The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NP-complete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in high-speed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.

We consider the following maximum disjoint paths problem (mdpp). We are given a large network, and pairs of nodes that wish to communicate over paths through the network --- the goal is to simultaneously connect as many of these pairs as possible in such a way that no two communication paths share an edge in the network. This classical problem has been brought into focus recently in papers discussing applications to routing in high-speed networks, where the current lack of understanding of the mdpp is an obstacle to the design of practical heuristics. We consider the class of densely embedded, nearly-Eulerian graphs, which includes the two-dimensional mesh and many other planar and locally planar interconnection networks. We obtain a constant-factor approximation algorithm for the maximum disjoint paths problem for this class of graphs; this improves on an O(log n)-approximation for the special case of the two-dimensional mesh due to Aumann--Rabani and the authors. For networks that ...

At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2-approximation to graph coloring, constant-factor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to half-integral multicommodity flow, subexponential fixed-parameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.

We solve an open problem posed by Eppstein in 1995 [14, 15] and re-enforced by Grohe [16, 17] concerning locally bounded treewidth in minor-closed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a function of r (not n). Eppstein characterized minor-closed families of graphs with bounded local treewidth as precisely minor-closed families that minor-exclude an apex graph, where an apex graph has one vertex whose removal leaves a planar graph. In particular, Eppstein showed that all apex-minor-free graphs have bounded local treewidth, but his bound is doubly exponential in r, leaving open whether a tighter bound could be obtained. We improve this doubly exponential bound to a linear bound, which is optimal. In particular, any minor-closed graph family with bounded local treewidth has linear local treewidth. Our bound generalizes previously known linear bounds for special classes of graphs proved by several authors. As a consequence of our result, we obtain substantially faster polynomial-time approximation schemes for a broad class of problems in apex-minor-free graphs, improving the running time from .

Preprocessing by data reduction is a simple but powerful technique used for practically solving di#erent network problems. A number of empirical studies shows that a set of reduction rules for solving Dominating Set problems introduced by Alber, Fellows & Niedermeier leads e#ciently to optimal solutions for many realistic networks. Despite of the encouraging experiments, the only class of graphs with proven performance guarantee of reductions rules was the class of planar graphs.

We present a modification of Bodlaender's linear time algorithm that, for constant k, determines whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a subgraph G^0 of G of treewidth greater than k is returned along with a tree decomposition of G^0 of width at most 2k. A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n^2) time. This is the primary motivation for this paper.

1 Introduction The recent theory of fixed-parameter algorithms and parameterized complex-ity [13] has attracted much attention in its less than 10 years of existence. In general the goal is to understand when NP-hard problems have algorithms thatare exponential only in a parameter k of the problem instead of the problemsize n. Fixed-parameter algorithms whose running time is polynomial for fixedparameter values--or more precisely

In the algorithm for the disjoint paths problem given in Graph Minors XIII, we used without proof a lemma that, in solving such a problem, a vertex which was sufficiently “insulated” from the rest of the graph by a large planar piece of the graph was irrelevant, and could be deleted without changing the problem. In this paper we prove the lemma.

In this work we relate the theory of quasi-orders to the theory of algorithms over some combinatorial objects. First we develope the theory of well-quasi-orderings, wqo's, and relate it to the theory of worst-case complexity. Then we give a general 0-1-law for hereditary properties that has implications for average case complexity. This result on average-case complexity is applied to the class of finite graphs equipped with the induced subgraph relation. We obtain that a wide class of problems, including e.g. perfectness, has average constant time algorithms. Then we show, by extending a result of Damaschke on cographs, that the classes of finite orders resp. graphs with bounded decomposition diameter form wqo's with respect to the induced suborder resp. induced subgraph relation. This leads to linear time algorithms for the recognition of any hereditary property on these objects. Our main result is then that the set of finite posets is a wqo with respect to a certain relation , calle...

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