5 research outputs found
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
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)
Stability-Preserving, Time-Efficient Mechanisms for School Choice in Two Rounds
We address the following dynamic version of the school choice question: a city, named City, admits students in two temporally-separated rounds, denoted R? and R?. In round R?, the capacity of each school is fixed and mechanism M? finds a student optimal stable matching. In round R?, certain parameters change, e.g., new students move into the City or the City is happy to allocate extra seats to specific schools. We study a number of Settings of this kind and give polynomial time algorithms for obtaining a stable matching for the new situations.
It is well established that switching the school of a student midway, unsynchronized with her classmates, can cause traumatic effects. This fact guides us to two types of results: the first simply disallows any re-allocations in round R?, and the second asks for a stable matching that minimizes the number of re-allocations. For the latter, we prove that the stable matchings which minimize the number of re-allocations form a sublattice of the lattice of stable matchings. Observations about incentive compatibility are woven into these results. We also give a third type of results, namely proofs of NP-hardness for a mechanism for round R? under certain settings