14 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
Parameterized Complexity of Finding a Spanning Tree with Minimum Reload Cost Diameter
We study the minimum diameter spanning tree problem under the reload cost model (DIAMETER-TREE for short) introduced by Wirth and Steffan (2001). In this problem, given an undirected edge-colored graph G, reload costs on a path arise at a node where the path uses consecutive edges of different colors. The objective is to find a spanning tree of G of minimum diameter with respect to the reload costs. We initiate a systematic study of the parameterized complexity of the DIAMETER-TREE problem by considering the following parameters: the cost of a solution, and the treewidth and the maximum degree Delta of the input graph. We prove that DIAMETER-TREE is para-np-hard for any combination of two of these three parameters, and that it is FPT parameterized by the three of them. We also prove that the problem can be solved in polynomial time on cactus graphs. This result is somehow surprising since we prove DIAMETER-TREE to be NP-hard on graphs of treewidth two, which is best possible as the problem can be trivially solved on forests. When the reload costs satisfy the triangle inequality, Wirth and Steffan (2001) proved that the problem can be solved in polynomial time on graphs with Delta=3, and Galbiati (2008) proved that it is NP-hard if Delta=4. Our results show, in particular, that without the requirement of the triangle inequality, the problem is NP-hard if Delta=3, which is also best possible. Finally, in the case where the reload costs are polynomially bounded by the size of the input graph, we prove that DIAMETER-TREE is in XP and W[1]-hard parameterized by the treewidth plus Delta
Precedence-Constrained Arborescences
The minimum-cost arborescence problem is a well-studied problem in the area
of graph theory, with known polynomial-time algorithms for solving it. Previous
literature introduced new variations on the original problem with different
objective function and/or constraints. Recently, the Precedence-Constrained
Minimum-Cost Arborescence problem was proposed, in which precedence constraints
are enforced on pairs of vertices. These constraints prevent the formation of
directed paths that violate precedence relationships along the tree. We show
that this problem is NP-hard, and we introduce a new scalable mixed integer
linear programming model for it. With respect to the previous models, the newly
proposed model performs substantially better. This work also introduces a new
variation on the minimum-cost arborescence problem with precedence constraints.
We show that this new variation is also NP-hard, and we propose several mixed
integer linear programming models for formulating the problem
A Dual Ascent Procedure for Large Scale Uncapacitated Network Design
The fixed-charge network design problem arises in a variety of problem contexts including transportation, communication, and production scheduling.We develop a family of dual ascent algorithms for this problem. This approach generalizes known ascent procedures for solving shortest path, plant location,Steiner network and directed spanning tree problems. Our computational results for several classes of test problems with up to 500 integer and 1.98 million continuous variables and constraints shows that the dual ascent procedure and an associated drop-add heuristic generates solutions that, in almost all cases, are guaranteed to be within 1 to 3 percent of optimality. Moreover, the procedure requires no more than 150 seconds on an IBM 3083 computer. The test problems correspond to dense and sparse networks,including some models arising in freight transport