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Electronic Colloquium on Computational Complexity

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All reports by Author Chris Umans:

TR22-063 | 30th April 2022
Vishwas Bhargava, Sumanta Ghosh, Zeyu Guo, Mrinal Kumar, Chris Umans

Fast Multivariate Multipoint Evaluation Over All Finite Fields

Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field $\mathbb{F}$ that outputs the evaluations of an $m$-variate polynomial of ... more >>>

TR11-067 | 25th April 2011
Noga Alon, Amir Shpilka, Chris Umans

On Sunflowers and Matrix Multiplication

Comments: 1

We present several variants of the sunflower conjecture of Erd\H{o}s and Rado and discuss the relations among them.

We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Winograd and Cohn et al. regarding possible approaches for obtaining fast matrix multiplication algorithms. ... more >>>

TR10-186 | 2nd December 2010
Bill Fefferman, Ronen Shaltiel, Chris Umans, Emanuele Viola

On beating the hybrid argument

The {\em hybrid argument}
allows one to relate
the {\em distinguishability} of a distribution (from
uniform) to the {\em
predictability} of individual bits given a prefix. The
argument incurs a loss of a factor $k$ equal to the
bit-length of the
distributions: $\epsilon$-distinguishability implies only
$\epsilon/k$-predictability. ... more >>>

TR09-145 | 20th December 2009
Shankar Kalyanaraman, Chris Umans

The Complexity of Rationalizing Network Formation

We study the complexity of {\em rationalizing} network formation. In
this problem we fix an underlying model describing how selfish
parties (the vertices) produce a graph by making individual
decisions to form or not form incident edges. The model is equipped
with a notion of stability (or equilibrium), and we ... more >>>

TR09-107 | 28th October 2009
Kevin Dick, Chris Umans

Improved inapproximability factors for some $\Sigma_2^p$ minimization problems

We give improved inapproximability results for some minimization problems in the second level of the Polynomial-Time Hierarchy. Extending previous work by Umans [Uma99], we show that several variants of DNF minimization are $\Sigma_2^p$-hard to approximate to within factors of $n^{1/3-\epsilon}$ and $n^{1/2-\epsilon}$ (where the previous results achieved $n^{1/4 - \epsilon}$), ... more >>>

TR09-068 | 1st September 2009
Dave Buchfuhrer, Chris Umans

Limits on the Social Welfare of Maximal-In-Range Auction Mechanisms

Many commonly-used auction mechanisms are ``maximal-in-range''. We show that any maximal-in-range mechanism for $n$ bidders and $m$ items cannot both approximate the social welfare with a ratio better than $\min(n, m^\eta)$ for any constant $\eta < 1/2$ and run in polynomial time, unless $NP \subseteq P/poly$. This significantly improves upon ... more >>>

TR08-021 | 3rd March 2008
Shankar Kalyanaraman, Chris Umans

The Complexity of Rationalizing Matchings

Given a set of observed economic choices, can one infer
preferences and/or utility functions for the players that are
consistent with the data? Questions of this type are called {\em
rationalization} or {\em revealed preference} problems in the
economic literature, and are the subject of a rich body of work.

... more >>>

TR07-069 | 29th July 2007
Ronen Shaltiel, Chris Umans

Low-end uniform hardness vs. randomness tradeoffs for AM

In 1998, Impagliazzo and Wigderson proved a hardness vs. randomness tradeoff for BPP in the {\em uniform setting}, which was subsequently extended to give optimal tradeoffs for the full range of possible hardness assumptions by Trevisan and Vadhan (in a slightly weaker setting). In 2003, Gutfreund, Shaltiel and Ta-Shma proved ... more >>>

TR07-059 | 6th July 2007
Shankar Kalyanaraman, Chris Umans

Algorithms for Playing Games with Limited Randomness

We study multiplayer games in which the participants have access to
only limited randomness. This constrains both the algorithms used to
compute equilibria (they should use little or no randomness) as well
as the mixed strategies that the participants are capable of playing
(these should be sparse). We frame algorithmic ... more >>>

TR06-134 | 18th October 2006
Venkatesan Guruswami, Chris Umans, Salil Vadhan

Extractors and condensers from univariate polynomials

Revisions: 1

We give new constructions of randomness extractors and lossless condensers that are optimal to within constant factors in both the seed length and the output length. For extractors, this matches the parameters of the current best known construction [LRVW03]; for lossless condensers, the previous best constructions achieved optimality to within ... more >>>

TR06-128 | 5th October 2006
Shankar Kalyanaraman, Chris Umans

On obtaining pseudorandomness from error-correcting codes.

A number of recent results have constructed randomness extractors
and pseudorandom generators (PRGs) directly from certain
error-correcting codes. The underlying construction in these
results amounts to picking a random index into the codeword and
outputting $m$ consecutive symbols (the codeword is obtained from
the weak random source in the case ... more >>>

TR04-086 | 12th October 2004
Ronen Shaltiel, Chris Umans

Pseudorandomness for Approximate Counting and Sampling

We study computational procedures that use both randomness and nondeterminism. Examples are Arthur-Merlin games and approximate counting and sampling of NP-witnesses. The goal of this paper is to derandomize such procedures under the weakest possible assumptions.

Our main technical contribution allows one to ``boost'' a given hardness assumption. One special ... more >>>

TR04-001 | 11th December 2003
Lance Fortnow, Russell Impagliazzo, Chris Umans

On the complexity of succinct zero-sum games

We study the complexity of solving succinct zero-sum games,
i.e., the
games whose payoff matrix $M$ is given implicitly by a Boolean circuit
$C$ such that $M(i,j)=C(i,j)$. We complement the known $\EXP$-hardness
of computing the \emph{exact} value of a succinct zero-sum game by
several results on \emph{approximating} the value. (1) ... more >>>

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