470 research outputs found
Approximating the Regular Graphic TSP in near linear time
We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an -vertex,
-regular graph, the algorithm computes a tour of length at most
, with high probability, in time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
factor approximation algorithm
running in time .Comment: 12 page
Approximability of Capacitated Network Design
In the capacitated survivable network design problem (Cap- SNDP), we are given an undirected multi-graph where each edge has a capacity and a cost. The goal is to find a minimum cost subset of edges that satisfies a given set of pairwise minimum-cut requirements. Unlike its classical special case of SNDP when all capacities are unit, the approximability of Cap-SNDP is not well understood; even in very restricted settings no known algorithm achieves a o(m) approximation, where m is the number of edges in the graph. In this paper, we obtain several new results and insights into the approximability of Cap-SNDP. We give an O(log n) approximation for a special case of Cap-SNDP where the global minimum cut is required to be at least R, by rounding the natural cut-based LP relaxation strengthened with valid knapsackcover inequalities. We then show that as we move away from global connectivity, the single pair case (that is, when only one pair (s, t) has positive connectivity requirement) captures much of the difficulty of Cap-SNDP: even strengthened with KC inequalities, the LP has an Ω(n) integrality gap. Furthermore, in directed graphs, we show that single pair Cap-SNDP is 2log1−3 n-hard to approximate for any fixed constant δ \u3e 0. We also consider a variant of the Cap-SNDP in which multiple copies of an edge can be bought: we give an O(log k) approximation for this case, where k is the number of vertex pairs with non-zero connectivity requirement. This improves upon the previously known O(min{k, log Rmax})-approximation for this problem when the largest minimumcut requirement, namely Rmax, is large. On the other hand, we observe that the multiple copy version of Cap-SNDP is Ω(log log n)-hard to approximate even for the single-source version of the problem
Spanner Approximations in Practice
A multiplicative -spanner is a subgraph of with the
same vertices and fewer edges that preserves distances up to the factor
, i.e., for all vertices , .
While many algorithms have been developed to find good spanners in terms of
approximation guarantees, no experimental studies comparing different
approaches exist. We implemented a rich selection of those algorithms and
evaluate them on a variety of instances regarding, e.g., their running time,
sparseness, lightness, and effective stretch
Spanner Approximations in Practice
A multiplicative ?-spanner H is a subgraph of G = (V,E) with the same vertices and fewer edges that preserves distances up to the factor ?, i.e., d_H(u,v) ? ?? d_G(u,v) for all vertices u, v. While many algorithms have been developed to find good spanners in terms of approximation guarantees, no experimental studies comparing different approaches exist. We implemented a rich selection of those algorithms and evaluate them on a variety of instances regarding, e.g., their running time, sparseness, lightness, and effective stretch
The Complexity of Finding Effectors
The NP-hard EFFECTORS problem on directed graphs is motivated by applications
in network mining, particularly concerning the analysis of probabilistic
information-propagation processes in social networks. In the corresponding
model the arcs carry probabilities and there is a probabilistic diffusion
process activating nodes by neighboring activated nodes with probabilities as
specified by the arcs. The point is to explain a given network activation state
as well as possible by using a minimum number of "effector nodes"; these are
selected before the activation process starts.
We correct, complement, and extend previous work from the data mining
community by a more thorough computational complexity analysis of EFFECTORS,
identifying both tractable and intractable cases. To this end, we also exploit
a parameterization measuring the "degree of randomness" (the number of "really"
probabilistic arcs) which might prove useful for analyzing other probabilistic
network diffusion problems as well.Comment: 28 page
Approximating the Smallest Spanning Subgraph for 2-Edge-Connectivity in Directed Graphs
Let be a strongly connected directed graph. We consider the following
three problems, where we wish to compute the smallest strongly connected
spanning subgraph of that maintains respectively: the -edge-connected
blocks of (\textsf{2EC-B}); the -edge-connected components of
(\textsf{2EC-C}); both the -edge-connected blocks and the -edge-connected
components of (\textsf{2EC-B-C}). All three problems are NP-hard, and thus
we are interested in efficient approximation algorithms. For \textsf{2EC-C} we
can obtain a -approximation by combining previously known results. For
\textsf{2EC-B} and \textsf{2EC-B-C}, we present new -approximation
algorithms that run in linear time. We also propose various heuristics to
improve the size of the computed subgraphs in practice, and conduct a thorough
experimental study to assess their merits in practical scenarios
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