Packing multiway cuts in capacitated graphs

We consider the following "multiway cut packing" problem in undirected graphs: we are given a graph G=(V,E) and k commodities, each corresponding to a set of terminals located at different vertices in the graph; our goal is to produce a collection of cuts {E_1,...,E_k} such that E_i is a multiway cut for commodity i and the maximum load on any edge is minimized. The load on an edge is defined to be the number of cuts in the solution crossing the edge. In the capacitated version of the problem the goal is to minimize the maximum relative load on any edge--the ratio of the edge's load to its capacity. Multiway cut packing arises in the context of graph labeling problems where we are given a partial labeling of a set of items and a neighborhood structure over them, and, informally, the goal is to complete the labeling in the most consistent way. This problem was introduced by Rabani, Schulman, and Swamy (SODA'08), who developed an O(log n/log log n) approximation for it in general graphs, as well as an improved O(log^2 k) approximation in trees. Here n is the number of nodes in the graph. We present the first constant factor approximation for this problem in arbitrary undirected graphs. Our approach is based on the observation that every instance of the problem admits a near-optimal laminar solution (that is, one in which no pair of cuts cross each other).Comment: The conference version of this paper is to appear at SODA 2009. This is the full versio

Linear-Time Algorithms for Edge-Based Problems

There is a dearth of algorithms that deal with edge-based problems in trees, specifically algorithms for edge sets that satisfy a particular parameter. The goal of this thesis is to create a methodology for designing algorithms for these edge-based problems. We will present a variant of the Wimer method [Wimer et al. 1985] [Wimer 1987] that can handle edge properties. We call this variant the Wimer edge variant. The thesis is divided into three sections, the first being a chapter devoted to defining and discussing the Wimer edge variant in depth, showing how to develop an algorithm using this variant, and an example of this process, including a run of an algorithm developed using this method. The second section involves algorithms developed using the Wimer edge variant. We will provide algorithms for a variety of edge parameters, including four different matching parameters (connected, disconnected, induced and 2-matching), three different domination parameters (edge, total edge and edge-vertex) and two covering parameters (edge cover and edge cover irredundance). Each of these algorithms are discussed in detail and run in linear time. The third section involves an attempt to characterize the Wimer edge variant. We show how the variant can be applied to three classes of graphs: weighted trees, unicyclic graphs and generalized series-parallel graphs. For each of these classes, we detail what adaptations are required (if any) and design an algorithm, including showing a run on an example graph. The fourth chapter is devoted to a discussion of what qualities a parameter has to have in order to be likely to have a solution using the Wimer edge variant. Also in this chapter we discuss classes of graphs that can utilize the Wimer edge variant. Other topics discussed in this thesis include a literature review, and a discussion of future work. There are plenty of options for future work on this topic, which hopefully this thesis can inspire. The intent of this thesis is to provide the foundation for future algorithms and other work in this area

Arc-disjoint in-trees in directed graphs

Given a directed graph D = (V, A) with a set of d specified vertices S = {s 1, …, s d } ⊆ V and a function f: S → ℕ where ℕ denotes the set of natural numbers, we present a necessary and sufficient condition such that there exist Σ i=1 d f(s i ) arc-disjoint in-trees denoted by T i, 1, T i, 2, …, $$T_{i, f(s_0 )}$$ for every i = 1, …, d such that T i, 1, …, $$T_{i, f(s_0 )}$$ are rooted at s i and each T i, j spans the vertices from which s i is reachable. This generalizes the result of Edmonds [2], i.e., the necessary and sufficient condition that for a directed graph D=(V, A) with a specified vertex s∈V, there are k arc-disjoint in-trees rooted at s each of which spans V. Furthermore, we extend another characterization of packing in-trees of Edmonds [1] to the one in our case