51 research outputs found
Parameterization Above a Multiplicative Guarantee
Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (at most) k+g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research.
We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k ? g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ?>0, multiplicative parameterization above g(I)^(1+?) of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth
On -Simple -Path
An -simple -path is a {path} in the graph of length that passes
through each vertex at most times. The -SIMPLE -PATH problem, given a
graph as input, asks whether there exists an -simple -path in . We
first show that this problem is NP-Complete. We then show that there is a graph
that contains an -simple -path and no simple path of length greater
than . So this, in a sense, motivates this problem especially
when one's goal is to find a short path that visits many vertices in the graph
while bounding the number of visits at each vertex.
We then give a randomized algorithm that runs in time that solves the -SIMPLE -PATH on a graph with
vertices with one-sided error. We also show that a randomized algorithm
with running time with gives a
randomized algorithm with running time \poly(n)\cdot 2^{cn} for the
Hamiltonian path problem in a directed graph - an outstanding open problem. So
in a sense our algorithm is optimal up to an factor
A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error
We consider one-sided error property testing of -minor freeness
in bounded-degree graphs for any finite family of graphs that
contains a minor of , the -circus graph, or the -grid
for any . This includes, for instance, testing whether a graph
is outerplanar or a cactus graph. The query complexity of our algorithm in
terms of the number of vertices in the graph, , is . Czumaj et~al.\ showed that cycle-freeness and -minor
freeness can be tested with query complexity by using
random walks, and that testing -minor freeness for any that contains a
cycles requires queries. In contrast to these results, we
analyze the structure of the graph and show that either we can find a subgraph
of sublinear size that includes the forbidden minor , or we can find a pair
of disjoint subsets of vertices whose edge-cut is large, which induces an
-minor.Comment: extended to testing outerplanarity, full version of ICALP pape
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
Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms
It is known that graphs on n vertices with minimum degree at least 3 have
spanning trees with at least n/4+2 leaves and that this can be improved to
(n+4)/3 for cubic graphs without the diamond K_4-e as a subgraph. We generalize
the second result by proving that every graph with minimum degree at least 3,
without diamonds and certain subgraphs called blossoms, has a spanning tree
with at least (n+4)/3 leaves, and generalize this further by allowing vertices
of lower degree. We show that it is necessary to exclude blossoms in order to
obtain a bound of the form n/3+c.
We use the new bound to obtain a simple FPT algorithm, which decides in
O(m)+O^*(6.75^k) time whether a graph of size m has a spanning tree with at
least k leaves. This improves the best known time complexity for MAX LEAF
SPANNING TREE.Comment: 25 pages, 27 Figure
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