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Electronic Colloquium on Computational Complexity

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REPORTS > KEYWORD > DISCREPANCY:
Reports tagged with Discrepancy:
TR94-025 | 12th December 1994
David P. Dobkin, Dimitrios Gunopulos

Computing the Maximum Bichromatic Discrepancy with applications to Computer Graphics and Machine Learning


Computing the maximum bichromatic discrepancy is an interesting
theoretical problem with important applications in computational
learning theory, computational geometry and computer graphics.
In this paper we give algorithms to compute the maximum
bichromatic discrepancy for simple geometric ranges, including
rectangles and halfspaces.
In addition, we give ... more >>>


TR07-050 | 25th May 2007
Arkadev Chattopadhyay

Discrepancy and the power of bottom fan-in in depth-three circuits

We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty 'Number on the Forehead' model. Our method is based on the notion of voting polynomial degree of functions and extends the Degree-Discrepancy Lemma in the recent work of Sherstov (STOC'07). ... more >>>


TR07-085 | 2nd September 2007
Ran Raz, Amir Yehudayoff

Multilinear Formulas, Maximal-Partition Discrepancy and Mixed-Sources Extractors

We study multilinear formulas, monotone arithmetic circuits, maximal-partition discrepancy, best-partition communication complexity and extractors constructions. We start by proving lower bounds for an explicit polynomial for the following three subclasses of syntactically multilinear arithmetic formulas over the field C and the set of variables {x1,...,xn}:

1. Noise-resistant. A syntactically multilinear ... more >>>


TR08-002 | 19th December 2007
Arkadev Chattopadhyay, Anil Ada

Multiparty Communication Complexity of Disjointness

Revisions: 3

We extend the 'Generalized Discrepancy' technique suggested by Sherstov to the `Number on the Forehead' model of multiparty communication. This allows us to prove strong lower bounds of n^{\Omega(1)} on the communication needed by k players to compute the Disjointness function, provided $k$ is a constant. In general, our method ... more >>>


TR08-014 | 26th February 2008
Matei David

Separating NOF communication complexity classes RP and NP

We provide a non-explicit separation of the number-on-forehead communication complexity classes RP and NP when the number of players is up to \delta log(n) for any \delta<1. Recent lower bounds on Set-Disjointness [LS08,CA08] provide an explicit separation between these classes when the number of players is only up to o(loglog(n)).

... more >>>

TR12-004 | 10th January 2012
Marcos Villagra, Masaki Nakanishi, Shigeru Yamashita, Yasuhiko Nakashima

Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication

Revisions: 3

In this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs. Here we focus on the first one. A strong quantum nondeterministic protocol accepts a correct input ... more >>>


TR13-159 | 20th November 2013
Per Austrin, Venkatesan Guruswami, Johan HÃ¥stad

$(2+\epsilon)$-SAT is NP-hard

Revisions: 2

We prove the following hardness result for a natural promise variant of the classical CNF-satisfiability problem: Given a CNF-formula where each clause has width $w$ and the guarantee that there exists an assignment satisfying at least $g = \lceil \frac{w}{2}\rceil -1$ literals in each clause, it is NP-hard to find ... more >>>


TR15-190 | 2nd November 2015
Esther Ezra, Shachar Lovett

On the Beck-Fiala Conjecture for Random Set Systems

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems $(X,\Sigma)$, where each element $x \in X$ lies in $t$ randomly selected sets of $\Sigma$, where $t$ is an integer parameter. We provide new bounds in two regimes of parameters. We ... more >>>


TR16-095 | 7th June 2016
Arkadev Chattopadhyay, Nikhil Mande

Small Error Versus Unbounded Error Protocols in the NOF Model

Revisions: 1 , Comments: 1

We show that a simple function has small unbounded error communication complexity in the $k$-party number-on-forehead (NOF) model but every probabilistic protocol that solves it with sub-exponential advantage over random guessing has cost essentially $\Omega\left(\frac{\sqrt{n}}{4^k}\right)$ bits. Such a separation was first shown for $k=2$ independently by Buhrman et al. ['07] ... more >>>


TR17-062 | 9th April 2017
Arkadev Chattopadhyay, Nikhil Mande

Dual polynomials and communication complexity of XOR functions

We show a new duality between the polynomial margin complexity of $f$ and the discrepancy of the function $f \circ$ XOR, called an XOR function. Using this duality,
we develop polynomial based techniques for understanding the bounded error (BPP) and the weakly-unbounded error (PP) communication complexities of XOR functions. ... more >>>




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