140 research outputs found
Complexity and approximation for Traveling Salesman Problems with profits
International audience; In TSP with profits we have to find an optimal tour and a set of customers satisfying a specific constraint. In this paper we analyze three different variants of TSP with profits that are NP-hard in general. We study their computational complexity on special structures of the underlying graph, both in the case with and without service times to the customers. We present polynomial algorithms for the polynomially solvable cases and fully polynomial time approximation schemes (FPTAS) for some NP-hard cases
Data Reductions and Combinatorial Bounds for Improved Approximation Algorithms
Kernelization algorithms in the context of Parameterized Complexity are often
based on a combination of reduction rules and combinatorial insights. We will
expose in this paper a similar strategy for obtaining polynomial-time
approximation algorithms. Our method features the use of
approximation-preserving reductions, akin to the notion of parameterized
reductions. We exemplify this method to obtain the currently best approximation
algorithms for \textsc{Harmless Set}, \textsc{Differential} and
\textsc{Multiple Nonblocker}, all of them can be considered in the context of
securing networks or information propagation
Graphs without a partition into two proportionally dense subgraphs
A proportionally dense subgraph (PDS) is an induced subgraph of a graph such that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the rest of the graph. In this paper, we study a partition of a graph into two proportionally dense subgraphs, namely a 2-PDS partition, with and without additional constraint of connectivity of the subgraphs. We present two infinite classes of graphs: one with graphs without a 2-PDS partition, and another with graphs that only admit a disconnected 2-PDS partition. These results answer some questions proposed by Bazgan et al. (2018)
On the complexity of finding a potential community
An independent 2-clique of a graph is a subset of vertices that is an independent set and such that any two vertices inside have a common neighbor outside. In this paper, we study the complexity of find-ing an independent 2-clique of maximum size in several graph classes and we compare its complexity with the complexity of maximum independent set. We prove that this problem is NP-hard on apex graphs, APX-hard on line graphs, not n1 /2−-approximable on bipartite graphs and not-approximable on split graphs, while it is polynomial-time solvable on graphs of bounded degree and their complements, graphs of bounded treewidth, planar graphs, (C3, C6)-free graphs, threshold graphs, interval graphs and cographs. © Springer International Publishing AG 2017
Building Clusters with Lower-Bounded Sizes
Classical clustering problems search for a partition of objects into a fixed number of clusters. In many scenarios however the number of clusters is not known or necessarily fixed. Further, clusters are sometimes only considered to be of significance if they have a certain size. We discuss clustering into sets of minimum cardinality k without a fixed number of sets and present a general model for these types of problems. This general framework allows the comparison of different measures to assess the quality of a clustering. We specifically consider nine quality-measures and classify the complexity of the resulting problems with respect to k. Further, we derive some polynomial-time solvable cases for k = 2 with connections to matching-type problems which, among other graph problems, then are used to compute approximations for larger values of k
Approximating Multiobjective Optimization Problems: How exact can you be?
It is well known that, under very weak assumptions, multiobjective
optimization problems admit -approximation
sets (also called -Pareto sets) of polynomial cardinality (in the
size of the instance and in ). While an approximation
guarantee of for any is the best one can expect
for singleobjective problems (apart from solving the problem to optimality),
even better approximation guarantees than
can be considered in the multiobjective case since the approximation might be
exact in some of the objectives.
Hence, in this paper, we consider partially exact approximation sets that
require to approximate each feasible solution exactly, i.e., with an
approximation guarantee of , in some of the objectives while still obtaining
a guarantee of in all others. We characterize the types of
polynomial-cardinality, partially exact approximation sets that are guaranteed
to exist for general multiobjective optimization problems. Moreover, we study
minimum-cardinality partially exact approximation sets concerning (weak)
efficiency of the contained solutions and relate their cardinalities to the
minimum cardinality of a -approximation
set
One-Exact Approximate Pareto Sets
Papadimitriou and Yannakakis show that the polynomial-time solvability of a
certain singleobjective problem determines the class of multiobjective
optimization problems that admit a polynomial-time computable -approximate Pareto set (also called an
-Pareto set). Similarly, in this article, we characterize the
class of problems having a polynomial-time computable approximate
-Pareto set that is exact in one objective by the efficient
solvability of an appropriate singleobjective problem. This class includes
important problems such as multiobjective shortest path and spanning tree, and
the approximation guarantee we provide is, in general, best possible.
Furthermore, for biobjective problems from this class, we provide an algorithm
that computes a one-exact -Pareto set of cardinality at most twice
the cardinality of a smallest such set and show that this factor of 2 is best
possible. For three or more objective functions, however, we prove that no
constant-factor approximation on the size of the set can be obtained
efficiently
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