11,235 research outputs found
Approximating Longest Path
We investigate the computational hardness of approximating the longest path and the longest cycle in undirected and directed graphs on n vertices. We show that * in any expander graph, we can find (n) long paths in polynomial time. * there is an algorithm that finds a path of length (log2 L/ log log L) in any undirected graph having a path of length L, in polynomial time. * there is an algorithm that finds a path of length (log2 n/ log log n) in any Hamiltonian directed graph of constant bounded outdegree, in polynomial time. * there cannot be an algorithm finding paths of length (n ) for any constant > 0 in a Hamiltonian directed graph of bounded outdegree in polynomial time, unless P = NP. * there cannot be an algorithm finding paths of length (log2+ n), or cycles of length (log1+ n) for any constant > 0 in a Hamiltonian directed graph of constant bounded outdegree in polynomial time, unless 3-Sat can be solved in subexponential time
Approximating the longest path length of a stochastic DAG by a normal distribution in linear time
AbstractThis paper presents a linear time algorithm for approximating, in the sense below, the longest path length of a given directed acyclic graph (DAG), where each edge length is given as a normally distributed random variable. Let F(x) be the distribution function of the longest path length of the DAG. Our algorithm computes the mean and the variance of a normal distribution whose distribution function F˜(x) satisfies F˜(x)⩽F(x) as long as F(x)⩾a, given a constant a (1/2⩽a<1). In other words, it computes an upper bound 1−F˜(x) on the tail probability 1−F(x), provided x⩾F−1(a). To evaluate the accuracy of the approximation of F(x) by F˜(x), we first conduct two experiments using a standard benchmark set ITC'99 of logical circuits, since a typical application of the algorithm is the delay analysis of logical circuits. We also perform a worst case analysis to derive an upper bound on the difference F˜−1(a)−F−1(a)
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
On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter
Conditional lower bounds for dynamic graph problems has received a great deal
of attention in recent years. While many results are now known for the
fully-dynamic case and such bounds often imply worst-case bounds for the
partially dynamic setting, it seems much more difficult to prove amortized
bounds for incremental and decremental algorithms. In this paper we consider
partially dynamic versions of three classic problems in graph theory. Based on
popular conjectures we show that:
-- No algorithm with amortized update time exists for
incremental or decremental maximum cardinality bipartite matching. This
significantly improves on the bound for sparse graphs
of Henzinger et al. [STOC'15] and bound of Kopelowitz,
Pettie and Porat. Our linear bound also appears more natural. In addition, the
result we present separates the node-addition model from the edge insertion
model, as an algorithm with total update time exists for the
former by Bosek et al. [FOCS'14].
-- No algorithm with amortized update time exists for
incremental or decremental maximum flow in directed and weighted sparse graphs.
No such lower bound was known for partially dynamic maximum flow previously.
Furthermore no algorithm with amortized update time
exists for directed and unweighted graphs or undirected and weighted graphs.
-- No algorithm with amortized update time exists
for incremental or decremental -approximating the diameter
of an unweighted graph. We also show a slightly stronger bound if node
additions are allowed. [...]Comment: To appear at ICALP'16. Abstract truncated to fit arXiv limit
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
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