Showing papers in "Theory of Computing in 2008"

A Quantum Algorithm for the Hamiltonian NAND Tree

Abstract: We give a quantum algorithm for the binary NAND tree problem in the Hamil- tonian oracle model. The algorithm uses a continuous time quantum walk with a running time proportional to p N. We also show a lower bound of W( p N) for the NAND tree problem in the Hamiltonian oracle model.

A Brief Introduction to Fourier Analysis on the Boolean Cube.

Abstract: We give a brief introduction to the basic notions of Fourier analysis on the Boolean cube, illustrated and motivated by a number of applications to theoretical computer science. ACM Classification: F.0, F.2 AMS Classification: 42-02, 68-02, 68Q17, 68Q25, 68Q32

Near-Optimal Network Design with Selfish Agents

Abstract: We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possible edges in the network have costs and each agent's goal is to pay as little as possible. Determining whether or not a Nash equilibrium exists in this game is NP-complete. However, when the goal of each player is to connect a terminal to a common source, we prove that there is a Nash equilibrium as cheap as the optimal network, and give a polynomial time algorithm to find a (1+e)-approximate Nash equilibrium that does not cost much more. For the general connection game we prove that there is a 3-approximate Nash equilibrium that is as cheap as the optimal network, and give an algorithm to find a (4.65+e)-approximate Nash equilibrium that does not cost much more.

Norms, XOR Lemmas, and Lower Bounds for Polynomials and Protocols

Abstract: This paper presents a unified and simple treatment of basic questions concern- ing two computational models: multiparty communication complexity and polynomials over GF(2). The key is the use of (known) norms on Boolean functions, which capture their proximity to each of these models (and are closely related to property testers of this proximity). The main contributions are new XOR lemmas. We show that if a Boolean function has correlation at most e 1/2 with either of these models, then the correlation of the parity of its values on m independent instances drops exponentially with m. More specifically: • For polynomials over GF(2) of degree d, the correlation drops to exp m/4 d . No

The One-Way Communication Complexity of Hamming Distance

Abstract: Consider the following version of the Hamming distance problem for ±1 vec- tors of length n: the promise is that the distance is either at least n + p n or at most n p n, and the goal is to find out which of these two cases occurs. Woodruff (Proc. ACM-SIAM Symposium on Discrete Algorithms, 2004) gave a linear lower bound for the randomized one-way communication complexity of this problem. In this note we give a simple proof of this result. Our proof uses a simple reduction from the indexing problem and avoids the VC-dimension arguments used in the previous paper. As shown by Woodruff (loc. cit.), this implies an W(1/e 2 )-space lower bound for approximating frequency moments within a factor 1+e in the data stream model.

Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators, and applications

Abstract: Ahlswede and Winter (IEEE Trans. Inf. Th. 2002) introduced a Chernoff bound for matrix-valued random variables, which is a non-trivial generalization of the usual Chernoff bound for real-valued random variables. We present an efficient derandomization of their bound using the method of pessimistic estimators (see Raghavan (JCSS 1988)). As a consequence, we derandomize an efficient construction by Alon and Roichman (RSA 1994) of an expanding Cayley graph of logarithmic degree on any (possibly non-abelian) group. This gives an optimal solution to the homomorphism testing problem of Shpilka and Wigderson (STOC 2004). We also apply these pessimistic estimators to the problem of solving semidefinite covering problems, thus giving a deterministic algorithm for the quantum hypergraph cover problem of Ahslwede and Winter.

Approximation Algorithms for Unique Games

Abstract: We present a polynomial time algorithm based on semidefinite programming that, given a unique game of value 1 - O(1/logn), satisfies a constant fraction of constraints, where n is the number of variables. For sufficiently large alphabets, it improves an algorithm of Khot (STOC'02) that satisfies a constant fraction of constraints in unique games of value 1 -O(1/(k/sup 10/(log k)/sup 5/)), where k is the size of the alphabet. We also present a simpler algorithm for the special case of unique games with linear constraints. Finally, we present a simple approximation algorithm for 2-to-1 games.

Optimal lower bounds for the Korkine-Zolotareff parameters of a lattice and for Schnorr's algorithm for the shortest vector problem

Abstract: Schnorr’s algorithm for finding an approximation for the shortest nonzero vector in an n-dimensional lattice depends on a parameter k. He proved that for a fixed k ≤ n his algorithm (block 2k-reduction) provides a lattice vector whose length is greater than the length of a shortest nonzero vector in the lattice by at most a factor of (2k)2n/k. (The time required by the algorithm depends on k.) We show that if k = o(n), this bound on the performance of Schnorr’s algorithm cannot be improved (apart from a constant factor in the exponent). Namely, we prove the existence of a basis in Rn which is KZ-reduced on all k-segments and where the ratio ‖b1‖/shortest(L) is at least kcn/k. Noting that such a basis renders all versions of Schnorr’s algorithm idle (output = input), it follows that the quantity kcn/k is a lower bound on the approximation ratio any version of Schnorr’s algorithm can achieve on the shortest vector problem. This proves that Schnorr’s analysis of ∗A preliminary version of this paper has appeared in the Proc. 35th ACM Symp. on Theory of Computing [2]. ACM Classification: F.2.2, G.2 AMS Classification: 68Q25, 68W40, 68W25, 11H55, 11H99, 52C07, 60D05

On the LP Relaxation of the Asymmetric Traveling Salesman Path Problem

Abstract: This is a comment on the article "An O(logn) Approximation Ratio for the Asymmetric Traveling Salesman Path Problem" by C. Chekuri and M. PTheory of Com- puting 3(2007), 197-209. We observe that the LP relaxation for the Asymmetric Traveling Salesman Path Problem suggested in Section 5 of that paper is not accurate, and state a corrected linear relaxation for the problem. The inaccuracy occurs in the statement of an open problem and does not affect the validity of any of the results in the Chekuri-Ppaper.