17,648 research outputs found

    Approximating subset kk-connectivity problems

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    A subset T⊆VT \subseteq V of terminals is kk-connected to a root ss in a directed/undirected graph JJ if JJ has kk internally-disjoint vsvs-paths for every v∈Tv \in T; TT is kk-connected in JJ if TT is kk-connected to every s∈Ts \in T. We consider the {\sf Subset kk-Connectivity Augmentation} problem: given a graph G=(V,E)G=(V,E) with edge/node-costs, node subset T⊆VT \subseteq V, and a subgraph J=(V,EJ)J=(V,E_J) of GG such that TT is kk-connected in JJ, find a minimum-cost augmenting edge-set F⊆E∖EJF \subseteq E \setminus E_J such that TT is (k+1)(k+1)-connected in JâˆȘFJ \cup F. The problem admits trivial ratio O(∣T∣2)O(|T|^2). We consider the case ∣T∣>k|T|>k and prove that for directed/undirected graphs and edge/node-costs, a ρ\rho-approximation for {\sf Rooted Subset kk-Connectivity Augmentation} implies the following ratios for {\sf Subset kk-Connectivity Augmentation}: (i) b(ρ+k)+(3∣T∣∣T∣−k)2H(3∣T∣∣T∣−k)b(\rho+k) + {(\frac{3|T|}{|T|-k})}^2 H(\frac{3|T|}{|T|-k}); (ii) ρ⋅O(∣T∣∣T∣−klog⁥k)\rho \cdot O(\frac{|T|}{|T|-k} \log k), where b=1 for undirected graphs and b=2 for directed graphs, and H(k)H(k) is the kkth harmonic number. The best known values of ρ\rho on undirected graphs are min⁥{∣T∣,O(k)}\min\{|T|,O(k)\} for edge-costs and min⁥{∣T∣,O(klog⁥∣T∣)}\min\{|T|,O(k \log |T|)\} for node-costs; for directed graphs ρ=∣T∣\rho=|T| for both versions. Our results imply that unless k=∣T∣−o(∣T∣)k=|T|-o(|T|), {\sf Subset kk-Connectivity Augmentation} admits the same ratios as the best known ones for the rooted version. This improves the ratios in \cite{N-focs,L}

    Approximating Source Location and Star Survivable Network Problems

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    In Source Location (SL) problems the goal is to select a mini-mum cost source set S⊆VS \subseteq V such that the connectivity (or flow) ψ(S,v)\psi(S,v) from SS to any node vv is at least the demand dvd_v of vv. In many SL problems ψ(S,v)=dv\psi(S,v)=d_v if v∈Sv \in S, namely, the demand of nodes selected to SS is completely satisfied. In a node-connectivity variant suggested recently by Fukunaga, every node vv gets a "bonus" pv≀dvp_v \leq d_v if it is selected to SS. Fukunaga showed that for undirected graphs one can achieve ratio O(kln⁥k)O(k \ln k) for his variant, where k=max⁥v∈Vdvk=\max_{v \in V}d_v is the maximum demand. We improve this by achieving ratio \min\{p^*\lnk,k\}\cdot O(\ln (k/q^*)) for a more general version with node capacities, where p∗=max⁥v∈Vpvp^*=\max_{v \in V} p_v is the maximum bonus and q∗=min⁥v∈Vqvq^*=\min_{v \in V} q_v is the minimum capacity. In particular, for the most natural case p∗=1p^*=1 considered by Fukunaga, we improve the ratio from O(kln⁥k)O(k \ln k) to O(ln⁥2k)O(\ln^2k). We also get ratio O(k)O(k) for the edge-connectivity version, for which no ratio that depends on kk only was known before. To derive these results, we consider a particular case of the Survivable Network (SN) problem when all edges of positive cost form a star. We give ratio O(min⁥{ln⁥n,ln⁥2k})O(\min\{\ln n,\ln^2 k\}) for this variant, improving over the best ratio known for the general case O(k3ln⁥n)O(k^3 \ln n) of Chuzhoy and Khanna

    Monte Carlo optimization approach for decentralized estimation networks under communication constraints

