3,587 research outputs found
Computational methods for finding long simple cycles in complex networks
© 2017 Elsevier B.V. Detection of long simple cycles in real-world complex networks finds many applications in layout algorithms, information flow modelling, as well as in bioinformatics. In this paper, we propose two computational methods for finding long cycles in real-world networks. The first method is an exact approach based on our own integer linear programming formulation of the problem and a data mining pipeline. This pipeline ensures that the problem is solved as a sequence of integer linear programs. The second method is a multi-start local search heuristic, which combines an initial construction of a long cycle using depth-first search with four different perturbation operators. Our experimental results are presented for social network samples, graphs studied in the network science field, graphs from DIMACS series, and protein-protein interaction networks. These results show that our formulation leads to a significantly more efficient exact approach to solve the problem than a previous formulation. For 14 out of 22 networks, we have found the optimal solutions. The potential of heuristics in this problem is also demonstrated, especially in the context of large-scale problem instances
Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph
A standard way to approximate the distance between any two vertices and
on a mesh is to compute, in the associated graph, a shortest path from
to that goes through one of sources, which are well-chosen vertices.
Precomputing the distance between each of the sources to all vertices of
the graph yields an efficient computation of approximate distances between any
two vertices. One standard method for choosing sources, which has been used
extensively and successfully for isometry-invariant surface processing, is the
so-called Farthest Point Sampling (FPS), which starts with a random vertex as
the first source, and iteratively selects the farthest vertex from the already
selected sources.
In this paper, we analyze the stretch factor of
approximate geodesics computed using FPS, which is the maximum, over all pairs
of distinct vertices, of their approximated distance over their geodesic
distance in the graph. We show that can be bounded in terms
of the minimal value of the stretch factor obtained using an
optimal placement of sources as , where is the ratio of the lengths of
the longest and the shortest edges of the graph. This provides some evidence
explaining why farthest point sampling has been used successfully for
isometry-invariant shape processing. Furthermore, we show that it is
NP-complete to find sources that minimize the stretch factor.Comment: 13 pages, 4 figure
Approximation Algorithms for Route Planning with Nonlinear Objectives
We consider optimal route planning when the objective function is a general
nonlinear and non-monotonic function. Such an objective models user behavior
more accurately, for example, when a user is risk-averse, or the utility
function needs to capture a penalty for early arrival. It is known that as
nonlinearity arises, the problem becomes NP-hard and little is known about
computing optimal solutions when in addition there is no monotonicity
guarantee. We show that an approximately optimal non-simple path can be
efficiently computed under some natural constraints. In particular, we provide
a fully polynomial approximation scheme under hop constraints. Our
approximation algorithm can extend to run in pseudo-polynomial time under a
more general linear constraint that sometimes is useful. As a by-product, we
show that our algorithm can be applied to the problem of finding a path that is
most likely to be on time for a given deadline.Comment: 9 pages, 2 figures, main part of this paper is to be appear in
AAAI'1
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