1,186 research outputs found

    Secluded Connectivity Problems

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    Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path PP can be measured as the number of nodes adjacent to it, denoted by N[P]N[P]. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected nn-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of O(2log⁑1βˆ’Ο΅n)O(2^{\log^{1-\epsilon}n}) for any Ο΅>0\epsilon>0 (under an appropriate complexity assumption), but is approximable with ratio Ξ”+3\sqrt{\Delta}+3, where Ξ”\Delta is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs

    On the Size and the Approximability of Minimum Temporally Connected Subgraphs

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    We consider temporal graphs with discrete time labels and investigate the size and the approximability of minimum temporally connected spanning subgraphs. We present a family of minimally connected temporal graphs with nn vertices and Ξ©(n2)\Omega(n^2) edges, thus resolving an open question of (Kempe, Kleinberg, Kumar, JCSS 64, 2002) about the existence of sparse temporal connectivity certificates. Next, we consider the problem of computing a minimum weight subset of temporal edges that preserve connectivity of a given temporal graph either from a given vertex r (r-MTC problem) or among all vertex pairs (MTC problem). We show that the approximability of r-MTC is closely related to the approximability of Directed Steiner Tree and that r-MTC can be solved in polynomial time if the underlying graph has bounded treewidth. We also show that the best approximation ratio for MTC is at least O(2log⁑1βˆ’Ο΅n)O(2^{\log^{1-\epsilon} n}) and at most O(min⁑{n1+Ο΅,(Ξ”M)2/3+Ο΅})O(\min\{n^{1+\epsilon}, (\Delta M)^{2/3+\epsilon}\}), for any constant Ο΅>0\epsilon > 0, where MM is the number of temporal edges and Ξ”\Delta is the maximum degree of the underlying graph. Furthermore, we prove that the unweighted version of MTC is APX-hard and that MTC is efficiently solvable in trees and 22-approximable in cycles

    Bicriteria Network Design Problems

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    We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur

    Approximating subset kk-connectivity problems

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    A subset TβŠ†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 TβŠ†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 FβŠ†Eβˆ–EJF \subseteq E \setminus E_J such that TT is (k+1)(k+1)-connected in JβˆͺFJ \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∣∣Tβˆ£βˆ’k)2H(3∣T∣∣Tβˆ£βˆ’k)b(\rho+k) + {(\frac{3|T|}{|T|-k})}^2 H(\frac{3|T|}{|T|-k}); (ii) ρ⋅O(∣T∣∣Tβˆ£βˆ’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=∣Tβˆ£βˆ’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}

    Spanning trees short or small

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    We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number kk of nodes are required to be connected in the solution. A prototypical example is the kkMST problem in which we require a tree of minimum weight spanning at least kk nodes in an edge-weighted graph. We show that the kkMST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio 2k2\sqrt{k} for the general edge-weighted case and O(k1/4)O(k^{1/4}) for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding kk-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page
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