16,190 research outputs found

    Approximating max-min linear programs with local algorithms

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    A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constant-size neighbourhood of the node. We study the applicability of local algorithms to max-min LPs where the objective is to maximise minkvckvxv\min_k \sum_v c_{kv} x_v subject to vaivxv1\sum_v a_{iv} x_v \le 1 for each ii and xv0x_v \ge 0 for each vv. Here ckv0c_{kv} \ge 0, aiv0a_{iv} \ge 0, and the support sets Vi={v:aiv>0}V_i = \{v : a_{iv} > 0 \}, Vk={v:ckv>0}V_k = \{v : c_{kv}>0 \}, Iv={i:aiv>0}I_v = \{i : a_{iv} > 0 \} and Kv={k:ckv>0}K_v = \{k : c_{kv} > 0 \} have bounded size. In the distributed setting, each agent vv is responsible for choosing the value of xvx_v, and the communication network is a hypergraph H\mathcal{H} where the sets VkV_k and ViV_i constitute the hyperedges. We present inapproximability results for a wide range of structural assumptions; for example, even if Vi|V_i| and Vk|V_k| are bounded by some constants larger than 2, there is no local approximation scheme. To contrast the negative results, we present a local approximation algorithm which achieves good approximation ratios if we can bound the relative growth of the vertex neighbourhoods in H\mathcal{H}.Comment: 16 pages, 2 figure

    Approximation Algorithms for Covering/Packing Integer Programs

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    Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing min {c.x: x in Z^n_+, Ax > a, Bx < b, x < d}. We give a bicriteria-approximation algorithm that, given epsilon in (0, 1], finds a solution of cost O(ln(m)/epsilon^2) times optimal, meeting the covering constraints (Ax > a) and multiplicity constraints (x < d), and satisfying Bx < (1 + epsilon)b + beta, where beta is the vector of row sums beta_i = sum_j B_ij. Here m denotes the number of rows of A. This gives an O(ln m)-approximation algorithm for CIP -- minimum-cost covering integer programs with multiplicity constraints, i.e., the special case when there are no packing constraints Bx < b. The previous best approximation ratio has been O(ln(max_j sum_i A_ij)) since 1982. CIP contains the set cover problem as a special case, so O(ln m)-approximation is the best possible unless P=NP.Comment: Preliminary version appeared in IEEE Symposium on Foundations of Computer Science (2001). To appear in Journal of Computer and System Science

    Successive Concave Sparsity Approximation for Compressed Sensing

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    In this paper, based on a successively accuracy-increasing approximation of the 0\ell_0 norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the 0\ell_0 norm can be controlled. We prove that the series of the approximations asymptotically coincides with the 1\ell_1 and 0\ell_0 norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization scheme is proposed which leads to a number of weighted 1\ell_1 minimization programs, whereas, in the presence of noise, we propose two iterative thresholding methods that are computationally appealing. A convergence guarantee for the iterative thresholding method is provided, and, for a particular function in the class of the approximating functions, we derive the closed-form thresholding operator. We further present some theoretical analyses via the restricted isometry, null space, and spherical section properties. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms.Comment: Submitted to IEEE Trans. on Signal Processin

    Quantum de Finetti Theorems under Local Measurements with Applications

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    Quantum de Finetti theorems are a useful tool in the study of correlations in quantum multipartite states. In this paper we prove two new quantum de Finetti theorems, both showing that under tests formed by local measurements one can get a much improved error dependence on the dimension of the subsystems. We also obtain similar results for non-signaling probability distributions. We give the following applications of the results: We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the exponential time hypothesis. We show that the maximum winning probability of free games can be estimated in polynomial time by linear programming. We also show that 3-SAT with m variables can be reduced to obtaining a constant error approximation of the maximum winning probability under entangled strategies of O(m^{1/2})-player one-round non-local games, in which the players communicate O(m^{1/2}) bits all together. We show that the optimization of certain polynomials over the hypersphere can be performed in quasipolynomial time in the number of variables n by considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy of semidefinite programs. As an application to entanglement theory, we find a quasipolynomial-time algorithm for deciding multipartite separability. We consider a result due to Aaronson -- showing that given an unknown n qubit state one can perform tomography that works well for most observables by measuring only O(n) independent and identically distributed (i.i.d.) copies of the state -- and relax the assumption of having i.i.d copies of the state to merely the ability to select subsystems at random from a quantum multipartite state. The proofs of the new quantum de Finetti theorems are based on information theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor improvements. v3: added some explanations, mostly about Theorem 1 and Conjecture 5. STOC version. v4, v5. small improvements and fixe
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