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We prove that, for any fixed k, one can construct a linear time algorithm that checks if a graph has branchwidth k and, if so, outputs a branch decomposition of minimum width. 1 Introduction This paper considers the problem of finding branch decompositions of graphs with small branchwidth. The notion of branchwidth has a close relationship to the more well-known notion of treewidth, a notion that has come to play a large role in many recent investigations in algorithmic graph theory. (See Section 2 for definitions of treewidth and branchwidth.) One reason for the interest in this notion is that many graph problems can be solved by linear time algorithms, when the inputs are restricted to graphs with some uniform upper bound on their treewidth. Most of these algorithms first try to find a tree decomposition of small width, and then utilize the advantages of the tree structure of the decomposition (see [1], [4]). The branchwidth of a graph differs from its treewidth by at most a multipl...

This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NP-hard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixed-parameter tractable algorithms and have been shown to be effective in a practical setting for NP-hard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.

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We present an algorithm that takes O(sort(N)) I/Os 1 to compute a tree decomposition of width at most k, for any graph G of treewidth at most k and size N. Given such a tree decomposition, we use a dynamic programming framework to solve a wide variety of problems on G in O(N/(DB)) I/Os, including the single-source shortest path problem and a number of problems that are NP-hard on general graphs. The tree decomposition can also be used to obtain an optimal separator decomposition of G. We use such a decomposition to perform depth-first search in G in O(N/(DB)) I/Os. As important tools that are used in the tree decomposition algorithm, we introduce flippable DAGs and present an algorithm that computes a perfect elimination ordering of a k-tree in O(sort(N)) I/Os. The second contribution of our paper, which is of independent interest, is a general and simple framework for obtaining I/O-efficient algorithms for a number of graph problems that can be solved using greedy algorithms in internal memory. We apply this framework in order to obtain an improved algorithm for finding a maximal matching and the first deterministic I/Oefficient algorithm for finding a maximal independent set of an arbitrary graph. Both algorithms take O(sort(|V |+|E|)) I/Os. The maximal matching algorithm is used in the tree decomposition algorithm.

We present a linear time algorithm which, given a tree, computes a linear layout optimal with respect to vertex separation. As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the running time of former algorithms from O(n log n) to O(n) and answers two related open questions raised in [7] and [15].

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.

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We show that for any fixed number of registers there is a linear-time algorithm which given a structured (j goto-free) program finds, if possible, an allocation of variables to registers without using intermediate storage. Our algorithm allows for rescheduling, i.e. that straightline sequences of statements may be reordered to achieve a better register allocation as long as the data dependencies of the program are not violated. If we also allow for registers of different types, e.g. for integers and floats, we can give only a polynomial time algorithm. In fact we show that the problem then becomes hard for the W-hierarchy which is a strong indication that no O(n c ) algorithm exists for it with c independent on the number of registers. However, if we do not allow for rescheduling then this non-uniform register case is also solved in linear time.