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Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation
TLDR
In general, the value of any polynomial in Valiant's class ${\sf VP}$ can be certified faster than "exhaustive summation" over all possible assignments.
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More Applications of the Polynomial Method to Algorithm Design
TLDR
This paper extends the polynomial method to solve a number of problems in combinatorial pattern matching and Boolean algebra, considerably faster than previously known methods.
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Distributed PCP Theorems for Hardness of Approximation in P
TLDR
Under SETH, it is shown that there are no truly-subquadratic approximation algorithms for the following problems: Maximum Inner Product over \{0,1\}-vectors, LCS Closest Pair over permutations, Approximate Partial Match,Approximate Regular Expression Matching, and Diameter in Product Metric.
View on IEEE
Polynomial Representations of Threshold Functions and Algorithmic Applications
TLDR
New polynomials for representing threshold functions in three different regimes are designed: probabilistic polynomic constructions of low degree, which need far less randomness than previous constructions, and polynomial threshold functions (PTFs) with "nice" threshold behavior and degree almost as low as the probabilists.
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Probabilistic rank and matrix rigidity
TLDR
A notion of Probabilistic rank and probabilistic sign-rank of a matrix, which measure the extent to which a matrix can be probabilistically represented by low-rank matrices, is considered, which shows that for every function f which is randomly self-reducible in a natural way, Bounding the communication complexity of f is equivalent to bounding the rigidity of the matrix of f, via an equivalence with probabilism rank.
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Subcubic Equivalences Between Path, Matrix, and Triangle Problems
TLDR
Generic equivalences between matrix products over a large class of algebraic structures used in optimization, verifying a matrix product over the same structure, and corresponding triangle detection problems over the structure are shown.
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Beating Brute Force for Systems of Polynomial Equations over Finite Fields
TLDR
The algorithm for systems of ΣΠΣ polynomials also introduces a new degree reduction method that takes an instance of the problem and outputs a subexponential-sized set of instances, in such a way that feasibility is preserved and every polynomial among the output instances has degree O(log(s/n).
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Beating Exhaustive Search for Quantified Boolean Formulas and Connections to Circuit Complexity
TLDR
Solving satisfiability of quantified CNF formulas with n variables, poly(n) size and at most q quantifier blocks can be solved in time 2n−nwq(1/q), then the complexity class NEXP does not have O(log n) depth circuits of polynomial size.
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On the connections between backdoors, restarts, and heavy-tailedness in combinatorial search
TLDR
Of the many enhancements of DPLL, this analysis will focus on the interplay between certain special features of problem instances, polytime propagation methods, and restart techniques.
On the (Non) NP-Hardness of Computing Circuit Complexity
TLDR
It is shown that MCSP is provably not NP-hard under O(n1/2-e)-time projections, and it is proved that the Σ2P-hardness of NMCSP, even under arbitrary polynomial-time reductions, would imply EXP ⊄ P/poly.
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