Max flow vitality in general and stst-planar graphs

The \emph{vitality} of an arc/node of a graph with respect to the maximum flow between two fixed nodes $s$ and $t$ is defined as the reduction of the maximum flow caused by the removal of that arc/node. In this paper we address the issue of determining the vitality of arcs and/or nodes for the maximum flow problem. We show how to compute the vitality of all arcs in a general undirected graph by solving only $2(n-1)$ max flow instances and, In $st$-planar graphs (directed or undirected) we show how to compute the vitality of all arcs and all nodes in $O(n)$ worst-case time. Moreover, after determining the vitality of arcs and/or nodes, and given a planar embedding of the graph, we can determine the vitality of a `contiguous' set of arcs/nodes in time proportional to the size of the set.Comment: 12 pages, 3 figure

Block linear majorants in quadratic 0–1 optimization

AbstractA usual technique to generate upper bounds on the optimum of a quadratic 0–1 maximization problem is to consider a linear majorant (LM) of the quadratic objective function f and then solve the corresponding linear relaxation. Several papers have considered LMs obtained by termwise bounding, but the possibility of bounding groups of terms simultaneously does not appear to have been given much attention so far. In the present paper a broad and flexible computational framework is developed for implementing such a strategy. Here is a brief overview of our approach: in the first place, a suitable collection of “elementary” quadratic functions of few variables (typically, 3 or 4) is generated. All the coefficients of any such function (block) are either 1 or −1, and agree in sign with the corresponding coefficients of the given quadratic function. Next, for each block, a tightest LM (i.e., one having the same value as the block in as many points as possible), or a closest LM (i.e., one minimizing the sum of slacks) is computed. This can be accomplished through the solution of a small mixed-integer program, or a small linear program, respectively. Finally, the objective function is written as a weighted sum of blocks, with non-negative weights. Replacing in this expression each block by the corresponding LM yields an LM of f. We shall choose the weights in this process so that the maximum value of the resulting linear function is as small as possible. This amounts to a large-scale (but still polynomial-size) linear program, which may be solved exactly or, for larger instances, approximately by truncated column generation. The results of a set of 480 numerical tests with up to 200 variables are presented: the upper bounds on the quadratic optimum obtained by the above procedure are (provably) never worse, and often turn out to be substantially sharper, than those resulting from termwise bounding. Moreover, our bounds turn out to be close to the optimum in many (although not all) instances of some well-known benchmarks

Max-flow vitality in undirected unweighted planar graphs

We show a fast algorithm for determining the set of relevant edges in a planar undirected unweighted graph with respect to the maximum flow. This is a special case of the \emph{max flow vitality} problem, that has been efficiently solved for general undirected graphs and $st$-planar graphs. The \emph{vitality} of an edge of a graph with respect to the maximum flow between two fixed vertices $s$ and $t$ is defined as the reduction of the maximum flow caused by the removal of that edge. In this paper we show that the set of edges having vitality greater than zero in a planar undirected unweighted graph with $n$ vertices, can be found in $O(n \log n)$ worst-case time and $O(n)$ space.Comment: 9 pages, 4 figure

Polynomial algorithms for partitioning a tree into single-center subtrees to minimize flat service costs

This paper deals with the following graph partitioning problem. Consider a connected graph with n nodes, p of which are centers, while the remaining ones are units. For each unit-center pair there is a fixed service cost and the goal is to find a partition into connected components such that each component contains only one center and the total service cost is minimum. This problem is known to be NP-hard on general graphs, and here we show that it remains such even if the service cost is monotone and the graph is bipartite. However, in this paper we derive some polynomial time algorithms for trees. For this class of graphs we provide several reformulations of the problem as integer linear programs proving the integrality of the corresponding polyhedra. As a consequence, the tree partitioning problem can be solved in polynomial time either by linear programming or by suitable convex nondifferentiable optimization algorithms. Moreover, we develop a dynamic programming algorithm, whose recursion is based on sequences of minimum weight closure problems, which solves the problem on trees in O(np) time

Finding the ℓ-core of a tree

AbstractAn ℓ-core of a tree T=(V,E) with |V|=n, is a path P with length at most ℓ that is central with respect to the property of minimizing the sum of the distances from the vertices in P to all the vertices of T not in P. The distance between two vertices is the length of the shortest path joining them. In this paper we present efficient algorithms for finding the ℓ-core of a tree. For unweighted trees we present an O(nℓ) time algorithm, while for weighted trees we give a procedure with time complexity of O(nlog2n). The algorithms use two different types of recursive principle in their operation

Connected Maximum Split Clustering of Ladder Graphs

Classification, Automation and New Media, Editors Gaul W., Ritter G