100,887 research outputs found
Algorithms for finding K-best perfect matchings
AbstractIn the K-best perfect matching problem (KM) one wants to find K pairwise different, perfect matchings M1,ā¦,Mk such that w(M1) ā„ w(M2) ā„ āÆ ā„ w(Mk) ā„ w(M), āM ā M1, M2,ā¦, Mk. The procedure discussed in this paper is based on a binary partitioning of the matching solution space. We survey different algorithms to perform this partitioning. The best complexity bound of the resulting algorithms discussed is O(Kn3), where n is the number of nodes in the graph
From matchings to independent sets
In 1965, Jack Edmonds proposed his celebrated polynomial-time algorithm to find a maximum matching in a graph. It is well-known that finding a maximum matching in G is equivalent to finding a maximum independent set in the line graph of G. For general graphs, the maximum independent set problem is NP-hard. What makes this problem easy in the class of line graphs and what other restrictions can lead to an efficient solution of the problem? In the present paper, we analyze these and related questions. We also review various techniques that may lead to efficient algorithms for the maximum independent set problem in restricted graph families, with a focus given to augmenting graphs and graph transformations. Both techniques have been used in the solution of Edmonds to the maximum matching problem, i.e. in the solution to the maximum independent set problem in the class of line graphs. We survey various results that exploit these techniques beyond the line graphs
Parallel Algorithms for Depth-First Search
In this paper we examine parallel algorithms for performing a depth-first search (DFS) of a directed or undirected graph in sub-linear time. this subject is interesting in part because DFS seemed at first to be an inherently sequential process, and for a long time many researchers believed that no such algorithms existed. We survey three seminal papers on the subject. The first one proves that a special case of DFS is (in all likelihood) inherently sequential; the second shows that DFS for planar undirected graphs is in NC; and the third shows that DFS for general undirected graphs is in RNC. We also discuss randomnized algorithms, P-completeness and matching, three topics that are essential for understanding and appreciating the results in these papers
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
Defining Equitable Geographic Districts in Road Networks via Stable Matching
We introduce a novel method for defining geographic districts in road
networks using stable matching. In this approach, each geographic district is
defined in terms of a center, which identifies a location of interest, such as
a post office or polling place, and all other network vertices must be labeled
with the center to which they are associated. We focus on defining geographic
districts that are equitable, in that every district has the same number of
vertices and the assignment is stable in terms of geographic distance. That is,
there is no unassigned vertex-center pair such that both would prefer each
other over their current assignments. We solve this problem using a version of
the classic stable matching problem, called symmetric stable matching, in which
the preferences of the elements in both sets obey a certain symmetry. In our
case, we study a graph-based version of stable matching in which nodes are
stably matched to a subset of nodes denoted as centers, prioritized by their
shortest-path distances, so that each center is apportioned a certain number of
nodes. We show that, for a planar graph or road network with nodes and
centers, the problem can be solved in time, which improves
upon the runtime of using the classic Gale-Shapley stable matching
algorithm when is large. Finally, we provide experimental results on road
networks for these algorithms and a heuristic algorithm that performs better
than the Gale-Shapley algorithm for any range of values of .Comment: 9 pages, 4 figures, to appear in 25th ACM SIGSPATIAL International
Conference on Advances in Geographic Information Systems (ACM SIGSPATIAL
2017) November 7-10, 2017, Redondo Beach, California, US
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