A Nearly Linear-Time PTAS for Explicit Fractional Packing and Covering Linear Programs

We give an approximation algorithm for 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 computes feasible primal and dual solutions whose costs are within a factor of 1+eps of the optimal cost in time O((r+c)log(n)/eps^2 + n).Comment: corrected version of FOCS 2007 paper: 10.1109/FOCS.2007.62. Accepted to Algorithmica, 201

Approximation Algorithms for Covering Problems

In this thesis we present sequential and distributed approximation algorithms for covering problems. First, we give a sequential $\ratio$-approximation algorithm for Monotone Covering, a generalization of many fundamental NP-hard covering problems. The approximation ratio $\ratio$ is the maximum number of variables on which any constraint depends. (For example, for vertex cover, $\ratio$ is 2.) The algorithm unifies, generalizes, and improvesmany previous algorithms for fundamental covering problems such as Vertex Cover, Set Cover, Facilities Location, and Covering Mixed-Integer Linear Programs with upper bound on the variables.The algorithm is also a $\ratio$-competitive algorithm for online Monotone Covering, which generalizes online versions of the above-mentioned covering problems as well as many fundamental online paging and caching problems. As such it also generalizes many classical online algorithms, including LRU, FIFO, FWF, Balance, Greedy-Dual, Greedy-Dual Size (a.k.a. Landlord), and algorithms for connection caching, where $\ratio$ is the cache size. It also gives new $\ratio$-competitive algorithms for upgradable variants of these problems, which model choosing the caching strategy andan appropriate hardware configuration (cache size, CPU, bus, network, etc.).Then we show distributed versions of the sequential algorithm. For Weighted Vertex Cover, we give a distributed 2-approximation algorithm taking $O(\log n)$ rounds. The algorithm generalizes to covering mixed integer linear programs (CMIP) with two variables per constraint ($\ratio=2$). For any Monotone Covering problem, we showa distributed $\ratio$-approximation algorithm takingO(\log^2 |\calC|) rounds, where |\calC| is the number of constraints. Last, we extend the distributed algorithms for covering to compute $\ratio$-approximate solutions for Fractional Packing and Maximum Weighted b-Matching in hypergraphs, where $\ratio$ is the maximum number of packing constraints in which a variable appears (for Maximum Weighted b-Matching $\ratio$ is the maximum edge degree --- for graphs $\ratio=2$)

Gaming the Jammer: Is Frequency Hopping Effective

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

Distributed and Parallel Algorithms for Weighted Vertex Cover . . .

The paper presents distributed and parallel δ-approximation algorithms for covering problems, where δ is the maximum number of variables on which any constraint depends (for example, δ = 2 for vertex cover). Specific results include the following. • For weighted vertex cover, the first distributed 2-approximation algorithm taking O(log n) rounds and the first parallel 2-approximation algorithm in RNC. The algorithms generalize to covering mixed integer linear programs (CMIP) with two variables per constraint (δ = 2). • For any covering problem with monotone constraints and submodular cost, a distributed δ-approximation algorithm taking O(log² |C|) rounds, where |C| is the number of constraints. (Special cases include CMIP, facility location, and probabilistic (two-stage) variants of these problems.