Showing papers in "Theory of Computing in 2007"

Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number

Abstract: A randomness extractor is an algorithm which extracts randomness from a low-quality random source, using some additional truly random bits. We construct new extractors which require only log n + O(1) additional random bits for sources with constant entropy rate. We further construct dispersers, which are similar to one-sided extractors, which use an arbitrarily small constant times log n additional random bits for sources with constant entropy rate. Our extractors and dispersers output 1-α fraction of the randomness, for any α>0.We use our dispersers to derandomize results of Hastad [23] and Feige-Kilian [19] and show that for all e>0, approximating MAX CLIQUE and CHROMATIC NUMBER to within n1-e are NP-hard. We also derandomize the results of Khot [29] and show that for some γ > 0, no quasi-polynomial time algorithm approximates MAX CLIQUE or CHROMATIC NUMBER to within n/2(log n)1-γ, unless NP = P.Our constructions rely on recent results in additive number theory and extractors by Bourgain-Katz-Tao [11], Barak-Impagliazzo-Wigderson [5], Barak-Kindler-Shaltiel-Sudakov-Wigderson [6], and Raz [36]. We also simplify and slightly strengthen key theorems in the second and third of these papers, and strengthen a related theorem by Bourgain [10].

The Randomized Communication Complexity of Set Disjointness

Abstract: We study the communication complexity of the disjointness function, in which each of two players holds a k-subset of a universe of size n and the goal is to determine whether the sets are disjoint. In the model of a common random string we prove that O(k) communication bits are sufficient, regardless of n. In the model of private random coins O(k+ log log n) bits suffice. Both results are asymptotically tight.

An Exponential Separation between Regular and General Resolution.

Abstract: Two distinct proofs of an exponential separation between regular resolution and unrestricted resolution are given. The previous best known separation between these systems was quasi-polynomial.

Approximation Algorithms and Online Mechanisms for Item Pricing

Abstract: We present approximation and online algorithms for a number of problems of pricing items for sale so as to maximize seller's revenue in an unlimited supply setting. Our first result is an O(k)-approximation algorithm for pricing items to single-minded bidders who each want at most k items. This improves over recent independent work of Briest and Krysta [5] who achieve an O(k2) bound. For the case k = 2, where we obtain a 4-approximation, this can be viewed as the following graph vertex pricing problem: given a (multi) graph G with valuations we on the edges, find prices pi ≥ 0 for the vertices to maximize Σ (pi + pj). {e=(i,j):we ≥ pi + pj.We also improve the approximation of Guruswami et al. [11] from O(log m + log n) to O(log n), where m is the number of bidders and n is the number of items, for the "highway problem" in which all desired subsets are intervals on a line.Our approximation algorithms can be fed into the generic reduction of Balcan et al. [2] to yield an incentive-compatible auction with nearly the same performance guarantees so long as the number of bidders is sufficiently large. In addition, we show how our algorithms can be combined with results of Blum and Hartline [3], Blum et al. [4], and Kalai and Vempala [13] to achieve good performance in the online setting, where customers arrive one at a time and each must be presented a set of item prices based only on knowledge of the customers seen so far.

Quantum Versus Classical Proofs and Advice

Abstract: This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms, whether QMA=QCMA. We prove three results about this question. First, we give a "quantum oracle separation" between QMA and QCMA. More concretely, we show that any quantum algorithm needs $\Omega(\sqrt{2^n/(m+1)})$ queries to find an $n$-qubit "marked state" $\lvert\psi\rangle$, even if given an $m$-bit classical description of $\lvert\psi\rangle$ together with a quantum black box that recognizes $\lvert\psi\rangle$. Second, we give an explicit QCMA protocol that nearly achieves this lower bound. Third, we show that, in the one previously-known case where quantum proofs seemed to provide an exponential advantage, classical proofs are basically just as powerful. In particular, Watrous gave a QMA protocol for verifying non-membership in finite groups. Under plausible group-theoretic assumptions, we give a QCMA protocol for the same problem. Even with no assumptions, our protocol makes only polynomially many queries to the group oracle. We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA.

An O(log n ) Approximation Ratio for the Asymmetric Traveling Salesman Path Problem

Abstract: Given an arc-weighted directed graph G = (V,A,l) and a pair of vertices s,t, we seek to find an s-twalk of minimum length that visits all the vertices in V. If l satisfies the asymmetric triangle inequality, the problem is equivalent to that of finding an s-tpath of minimum length that visits all the vertices. We refer to this problem as ATSPP. When s = t this is the well known asymmetric traveling salesman tour problem (ATSP). Although an O(logn) approximation ratio has long been known for ATSP, the best known ratio for ATSPP is $O(\sqrt{n})$. In this paper we present a polynomial time algorithm for ATSPP that has approximation ratio of O(logn). The algorithm generalizes to the problem of finding a minimum length path or cycle that is required to visit a subset of vertices in a given order.

