1,175 research outputs found
Static Data Structure Lower Bounds Imply Rigidity
We show that static data structure lower bounds in the group (linear) model
imply semi-explicit lower bounds on matrix rigidity. In particular, we prove
that an explicit lower bound of on the cell-probe
complexity of linear data structures in the group model, even against
arbitrarily small linear space , would already imply a
semi-explicit () construction of rigid matrices with
significantly better parameters than the current state of art (Alon, Panigrahy
and Yekhanin, 2009). Our results further assert that polynomial () data structure lower bounds against near-optimal space, would
imply super-linear circuit lower bounds for log-depth linear circuits (a
four-decade open question). In the succinct space regime , we show
that any improvement on current cell-probe lower bounds in the linear model
would also imply new rigidity bounds. Our results rely on a new connection
between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak,
2006), and on a new reduction from worst-case to average-case rigidity, which
is of independent interest
Range Avoidance for Constant Depth Circuits: Hardness and Algorithms
Range Avoidance (Avoid) is a total search problem where, given a Boolean circuit ?: {0,1}? ? {0,1}^m, m > n, the task is to find a y ? {0,1}^m outside the range of ?. For an integer k ? 2, NC?_k-Avoid is a special case of Avoid where each output bit of ? depends on at most k input bits. While there is a very natural randomized algorithm for Avoid, a deterministic algorithm for the problem would have many interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high formula complexity, rigid matrices, and optimal linear codes, reduce to NC??-Avoid, thus establishing conditional hardness of the NC??-Avoid problem. On the other hand, NC??-Avoid admits polynomial-time algorithms, leaving the question about the complexity of NC??-Avoid open.
We give the first reduction of an explicit construction question to NC??-Avoid. Specifically, we prove that a polynomial-time algorithm (with an NP oracle) for NC??-Avoid for the case of m = n+n^{2/3} would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits.
We also give deterministic polynomial-time algorithms for all NC?_k-Avoid problems for m ? n^{k-1}/log(n). Prior work required an NP oracle, and required larger stretch, m ? n^{k-1}
Range Avoidance for Constant-Depth Circuits: Hardness and Algorithms
Range Avoidance (AVOID) is a total search problem where, given a Boolean
circuit , , the task is to find a
outside the range of . For an integer ,
-AVOID is a special case of AVOID where each output bit of
depends on at most input bits. While there is a very natural randomized
algorithm for AVOID, a deterministic algorithm for the problem would have many
interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami,
Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of
high formula complexity, rigid matrices, and optimal linear codes, reduce to
-AVOID, thus establishing conditional hardness of the
-AVOID problem. On the other hand, -AVOID
admits polynomial-time algorithms, leaving the question about the complexity of
-AVOID open.
We give the first reduction of an explicit construction question to
-AVOID. Specifically, we prove that a polynomial-time
algorithm (with an oracle) for -AVOID for the
case of would imply an explicit construction of a rigid matrix,
and, thus, a super-linear lower bound on the size of log-depth circuits.
We also give deterministic polynomial-time algorithms for all
-AVOID problems for . Prior work
required an oracle, and required larger stretch, .Comment: 19 page
Rigid Matrices From Rectangular PCPs
We introduce a variant of PCPs, that we refer to as rectangular PCPs, wherein
proofs are thought of as square matrices, and the random coins used by the
verifier can be partitioned into two disjoint sets, one determining the row of
each query and the other determining the column.
We construct PCPs that are efficient, short, smooth and (almost-)rectangular.
As a key application, we show that proofs for hard languages in ,
when viewed as matrices, are rigid infinitely often. This strengthens and
simplifies a recent result of Alman and Chen [FOCS, 2019] constructing explicit
rigid matrices in FNP. Namely, we prove the following theorem:
- There is a constant such that there is an FNP-machine
that, for infinitely many , on input outputs matrices
with entries in that are -far (in Hamming distance)
from matrices of rank at most .
Our construction of rectangular PCPs starts with an analysis of how
randomness yields queries in the Reed--Muller-based outer PCP of Ben-Sasson,
Goldreich, Harsha, Sudan and Vadhan [SICOMP, 2006; CCC, 2005]. We then show how
to preserve rectangularity under PCP composition and a smoothness-inducing
transformation. This warrants refined and stronger notions of rectangularity,
which we prove for the outer PCP and its transforms.Comment: 36 pages, 3 figure
Minimum Circuit Size, Graph Isomorphism, and Related Problems
We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions from supposedly-intractable problems to MCSP / MKTP hinged on the power of MCSP / MKTP to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive proof system for the complement of Graph Isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include Linear Code Equivalence, Permutation Group Conjugacy, and Matrix Subspace Conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest
Matrix Multiplication Verification Using Coding Theory
We study the Matrix Multiplication Verification Problem (MMV) where the goal
is, given three matrices , , and as input, to decide
whether . A classic randomized algorithm by Freivalds (MFCS, 1979)
solves MMV in time, and a longstanding challenge is to
(partially) derandomize it while still running in faster than matrix
multiplication time (i.e., in time).
To that end, we give two algorithms for MMV in the case where is
sparse. Specifically, when has at most non-zero
entries for a constant , we give (1) a deterministic
-time algorithm for constant , and (2) a randomized -time
algorithm using random bits. The former
algorithm is faster than the deterministic algorithm of K\"{u}nnemann (ESA,
2018) when , and the latter algorithm uses fewer random bits
than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same
time and uses random bits (in turn fewer than Freivalds's
algorithm).
We additionally study the complexity of MMV. We first show that all
algorithms in a natural class of deterministic linear algebraic algorithms for
MMV (including ours) require time. We also show a barrier
to proving a super-quadratic running time lower bound for matrix multiplication
(and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally,
we study relationships between natural variants and special cases of MMV (with
respect to deterministic -time reductions)
Smooth and Strong PCPs
Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs:
- A PCP is strong if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim.
- A PCP is smooth if each location in a proof is queried with equal probability.
We prove that all sets in NP have PCPs that are both smooth and strong, are of polynomial length, and can be verified based on a constant number of queries. This is achieved by following the proof of the PCP theorem of Arora, Lund, Motwani, Sudan and Szegedy (JACM, 1998), providing a stronger analysis of the Hadamard and Reed - Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in NP has a smooth strong canonical PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of NP witnesses to correct proofs. This improves on the recent construction of Dinur, Gur and Goldreich (ITCS, 2019) of PCPPs that are strong canonical but inherently non-smooth.
Our result implies the hardness of approximating the satisfiability of "stable" 3CNF formulae with bounded variable occurrence, where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (SODA, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems
- …