286,791 research outputs found
Supermodularity in Unweighted Graph Optimization I: Branchings and Matchings
The main result of this paper is motivated by the following two apparently unrelated graph optimization problems: (A) As an extension of Edmonds' disjoint branchings theorem, characterize digraphs comprising k disjoint branchings B-i each having a specified number mu(i) of arcs. (B) As an extension of Ryser's maximum term rank formula, determine the largest possible matching number of simple bipartite graphs complying with degree-constraints. The solutions to these problems and to their generalizations will be obtained from a new min-max theorem on covering a supermodular function by a simple degree-constrained bipartite graph. A specific feature of the result is that its minimum cost extension is already NP-hard. Therefore classic polyhedral tools themselves definitely cannot be sufficient for solving the problem, even though they make some good service in our approach
Parameterized Rural Postman Problem
The Directed Rural Postman Problem (DRPP) can be formulated as follows: given
a strongly connected directed multigraph with nonnegative integral
weights on the arcs, a subset of and a nonnegative integer ,
decide whether has a closed directed walk containing every arc of and
of total weight at most . Let be the number of weakly connected
components in the the subgraph of induced by . Sorge et al. (2012) ask
whether the DRPP is fixed-parameter tractable (FPT) when parameterized by ,
i.e., whether there is an algorithm of running time where is a
function of only and the notation suppresses polynomial factors.
Sorge et al. (2012) note that this question is of significant practical
relevance and has been open for more than thirty years. Using an algebraic
approach, we prove that DRPP has a randomized algorithm of running time
when is bounded by a polynomial in the number of vertices in
. We also show that the same result holds for the undirected version of
DRPP, where is a connected undirected multigraph
Coloring Sums of Extensions of Certain Graphs
Recall that the minimum number of colors that allow a proper coloring of
graph is called the chromatic number of and denoted by In
this paper the concepts of '-chromatic sum and -chromatic sum are
introduced. The extended graph of a graph was recently introduced for
certain regular graphs. We further the concepts of '-chromatic sum and
-chromatic sum to extended paths and cycles. The paper concludes with
\emph{patterned structured} graphs.Comment: 12 page
Algorithm and Complexity for a Network Assortativity Measure
We show that finding a graph realization with the minimum Randi\'c index for
a given degree sequence is solvable in polynomial time by formulating the
problem as a minimum weight perfect b-matching problem. However, the
realization found via this reduction is not guaranteed to be connected.
Approximating the minimum weight b-matching problem subject to a connectivity
constraint is shown to be NP-Hard. For instances in which the optimal solution
to the minimum Randi\'c index problem is not connected, we describe a heuristic
to connect the graph using pairwise edge exchanges that preserves the degree
sequence. In our computational experiments, the heuristic performs well and the
Randi\'c index of the realization after our heuristic is within 3% of the
unconstrained optimal value on average. Although we focus on minimizing the
Randi\'c index, our results extend to maximizing the Randi\'c index as well.
Applications of the Randi\'c index to synchronization of neuronal networks
controlling respiration in mammals and to normalizing cortical thickness
networks in diagnosing individuals with dementia are provided.Comment: Added additional section on application
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