12,559 research outputs found

    Approximating k-Connected m-Dominating Sets

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    A subset SS of nodes in a graph GG is a kk-connected mm-dominating set ((k,m)(k,m)-cds) if the subgraph G[S]G[S] induced by SS is kk-connected and every vVSv \in V \setminus S has at least mm neighbors in SS. In the kk-Connected mm-Dominating Set ((k,m)(k,m)-CDS) problem the goal is to find a minimum weight (k,m)(k,m)-cds in a node-weighted graph. For mkm \geq k we obtain the following approximation ratios. For general graphs our ratio O(klnn)O(k \ln n) improves the previous best ratio O(k2lnn)O(k^2 \ln n) and matches the best known ratio for unit weights. For unit disc graphs we improve the ratio O(klnk)O(k \ln k) to min{mmk,k2/3}O(ln2k)\min\left\{\frac{m}{m-k},k^{2/3}\right\} \cdot O(\ln^2 k) -- this is the first sublinear ratio for the problem, and the first polylogarithmic ratio O(ln2k)/ϵO(\ln^2 k)/\epsilon when m(1+ϵ)km \geq (1+\epsilon)k; furthermore, we obtain ratio min{mmk,k}O(ln2k)\min\left\{\frac{m}{m-k},\sqrt{k}\right\} \cdot O(\ln^2 k) for uniform weights. These results are obtained by showing the same ratios for the Subset kk-Connectivity problem when the set TT of terminals is an mm-dominating set with mkm \geq k

    Ramsey numbers R(K3,G) for graphs of order 10

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    In this article we give the generalized triangle Ramsey numbers R(K3,G) of 12 005 158 of the 12 005 168 graphs of order 10. There are 10 graphs remaining for which we could not determine the Ramsey number. Most likely these graphs need approaches focusing on each individual graph in order to determine their triangle Ramsey number. The results were obtained by combining new computational and theoretical results. We also describe an optimized algorithm for the generation of all maximal triangle-free graphs and triangle Ramsey graphs. All Ramsey numbers up to 30 were computed by our implementation of this algorithm. We also prove some theoretical results that are applied to determine several triangle Ramsey numbers larger than 30. As not only the number of graphs is increasing very fast, but also the difficulty to determine Ramsey numbers, we consider it very likely that the table of all triangle Ramsey numbers for graphs of order 10 is the last complete table that can possibly be determined for a very long time.Comment: 24 pages, submitted for publication; added some comment

    On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint

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    In the problem of minimum connected dominating set with routing cost constraint, we are given a graph G=(V,E)G=(V,E), and the goal is to find the smallest connected dominating set DD of GG such that, for any two non-adjacent vertices uu and vv in GG, the number of internal nodes on the shortest path between uu and vv in the subgraph of GG induced by D{u,v}D \cup \{u,v\} is at most α\alpha times that in GG. For general graphs, the only known previous approximability result is an O(logn)O(\log n)-approximation algorithm (n=Vn=|V|) for α=1\alpha = 1 by Ding et al. For any constant α>1\alpha > 1, we give an O(n11α(logn)1α)O(n^{1-\frac{1}{\alpha}}(\log n)^{\frac{1}{\alpha}})-approximation algorithm. When α5\alpha \geq 5, we give an O(nlogn)O(\sqrt{n}\log n)-approximation algorithm. Finally, we prove that, when α=2\alpha =2, unless NPDTIME(npolylogn)NP \subseteq DTIME(n^{poly\log n}), for any constant ϵ>0\epsilon > 0, the problem admits no polynomial-time 2log1ϵn2^{\log^{1-\epsilon}n}-approximation algorithm, improving upon the Ω(logn)\Omega(\log n) bound by Du et al. (albeit under a stronger hardness assumption)

    Near-Optimal Distributed Approximation of Minimum-Weight Connected Dominating Set

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    This paper presents a near-optimal distributed approximation algorithm for the minimum-weight connected dominating set (MCDS) problem. The presented algorithm finds an O(logn)O(\log n) approximation in O~(D+n)\tilde{O}(D+\sqrt{n}) rounds, where DD is the network diameter and nn is the number of nodes. MCDS is a classical NP-hard problem and the achieved approximation factor O(logn)O(\log n) is known to be optimal up to a constant factor, unless P=NP. Furthermore, the O~(D+n)\tilde{O}(D+\sqrt{n}) round complexity is known to be optimal modulo logarithmic factors (for any approximation), following [Das Sarma et al.---STOC'11].Comment: An extended abstract version of this result appears in the proceedings of 41st International Colloquium on Automata, Languages, and Programming (ICALP 2014

    Line-distortion, Bandwidth and Path-length of a graph

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    We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson-Seymour's path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its path-decompositions. In particular, we show: - if a graph GG can be embedded into the line with distortion kk, then GG admits a Robertson-Seymour's path-decomposition with bags of diameter at most kk in GG; - for every class of graphs with path-length bounded by a constant, there exist an efficient constant-factor approximation algorithm for the minimum line-distortion problem and an efficient constant-factor approximation algorithm for the minimum bandwidth problem; - there is an efficient 2-approximation algorithm for computing the path-length of an arbitrary graph; - AT-free graphs and some intersection families of graphs have path-length at most 2; - for AT-free graphs, there exist a linear time 8-approximation algorithm for the minimum line-distortion problem and a linear time 4-approximation algorithm for the minimum bandwidth problem
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