A Simple PromiseBQP-complete Matrix Problem

Abstract: Let A be a real symmetric matrix of size N such that the number of non-zero entries in each row is polylogarithmic in N and the positions and the values of these entries are specified by an efficiently computable function. We consider the problem of estimating an arbitrary diagonal entry (A m ) j j of the matrix A m up to an error of e b m , where b is an a priori given upper bound on the norm of A and m and e are polylogarithmic and inverse polylogarithmic in N, respectively. We show that this problem is PromiseBQP-complete. It can be solved efficiently on a quantum computer by repeatedly applying measurements of A to the jth basis vector and raising the outcome to the mth power. Conversely, every uniform quantum circuit of polynomial length can be encoded into a sparse matrix such that some basis vector | ji corresponding to the input induces two different spectral measures depending on whether the input is accepted or not. These measures can be distinguished by estimating the mth statistical moment for some appropriately chosen m, i. e., by the jth diagonal entry of A m . The problem remains PromiseBQP-hard when restricted to matrices having only 1, 0, and 1 as entries. Estimating off-diagonal entries is also PromiseBQP-complete.

Easily refutable subformulas of large random 3CNF formulas

Abstract: A simple nonconstructive argument shows that most 3CNF formulas with cn clauses (where c is a large enough constant) are not satisfiable. It is an open question whether there is an efficient refutation algorithm that for most formulas with cn clauses proves that they are not satisfiable. We present a polynomial time algorithm that for most 3CNF formulas with cn 3/2 clauses (where c is a large enough constant) finds a subformula with O(c 2 n) clauses and then proves that this subformula is not satisfiable (and hence that the original formula is not satisfiable). Previously, it was only known how to efficiently certify the unsatisfiability of random 3CNF formulas with at least poly(log(n)) · n 3/2 clauses. Our algorithm is simple enough to run in practice. We present some preliminary experimental results.

Censorship Resistant Peer-to-Peer Networks

Abstract: We present a censorship resistant peer-to-peer network for accessing n data items in a network of n nodes Each search for a data item in the network takes O(log n) time and requires at most O(log 2 n) messages Our network is censorship resistant in the sense that even after adversarial removal of an arbitrarily large constant fraction of the nodes in the network, all but an arbitrarily small fraction of the remaining nodes can obtain all but an arbitrarily small fraction of the original data items The network can be created in a fully distributed fashion It requires only O(log n) memory in each node We also give a variant of our scheme that has the property that it is highly spam resistant: an adversary can take over complete control of a constant fraction of the nodes in the network and yet will still be unable to generate spam

An Ω( n 1/3 ) Lower Bound for Bilinear Group Based Private Information Retrieval

Abstract: A two server private information retrieval (PIR) scheme allows a user U to retrieve the i-th bit of an n-bit string x replicated between two servers while each server individually learns no information about i. The main parameter of interest in a PIR scheme is its communication complexity, namely the number of bits exchanged by the user and the servers. A large amount of effort has been invested by researchers over the last decade in search for efficient PIR schemes. A number of different schemes [6, 4, 19] have been proposed, however all of them ended up with the same communication complexity of O(n1/3). The best known lower bound to date is 5logn by [17]. The tremendous gap between upper and lower bounds is the focus of our paper. We show an Ω(n1/3) lower bound in a restricted model that nevertheless captures all known upper bound techniques. Our lower bound applies to bilinear group based PIR schemes. Abilinear PIR scheme is a one round PIR scheme, where user computes the dot product of servers’ responses to obtain the desired value of the i-th bit. Every linear scheme can be turned into a bilinear one with an asymptotically negligible communication overhead. A group based PIR scheme is a PIR scheme that involves servers representing database by a function on a certain finite group G, and allows user to retrieve the value of this function at any group element using the natural secret sharing scheme based on G. Our proof relies on representation theory of finite groups.

Removing Degeneracy May Require a Large Dimension Increase

Abstract: Many geometric algorithms are formulated for input objects in general position; sometimes this is for convenience and simplicity, and sometimes it is essential for the al- gorithm to work at all. For arbitrary inputs this requires removing degeneracies, which has usually been solved by relatively complicated and computationally demanding perturbation methods. The result of this paper can be regarded as an indication that the problem of removing degeneracies has no simple "abstract" solution. We consider LP-type problems, a successful axiomatic framework for optimization problems capturing, e. g., linear programming and the smallest enclosing ball of a point set. For infinitely many integers D we construct a D- dimensional LP-type problem such that in order to remove degeneracies from it, we have to increase the dimension to at least (1+e)D, where e > 0 is an absolute constant. The proof consists of showing that certain posets cannot be covered by pairwise disjoint copies of Boolean algebras under some restrictions on their placement. To this end, we prove that certain systems of linear inequalities are unsolvable, which seems to require surprisingly precise calculations.

On the Hardness of Satisfiability with Bounded Occurrences in the Polynomial-Time Hierarchy

Abstract: In 1991, Papadimitriou and Yannakakis gave a reduction implying the NP- hardness of approximating the problem 3-SAT with bounded occurrences. Their reduction is based on expander graphs. We present an analogue of this result for the second level of the polynomial-time hierarchy based on superconcentrator graphs. This resolves an open question of Ko and Lin (1995) and should be useful in deriving inapproximability results in the polynomial-time hierarchy. More precisely, we show that given an instance of 89-3-SAT in which every variable occurs at most B times (for some absolute constant B), it is Π2-hard to distinguish between the following two cases: YES instances, in which for any assignment to the universal vari- ables there exists an assignment to the existential variables that satisfies all the clauses, and NO instances in which there exists an assignment to the universal variables such that any