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    We consider designing decentralized estimation schemes over bandwidth limited communication links with a particular interest in the tradeoff between the estimation accuracy and the cost of communications due to, e.g., energy consumption. We take two classes of in–network processing strategies into account which yield graph representations through modeling the sensor platforms as the vertices and the communication links by edges as well as a tractable Bayesian risk that comprises the cost of transmissions and penalty for the estimation errors. This approach captures a broad range of possibilities for “online” processing of observations as well as the constraints imposed and enables a rigorous design setting in the form of a constrained optimization problem. Similar schemes as well as the structures exhibited by the solutions to the design problem has been studied previously in the context of decentralized detection. Under reasonable assumptions, the optimization can be carried out in a message passing fashion. We adopt this framework for estimation, however, the corresponding optimization schemes involve integral operators that cannot be evaluated exactly in general. We develop an approximation framework using Monte Carlo methods and obtain particle representations and approximate computational schemes for both classes of in–network processing strategies and their optimization. The proposed Monte Carlo optimization procedures operate in a scalable and efficient fashion and, owing to the non-parametric nature, can produce results for any distributions provided that samples can be produced from the marginals. In addition, this approach exhibits graceful degradation of the estimation accuracy asymptotically as the communication becomes more costly, through a parameterized Bayesian risk

    Monte Carlo optimization approach for decentralized estimation networks under communication constraints

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    We consider designing decentralized estimation schemes over bandwidth limited communication links with a particular interest in the tradeoff between the estimation accuracy and the cost of communications due to, e.g., energy consumption. We take two classes of in–network processing strategies into account which yield graph representations through modeling the sensor platforms as the vertices and the communication links by edges as well as a tractable Bayesian risk that comprises the cost of transmissions and penalty for the estimation errors. This approach captures a broad range of possibilities for “online” processing of observations as well as the constraints imposed and enables a rigorous design setting in the form of a constrained optimization problem. Similar schemes as well as the structures exhibited by the solutions to the design problem has been studied previously in the context of decentralized detection. Under reasonable assumptions, the optimization can be carried out in a message passing fashion. We adopt this framework for estimation, however, the corresponding optimization schemes involve integral operators that cannot be evaluated exactly in general. We develop an approximation framework using Monte Carlo methods and obtain particle representations and approximate computational schemes for both classes of in–network processing strategies and their optimization. The proposed Monte Carlo optimization procedures operate in a scalable and efficient fashion and, owing to the non-parametric nature, can produce results for any distributions provided that samples can be produced from the marginals. In addition, this approach exhibits graceful degradation of the estimation accuracy asymptotically as the communication becomes more costly, through a parameterized Bayesian risk

    Monte Carlo optimization of decentralized estimation networks over directed acyclic graphs under communication constraints

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    Motivated by the vision of sensor networks, we consider decentralized estimation networks over bandwidth–limited communication links, and are particularly interested in the tradeoff between the estimation accuracy and the cost of communications due to, e.g., energy consumption. We employ a class of in–network processing strategies that admits directed acyclic graph representations and yields a tractable Bayesian risk that comprises the cost of communications and estimation error penalty. This perspective captures a broad range of possibilities for processing under network constraints and enables a rigorous design problem in the form of constrained optimization. A similar scheme and the structures exhibited by the solutions have been previously studied in the context of decentralized detection. Under reasonable assumptions, the optimization can be carried out in a message passing fashion. We adopt this framework for estimation, however, the corresponding optimization scheme involves integral operators that cannot be evaluated exactly in general. We develop an approximation framework using Monte Carlo methods and obtain particle representations and approximate computational schemes for both the in–network processing strategies and their optimization. The proposed Monte Carlo optimization procedure operates in a scalable and efficient fashion and, owing to the non-parametric nature, can produce results for any distributions provided that samples can be produced from the marginals. In addition, this approach exhibits graceful degradation of the estimation accuracy asymptotically as the communication becomes more costly, through a parameterized Bayesian risk

    Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs

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    We present a 6-approximation algorithm for the minimum-cost kk-node connected spanning subgraph problem, assuming that the number of nodes is at least k3(k−1)+kk^3(k-1)+k. We apply a combinatorial preprocessing, based on the Frank-Tardos algorithm for kk-outconnectivity, to transform any input into an instance such that the iterative rounding method gives a 2-approximation guarantee. This is the first constant-factor approximation algorithm even in the asymptotic setting of the problem, that is, the restriction to instances where the number of nodes is lower bounded by a function of kk.Comment: 20 pages, 1 figure, 28 reference

    On Generalizations of Network Design Problems with Degree Bounds

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    Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing spanning tree), and (2) by incorporating `degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure
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