147 research outputs found
Parameterized complexity of the MINCCA problem on graphs of bounded decomposability
In an edge-colored graph, the cost incurred at a vertex on a path when two
incident edges with different colors are traversed is called reload or
changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem
consists in finding an arborescence with a given root vertex such that the
total changeover cost of the internal vertices is minimized. It has been
recently proved by G\"oz\"upek et al. [TCS 2016] that the problem is FPT when
parameterized by the treewidth and the maximum degree of the input graph. In
this article we present the following results for the MINCCA problem:
- the problem is W[1]-hard parameterized by the treedepth of the input graph,
even on graphs of average degree at most 8. In particular, it is W[1]-hard
parameterized by the treewidth of the input graph, which answers the main open
problem of G\"oz\"upek et al. [TCS 2016];
- it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the
input multigraph;
- it is FPT parameterized by the star tree-cutwidth of the input graph, which
is a slightly restricted version of tree-cutwidth. This result strictly
generalizes the FPT result given in G\"oz\"upek et al. [TCS 2016];
- it remains NP-hard on planar graphs even when restricted to instances with
at most 6 colors and 0/1 symmetric costs, or when restricted to instances with
at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.Comment: 25 pages, 11 figure
SDP-based bounds for the Quadratic Cycle Cover Problem via cutting plane augmented Lagrangian methods and reinforcement learning
We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a
node-disjoint cycle cover in a directed graph with minimum interaction cost
between successive arcs. We derive several semidefinite programming (SDP)
relaxations and use facial reduction to make these strictly feasible. We
investigate a nontrivial relationship between the transformation matrix used in
the reduction and the structure of the graph, which is exploited in an
efficient algorithm that constructs this matrix for any instance of the
problem. To solve our relaxations, we propose an algorithm that incorporates an
augmented Lagrangian method into a cutting plane framework by utilizing
Dykstra's projection algorithm. Our algorithm is suitable for solving SDP
relaxations with a large number of cutting planes. Computational results show
that our SDP bounds and our efficient cutting plane algorithm outperform other
QCCP bounding approaches from the literature. Finally, we provide several
SDP-based upper bounding techniques, among which a sequential Q-learning method
that exploits a solution of our SDP relaxation within a reinforcement learning
environment
The Quadratic Shortest Path Problem:Complexity, Approximability, and Solution Methods
We consider the problem of finding a shortest path in a directed graph with a quadratic objective function (the QSPP). We show that the QSPP cannot be approximated unless P=NP . For the case of a convex objective function, an n-approximation algorithm is presented, where n is the number of nodes in the graph, and APX-hardness is shown. Furthermore, we prove that even if only adjacent arcs play a part in the quadratic objective function, the problem still cannot be approximated unless P=NP. In order to solve the problem we first propose a mixed integer programming formulation, and then devise an efficient exact Branch-and-Bound algorithm for the general QSPP, where lower bounds are computed by considering a reformulation scheme that is solvable through a number of minimum cost flow problems. In our computational experiments we solve to optimality different classes of instances with up to 1000 nodes
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