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Electronic Colloquium on Computational Complexity

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Reports tagged with degree:
TR03-020 | 27th March 2003
Elad Hazan, Shmuel Safra, Oded Schwartz

On the Hardness of Approximating k-Dimensional Matching

We study bounded degree graph problems, mainly the problem of
k-Dimensional Matching \emph{(k-DM)}, namely, the problem of
finding a maximal matching in a k-partite k-uniform balanced
hyper-graph. We prove that k-DM cannot be efficiently approximated
to within a factor of $ O(\frac{k}{ \ln k}) $ unless $P = NP$.
This ... more >>>

TR06-097 | 9th August 2006
Emanuele Viola

New correlation bounds for GF(2) polynomials using Gowers uniformity

We study the correlation between low-degree GF(2) polynomials p and explicit functions. Our main results are the following:

(I) We prove that the Mod_m unction on n bits has correlation at most exp(-Omega(n/4^d)) with any GF(2) polynomial of degree d, for any fixed odd integer m. This improves on the ... more >>>

TR07-132 | 8th December 2007
Emanuele Viola

The sum of d small-bias generators fools polynomials of degree d

We prove that the sum of $d$ small-bias generators $L
: \F^s \to \F^n$ fools degree-$d$ polynomials in $n$
variables over a prime field $\F$, for any fixed
degree $d$ and field $\F$, including $\F = \F_2 =

Our result improves on both the work by Bogdanov and
Viola ... more >>>

TR08-048 | 8th April 2008
Meena Mahajan, B. V. Raghavendra Rao

Arithmetic circuits, syntactic multilinearity, and the limitations of skew formulae

Functions in arithmetic NC1 are known to have equivalent constant
width polynomial degree circuits, but the converse containment is
unknown. In a partial answer to this question, we show that syntactic
multilinear circuits of constant width and polynomial degree can be
depth-reduced, though the resulting circuits need not be ... more >>>

TR14-081 | 13th June 2014
Yuval Filmus, Massimo Lauria, Mladen Mikša, Jakob Nordström, Marc Vinyals

From Small Space to Small Width in Resolution

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of CNF formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools ... more >>>

TR14-118 | 9th September 2014
Albert Atserias, Massimo Lauria, Jakob Nordström

Narrow Proofs May Be Maximally Long

We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n^Omega(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size n^O(w) is essentially tight. ... more >>>

TR15-053 | 7th April 2015
Massimo Lauria, Jakob Nordström

Tight Size-Degree Bounds for Sums-of-Squares Proofs

We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size n^{Omega(d)} for values of d = d(n) from constant all the way up to n^{delta} for some universal constant delta. This shows that ... more >>>

TR15-078 | 4th May 2015
Mladen Mikša, Jakob Nordström

A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03] established that if the clause-variable incidence graph of a CNF formula F is a good ... more >>>

TR17-192 | 15th December 2017
Krishnamoorthy Dinesh, Jayalal Sarma

Alternation, Sparsity and Sensitivity : Bounds and Exponential Gaps

Revisions: 1

The well-known Sensitivity Conjecture regarding combinatorial complexity measures on Boolean functions states that for any Boolean function $f:\{0,1\}^n \to \{0,1\}$, block sensitivity of $f$ is polynomially related to sensitivity of $f$ (denoted by $\mathbf{sens}(f)$). From the complexity theory side, the XOR Log-Rank Conjecture states that for any Boolean function, $f:\{0,1\}^n ... more >>>

TR18-152 | 30th August 2018
Krishnamoorthy Dinesh, Jayalal Sarma

Sensitivity, Affine Transforms and Quantum Communication Complexity

In this paper, we study the Boolean function parameters sensitivity ($\mathbf{s}$), block sensitivity ($\mathbf{bs}$), and alternation ($\mathbf{alt}$) under specially designed affine transforms and show several applications. For a function $f:\mathbb{F}_2^n \to \{0,1\}$, and $A = Mx+b$ for $M \in \mathbb{F}_2^{n \times n}$ and $b \in \mathbb{F}_2^n$, the result of the ... more >>>

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