836 research outputs found
Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness
We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field ?, in the blackbox setting.
Our algorithm works for every field ? with | ? | ? d+1, and uses only d log n + log (1/ ?) + O(d log log n) random bits to achieve a success probability 1 - ? for some ? > 0. In the low degree regime that is d ? n, it hits the information theoretic lower bound and differs from it only in the lower order terms. Previous best known algorithms achieve the number of random bits (Guruswami-Xing, CCC\u2714 and Bshouty, ITCS\u2714) that are constant factor away from our bound. Like Bshouty, we use Sidon sets for our algorithm. However, we use a new construction of Sidon sets to achieve the improved bound.
We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials. Our contribution is that we give new, very simple proofs for both the constructions
Sharp Concentration of Hitting Size for Random Set Systems
Consider the random set system of {1,2,...,n}, where each subset in the power
set is chosen independently with probability p. A set H is said to be a hitting
set if it intersects each chosen set. The second moment method is used to
exhibit the sharp concentration of the minimal size of H for a variety of
values of p.Comment: 11 page
08381 Abstracts Collection -- Computational Complexity of Discrete Problems
From the 14th of September to the 19th of September, the Dagstuhl Seminar
08381 ``Computational Complexity of Discrete Problems\u27\u27 was held in Schloss Dagstuhl - Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work as well as open problems were discussed.
Abstracts of the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this report. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
The asymptotic value in finite stochastic games
We provide a direct, elementary proof for the existence of , where is the value of a -discounted
finite two-person zero-sum stochastic game
Lower Bounds for Matrix Factorization
We study the problem of constructing explicit families of matrices which
cannot be expressed as a product of a few sparse matrices. In addition to being
a natural mathematical question on its own, this problem appears in various
incarnations in computer science; the most significant being in the context of
lower bounds for algebraic circuits which compute linear transformations,
matrix rigidity and data structure lower bounds.
We first show, for every constant , a deterministic construction in
subexponential time of a family of matrices which cannot
be expressed as a product where the total sparsity of
is less than . In other words, any depth-
linear circuit computing the linear transformation has size at
least . This improves upon the prior best lower bounds for
this problem, which are barely super-linear, and were obtained by a long line
of research based on the study of super-concentrators (albeit at the cost of a
blow up in the time required to construct these matrices).
We then outline an approach for proving improved lower bounds through a
certain derandomization problem, and use this approach to prove asymptotically
optimal quadratic lower bounds for natural special cases, which generalize many
of the common matrix decompositions
The Tensor Track, III
We provide an informal up-to-date review of the tensor track approach to
quantum gravity. In a long introduction we describe in simple terms the
motivations for this approach. Then the many recent advances are summarized,
with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion,
Osterwalder-Schrader positivity...) which, while important for the tensor track
program, are not detailed in the usual quantum gravity literature. We list open
questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure
IST Austria Technical Report
We consider two-player stochastic games played on a finite state space for an infinite num- ber of rounds. The games are concurrent: in each round, the two players (player 1 and player 2) choose their moves independently and simultaneously; the current state and the two moves determine a probability distribution over the successor states. We also consider the important special case of turn-based stochastic games where players make moves in turns, rather than concurrently. We study concurrent games with ω-regular winning conditions specified as parity objectives. The value for player 1 for a parity objective is the maximal probability with which the player can guarantee the satisfaction of the objective against all strategies of the opponent. We study the problem of continuity and robustness of the value function in concurrent and turn-based stochastic parity games with respect to imprecision in the transition probabilities. We present quantitative bounds on the difference of the value function (in terms of the imprecision of the transition probabilities) and show the value continuity for structurally equivalent concurrent games (two games are structurally equivalent if the support of the transition func- tion is same and the probabilities differ). We also show robustness of optimal strategies for structurally equivalent turn-based stochastic parity games. Finally we show that the value continuity property breaks without the structurally equivalent assumption (even for Markov chains) and show that our quantitative bound is asymptotically optimal. Hence our results are tight (the assumption is both necessary and sufficient) and optimal (our quantitative bound is asymptotically optimal)
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