46 research outputs found
The Structure of Minimum Vertex Cuts
In this paper we continue a long line of work on representing the cut structure of graphs. We classify the types of minimum vertex cuts, and the possible relationships between multiple minimum vertex cuts.
As a consequence of these investigations, we exhibit a simple O(? n)-space data structure that can quickly answer pairwise (?+1)-connectivity queries in a ?-connected graph. We also show how to compute the "closest" ?-cut to every vertex in near linear O?(m+poly(?)n) time
Global and Fixed-Terminal Cuts in Digraphs
The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut.
1. Fixed-terminal edge-weighted double cut is known to be solvable efficiently. We show that fixed-terminal node-weighted double cut cannot be approximated to a factor smaller than 2 under the Unique Games Conjecture (UGC), and we also give a 2-approximation algorithm. For the global version of the problem, we prove an inapproximability bound of 3/2 under UGC.
2. Fixed-terminal edge-weighted bicut is known to have an approximability factor of 2 that is tight under UGC. We show that the global edge-weighted bicut is approximable to
a factor strictly better than 2, and that the global node-weighted bicut cannot be approximated to a factor smaller than 3/2 under UGC.
3. In relation to these investigations, we also prove two results on undirected graphs which are of independent interest. First, we show NP-completeness and a tight inapproximability bound of 4/3 for the node-weighted 3-cut problem under UGC. Second, we show that for constant k, there exists an efficient algorithm to solve the minimum {s,t}-separating k-cut problem.
Our techniques for the algorithms are combinatorial, based on LPs and based on the enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances
The Firebreak Problem
Suppose we have a network that is represented by a graph . Potentially a
fire (or other type of contagion) might erupt at some vertex of . We are
able to respond to this outbreak by establishing a firebreak at other
vertices of , so that the fire cannot pass through these fortified vertices.
The question that now arises is which vertices will result in the greatest
number of vertices being saved from the fire, assuming that the fire will
spread to every vertex that is not fully behind the vertices of the
firebreak. This is the essence of the {\sc Firebreak} decision problem, which
is the focus of this paper. We establish that the problem is intractable on the
class of split graphs as well as on the class of bipartite graphs, but can be
solved in linear time when restricted to graphs having constant-bounded
treewidth, or in polynomial time when restricted to intersection graphs. We
also consider some closely related problems
Edges not contained in triangles and the distribution of contractible edges in a 4-connected graph
AbstractWe prove results concerning the distribution of 4-contractible edges in a 4-connected graph G in connection with the edges of G not contained in a triangle. As a corollary, we show that if G is 4-regular 4-connected graph, then the number of 4-contractible edges of G is at least one half of the number of edges of G not contained in a triangle