NP is easy on average
An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams
We present an algorithm for the maximum matching problem in dynamic
(insertion-deletions) streams with *asymptotically optimal* space complexity:
for any -vertex graph, our algorithm with high probability outputs an
-approximate matching in a single pass using bits of
space.
A long line of work on the dynamic streaming matching problem has reduced the
gap between space upper and lower bounds first to factors
[Assadi-Khanna-Li-Yaroslavtsev; SODA 2016] and subsequently to
factors [Dark-Konrad; CCC 2020]. Our upper bound now
matches the Dark-Konrad lower bound up to factors, thus completing this
research direction.
Our approach consists of two main steps: we first (provably) identify a
family of graphs, similar to the instances used in prior work to establish the
lower bounds for this problem, as the only "hard" instances to focus on. These
graphs include an induced subgraph which is both sparse and contains a large
matching. We then design a dynamic streaming algorithm for this family of
graphs which is more efficient than prior work. The key to this efficiency is a
novel sketching method, which bypasses the typical loss of
-factors in space compared to standard -sampling
primitives, and can be of independent interest in designing optimal algorithms
for other streaming problems.Comment: Full version of the paper accepted to ITCS 2022. 42 pages, 5 Figure