This paper describes a simple greedy ∆-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most ∆ variables of the problem. (A simple example is VERTEX COVER, with ∆ = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems.
We give an approximation algorithm for fractional packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm (with high probability) computes feasible primal and dual solutions whose costs are within a factor of 1 + ε of opt (the optimal cost) in time O((r + c) log(n)/ε 2 + n). 1 IntroductionA packing problem is a linear program of the form max{a · x : M x ≤ b, x ∈ P }, where the entries of the constraint matrix M are non-negative and P is a convex polytope admitting some form of optimization oracle. A covering problem is of the form min{a ·x :This paper focuses on explicitly given packing and covering problems, that is, max{a · x : M x ≤ b, x ≥ 0} and min{a·x : Mx ≥ b,x ≥ 0}, where the polytope P is just the positive orthant. Explicitly given packing and covering are important special cases of linear programming, including, for example, fractional set cover, multicommodity flow problems with given paths, two-player zero-sum matrix games with non-negative payoffs, and variants of these problems.The paper gives a (1 + ε)-approximation algorithm -that is, an algorithm that returns feasible primal and dual solutions whose costs are within a given factor 1 + ε of opt. With high probability, it runs in time O((r + c) log(n)/ε 2 + n), where n -the input size -is the number of non-zero entries in the constraint matrix and r + c is the number of rows plus columns (i.e., constraints plus variables).For dense instances, r + c can be as small as O( √ n). For moderately dense instances -as long as r + c = o(n/ log n) -the 1/ε 2 factor multiplies a sub-linear term. Generally, the time is linear in the input size n as long as ε ≥ Ω( (r + c) log(n)/n). Related workThe algorithm is a Lagrangian-relaxation (a.k.a. price-directed decomposition, multiplicative weights) algorithm. Broadly, these algorithms work by replacing a set of hard constraints by a sum of smooth penalties, one per constraint, and then iteratively augmenting a solution while trading off the increase in the objective against the increase in the sum of penalties. Here the penalties are exponential in the constraint violation, and, in each iteration, only the first-order (linear) approximation is used to estimate the change in the sum of penalties. Such algorithms, which can provide useful alternatives to interior-point and Simplex methods, have a long history and a large literature. Bienstock gives an implementation-oriented, operations-research perspective [2]. Arora et al. discuss them from a computer-science perspective, highlighting connections to other fields such as learning theory [1]. An overview by Todd places them in the context of general linear programming [18].The running times of algorithms of this type increase as the approximation parameter ε gets small. For algorithms that rely on linear approximation of the penalty changes in each iteration, the running times grow at least quadratically in 1/ε (times a polynomial in the other parameters)....
We give an approximation algorithm for fractional packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm (with high probability) computes feasible primal and dual solutions whose costs are within a factor of 1 + ε of opt (the optimal cost) in time O((r + c) log(n)/ε 2 + n). 1 1 Introduction A packing problem is a linear program of the form max{a • x : M x ≤ b, x ∈ P }, where the entries of the constraint matrix M are non-negative and P is a convex polytope admitting some form of optimization oracle. A covering problem is of the form min{a •x : Mx ≥ b,x ∈ P }. This paper focuses on explicitly given packing and covering problems, that is, max{a • x : M x ≤ b, x ≥ 0} and min{a•x : Mx ≥ b,x ≥ 0}, where the polytope P is just the positive orthant. Explicitly given packing and covering are important special cases of linear programming, including, for example, fractional set cover, multicommodity flow problems with given paths, two-player zero-sum matrix games with non-negative payoffs, and variants of these problems. The paper gives a (1 + ε)-approximation algorithm-that is, an algorithm that returns feasible primal and dual solutions whose costs are within a given factor 1 + ε of opt. With high probability, it runs in time O((r + c) log(n)/ε 2 + n), where n-the input size-is the number of non-zero entries in the constraint matrix and r + c is the number of rows plus columns (i.e., constraints plus variables). For dense instances, r + c can be as small as O(√ n). For moderately dense instances-as long as r + c = o(n/ log n)-the 1/ε 2 factor multiplies a sub-linear term. Generally, the time is linear in the input size n as long as ε ≥ Ω((r + c) log(n)/n).
Abstract-Frequency hopping has been the most popularly considered approach for alleviating the effects of jamming attacks. In this paper, we provide a novel, measurement-driven, game theoretic framework that captures the interactions between a communication link and an adversarial jammer, possibly with multiple jamming devices, in a wireless network employing frequency hopping (FH). The framework can be used to quantify the efficacy of FH as a jamming countermeasure. Our model accounts for two important factors that affect the aforementioned interactions: (a) the number of orthogonal channels available for use and (b) the frequency separation between these orthogonal bands. If the latter is small, then the energy spill over between two adjacent channels (considered orthogonal) is high; as a result a jammer on an orthogonal band that is adjacent to that used by a legitimate communication, can be extremely effective. We account for both these factors and using our framework we provide bounds on the performance of proactive frequency hopping in alleviating the impact of a jammer. The main contributions of our work are: (a) Construction of a measurement driven game theoretic framework which models the interactions between a jammer and a communication link that employ FH. (b) Extensive experimentation on our indoor testbed in order to quantify the impact of a jammer in a 802.11a/g network. (c) Application of our framework to quantify the efficacy of proactive FH across a variety of 802.11 network configurations. (d) Formal derivation of the optimal strategies for both the link and the jammer in 802.11 networks. Our results demonstrate that frequency hopping is largely inadequate in coping with jamming attacks in current 802.11 networks. In particular, we show that if current systems were to support hundreds of additional channels, FH would form a robust jamming countermeasure 1 .
We give an approximation algorithm for packing and covering linear programs (linear programs with nonnegative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm (with high probability) computes feasible primal and dual solutions whose costs are within a factor of 1 + ε of OPT (the optimal cost) in time O(n + (r + c) log(n)/ε 2 ).For dense problems (with r, c = O( √ n)) the time is O(n + √ n log(n)/ε 2 ) -linear even as ε → 0. In comparison, previous Lagrangian-relaxation algorithms generally take at least Ω(n log(n)/ε 2 ) time, while (for small ε) the Simplex algorithm typically takes at least Ω(n min(r, c)) time.
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