22 research outputs found
Computing a Minimum-Dilation Spanning Tree is NP-hard
In a geometric network G = (S, E), the graph distance between two vertices u,
v in S is the length of the shortest path in G connecting u to v. The dilation
of G is the maximum factor by which the graph distance of a pair of vertices
differs from their Euclidean distance. We show that given a set S of n points
with integer coordinates in the plane and a rational dilation delta > 1, it is
NP-hard to determine whether a spanning tree of S with dilation at most delta
exists
Three problems on well-partitioned chordal graphs
In this work, we solve three problems on well-partitioned chordal graphs. First, we show that every connected (resp., 2-connected) well-partitioned chordal graph has a vertex that intersects all longest paths (resp., longest cycles). It is an open problem [Balister et al., Comb. Probab. Comput. 2004] whether the same holds for chordal graphs. Similarly, we show that every connected well-partitioned chordal graph admits a (polynomial-time constructible) tree 3-spanner, while the complexity status of the Tree 3-Spanner problem remains open on chordal graphs [Brandstädt et al., Theor. Comput. Sci. 2004]. Finally, we show that the problem of finding a minimum-size geodetic set is polynomial-time solvable on well-partitioned chordal graphs. This is the first example of a problem that is NP -hard on chordal graphs and polynomial-time solvable on well-partitioned chordal graphs. Altogether, these results reinforce the significance of this recently defined graph class as a tool to tackle problems that are hard or unsolved on chordal graphs.acceptedVersio
Better Hardness Results for the Minimum Spanning Tree Congestion Problem
In the spanning tree congestion problem, given a connected graph , the
objective is to compute a spanning tree in for which the maximum edge
congestion is minimized, where the congestion of an edge of is the
number of vertex pairs adjacent in for which the path connecting them in
traverses . The problem is known to be NP-hard, but its approximability
is still poorly understood, and it is not even known whether the optimum can be
efficiently approximated with ratio . In the decision version of this
problem, denoted STC-, we need to determine if has a spanning tree with
congestion at most . It is known that STC- is NP-complete for ,
and this implies a lower bound of on the approximation ratio of
minimizing congestion. On the other hand, -STC can be solved in polynomial
time, with the complexity status of this problem for
remaining an open problem. We substantially improve the earlier hardness result
by proving that STC- is NP-complete for . This leaves only the case
open, and improves the lower bound on the approximation ratio to
Parameterized complexity of the spanning tree congestion problem
We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k. We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k≥8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k≤3 the problem becomes polynomially time solvable.publishedVersio
Complexity Results for the Spanning Tree Congestion Problem
We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the complexity of this problem. First, we show that for every fixed k and d the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k ≥ 10. For very small values of k however, the problem becomes polynomially solvable. We also show that it is NP-hard to approximate the spanning tree congestion within a factor better than 11/10. On planar graphs, we prove the problem is NP-hard in general, but solvable in linear time for fixed k