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 study the approximability of edge-disjoint paths and related problems. In the edge-disjoint paths problem (EDP), we are given a network G with source-sink pairs (si, ti), 1 ≤ i ≤ k, and the goal is to find a largest subset of source-sink pairs that can be simultaneously connected in an edge-disjoint manner. We show that in directed networks, for any ɛ> 0, EDP is NP-hard to approximate within m 1/2−ɛ. We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any ɛ> 0, bounded length EDP is hard to approximate within m 1/2−ɛ even in undirected networks, and give an O ( √ m)-approximation algorithm for it. For directed networks, we show that even the single source-sink pair case (i.e. find the maximum number of paths of bounded length between a given sourcesink pair) is hard to approximate within m 1/2−ɛ, for any ɛ> 0.

The max-flow min-cut theorem of Ford and Fulkerson is based on an even more foundational result, namely Menger's theorem on graph connectivity. Menger's theorem provides a good characterization for the following single-source disjoint paths problem: given a graph G, with a source vertex s and terminals t 1 , ..., t k , decide whether there exist edge-disjoint s-t i paths, for i = 1, ..., k. We consider a natural, NP-hard generalization of this problem, which we call the single-source unsplittable flow problem. We are given a source and terminals as before; but now each terminal t i has a demand ae i 1, and each edge e of G has a capacity c e 1. The problem is to decide whether one can choose a single s-t i path, for each i, so that the resulting set of paths respects the capacity constraints --- the total amount of demand routed across any edge e must be bounded by the capacity c e . The main results of this paper are constant-factor approximation algorithms for three n...

The approximability of the maximum edge disjoint paths problem (EDP) in directed graphs was seemingly settled by the )-hardness result of Guruswami et al. [10] and the O( m) approximation achievable via both the natural LP relaxation [19] and the greedy algorithm [11, 12]. Here m is the number of edges in the graph. However, we observe that the hardness of approximation shown in [10] applies to sparse graphs and hence when expressed as a function of n, the number of vertices, only an \Omega\Gamma n )-hardness follows. On the other hand, the O( m)-approximation algorithms do not guarantee a sub-linear (in terms of n) approximation algorithm for dense graphs. We note that a similar gap exists in the known results on the integrality gap of the natural LP relaxation: an \Omega\Gamma n) lower bound and an O( m) upper bound. Motivated by this discrepancy in the upper and lower bounds we study algorithms for the EDP in directed and undirected graphs obtaining improved approximation ratios. We show that the greedy algorithm has an approximation ratio of O(min(n m)) in undirected graphs and a ratio of O(min(n m)) in directed graphs. For ayclic graphs we give an O( n log n) approximation via LP rounding. These are the first sub-linear approximation ratios for EDP. Our results also extend to EDP with weights and to the unsplittable flow problem with uniform edge capacities.

) Stavros G. Kolliopoulos 1 Clifford Stein 1 Abstract In the single-source unsplittable flow problem we are given a graph G; a source vertex s and a set of sinks t 1 ; : : : ; t k with associated demands. We seek a single s-t i flow path for each commodity i so that the demands are satisfied and the total flow routed across any edge e is bounded by its capacity c e : The problem is an NP-hard variant of max flow and a generalization of single-source edge-disjoint paths with applications to scheduling, load balancing and virtual-circuit routing problems. In a significant development, Kleinberg gave recently constant-factor approximation algorithms for several natural optimization versions of the problem [18]. In this paper we give a generic framework that yields simpler algorithms and significant improvements upon the constant factors. Our framework, with appropriate subroutines, applies to all optimization versions previously considered and treats in a unified manner directed and u...

We provide the first strongly polynomial algorithms with the best approximation ratio for all three variants of the unsplittable ow problem (UFP). In this problem we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path as to maximize the total profit of the satisfied terminal pairs subject to the capacity constraints. Classical UFP, in which demands must be lower than edge capacities, is known to have an O( m) approximation algorithm. We provide the same result with a strongly polynomial combinatorial algorithm. The extended UFP case is when some demands might be higher than edge capacities. For that case we both improve the current best approximation ratio and use strongly polynomial algorithms.

m)), the same as that for edp [10]. Our algorithm extends to the case where each pair siti has a demand di associated with it and we need to completely route di to get credit for pair i. We also consider the online admission control version where pairs arrive online and the algorithm has to decide immediately on its arrival whether to accept it or not. We obtain a randomized algorithm with a competitive ratio that is similar to the approximation ratio for the offline algorithm. \Lambda

Abstract | An increasing number of components in embedded systems are implemented by software running on embedded processors. This trend creates a need for compilers for embedded processors capable of generating high quality machine code. Particularly for DSPs, such compilers are hardly available, and novel DSP-speci c code optimization techniques are required. In this paper we focus on e cient address computation for array accesses in loops. Based on previous work, we present a new and optimal algorithm for address register allocation and provide an experimental evaluation of di erent algorithms. Furthermore, an e cient and close-to-optimum heuristic is proposed for large problems. 1 I.

We consider the problem of finding long paths and cycles in Hamiltonian graphs. The focus of our work is on sparse graphs, e.g., cubic graphs, that satisfy some property known to hold for Hamiltonian graphs, e.g., k-cyclability. We first consider the problem of finding long cycles in 3-connected cubic graphs whose edges have weights wi * 0. We find cycles of weight at least (P wai) 1 a for a = log2 3. Based on this result, we develop an algorithm for finding a cycle of length at least m(log3 2)=2 in 3-cyclable graphs with vertices of degree at most 3. As a corollary of this result, for arbitrary graphs with vertices of degree at most 3 that have a cycle of length l (or more generally a 3-cyclable minor with degrees at most 3 and with l edges), we find a cycle of length at least l(log3 2)=2. We consider the graph property of 1-toughness that is common to Hamiltonian graphs and 3- connected cubic graphs, and try to determine if 1-toughness implies the existence of long cycles. We show that 2-connectivity and 1-toughness, for constant degree graphs, may give cycles that are only of logarithmic length. However, we exhibit a class of 3-connected 1-tough graphs with degrees up to 6 where we can find cycles of length at least mlog3 2=2.

In a packing integer program, we are given a matrix A and column vectors b; c with nonnegative entries. We seek a vector x of nonnegative integers, which maximizes c^T x; subject to Ax ≤ b: The edge and vertex-disjoint path problems together with their unsplittable ow generalization are NP-hard problems with a multitude of applications in areas such as routing, scheduling and bin packing. These two categories of problems are known to be conceptually related, but this connection has largely been ignored in terms of approximation algorithms. We explore the topic of approximating disjoint-path problems using polynomial-size packing integer programs. Motivated by the...