### Edge-Disjoint Paths and Unsplittable Flow [chapter]

Stavros Kolliopoulos
2007 Handbook of Approximation Algorithms and Metaheuristics
Finding disjoint paths in graphs is a problem that has attracted considerable attention from at least three perspectives: graph theory, VLSI design and network routing/flow. The corresponding literature is extensive. In this chapter we limit ourselves mostly to results on offline approximation algorithms for problems on general graphs as influenced from the network flow perspective. Surveys examining the underlying graph theory, combinatorial problems in VLSI, and disjoint paths on special
more » ... classes can be found in [35, 36, 86, 88, 76, 83, 75, 51 ]. An instance of disjoint paths consists of a (directed or undirected) graph G = (V, E) and a multiset T = {(s i , t i ) : s i ∈ V, t i ∈ V, i = 1, . . . , k} of k source-sink pairs. Any source or sink is called a terminal. An element of T is also called a commodity. One seeks a set of edge-(or vertex-)disjoint paths P 1 , P 2 , . . . , P k , where P i is an s i −t i path, i = 1, . . . , k. In the case of vertex-disjoint paths we are interested in paths that are internally disjoint, i.e., a terminal may appear in more than one pair in T . We abbreviate the edge-disjoint paths problem by Edp. The notation introduced will be used throughout the chapter to refer to an input instance. We will also denote |V | by n and |E| by m for the corresponding graph. Based on whether G is directed or undirected and the edge-or vertex-disjointness condition one obtains 1 INTRODUCTION 2 four basic problem versions. The following polynomial-time reductions exist among them. Any undirected problem can be reduced to its directed counterpart by replacing an undirected edge with an appropriate gadget; both reductions maintain planarity. See  and [88, Chapter 70] for details. An edge-disjoint problem can be reduced to its vertex-disjoint counterpart by replacing G with its line graph (or digraph as the case may be). Directed vertex-disjoint paths reduce to directed edge-disjoint paths by replacing every vertex with a pair of new vertices connected by an edge. There is no known reduction from a directed to an undirected problem. The reader should bear in mind these transformations throughout the chapter. They can serve for translating approximation guarantees or hardness results from the edge-disjoint to the vertex-disjoint setting and vice versa. The unsplittable flow problem (Ufp ) is the generalization of Edp where every edge e ∈ E has a positive capacity u e , and every commodity i has a demand d i > 0. The demand from s i to t i has to be routed in an unsplittable manner, i.e., along a single path from s i to t i . For every edge e the total demand routed through that edge should be at most u e . We will often refer to a capacitated graph as a network. In a similar manner a vertex-capacitated generalization of vertex-disjoint paths can be defined. Ufp was introduced in the PhD thesis of Kleinberg  . Versions of the problem had been studied before though not under the Ufp moniker (see, e.g., [22, 6] ). If one relaxes the requirement that every commodity should use exactly one path, one obtains the multicommodity flow problem which is well known to be solvable in polynomial time. When all the sources of a multicommodity flow instance coincide at a vertex s and all the sinks at a vertex t, we obtain the classical maximum flow problem to which we also refer to as s-t flow. The relation between Ufp and multicommodity flow is an important one to which we shall return often in this survey. We will denote a solution to either problem as a flow vector f, defined on the edges or the paths of G as appropriate. 1.1 Complexity of disjoint-path problems. For general k all four basic problems are N P -complete. The undirected vertex-disjoint paths problem was shown to be N P -complete by Knuth in 1974 (see  ), via a reduction from SAT, and by Lynch . This implies the N P -completeness of directed vertex-disjoint paths and directed edge-disjoint paths. Even, Itai and Shamir  showed that both problems remain N Pcomplete on directed acyclic graphs (DAGs). In the same paper the undirected edge-disjoint paths problem 5 exact formulation for maximum demand Ufp. A similar LP, corresponding to the concurrent flow problem, can be written for minimizing congestion. See  for details. We call an LP solution for the optimization problem of interest fractional. Several early approximation algorithms for Ufp, and more generally integer multicommodity flow, work in two stages. First a fractional solution f is computed. Then f is rounded to an unsplittable solutionf through procedures of varying intricacy, most commonly by randomized rounding as shown by Raghavan and Thompson . The randomized rounding stage can usually be derandomized using the method of conditional probabilities [28, 95, 81]. The derandomization component has gradually become very important in the literature for two reasons. First, through the key work of Srinivasan [98, 96] on pessimistic estimators, good deterministic approximation algorithms were designed even in cases where the success probability of the randomized experiment was small. See [97, 11] for applications to disjoint paths. Second, in some cases the above two-stage scheme can be implemented rather surprisingly without solving first the linear program. Instead one designs directly a suitable Langrangean relaxation algorithm implementing the derandomization part. See the work of Young  and Chapter R-2 in this volume. We note that some of the approximation ratios obtained through the LP-rounding method can nowadays be matched (or surpassed) by simple combinatorial algorithms. By combinatorial one usually means algorithms restricted to ordered ring operations as opposed to ordered field ones. Two distinct greedy algorithms for Edp were given by Kleinberg  (see also ), and Kolliopoulos and Stein  (see also ). Most of the subsequent work on combinatorial algorithms uses these two approaches as a basis. Still the influence of rounding methods on the development of algorithms for disjoint-path problems can hardly be overstated. See Chapters 6 and 7 in this volume for further background on LP-based approximation algorithms. Approximate max-flow min-multicut theorems. One of the first results on disjoint paths and in fact one of the cornerstones of graph theory is Menger's Theorem : an undirected graph is k vertex-connected if and only if there are k vertex-disjoint paths between any two vertices. The edge analogue holds as well and the min-max relation behind the theorem has resurfaced in a number of guises, most notably as the max-flow min-cut theorem for s-t flows. Let G = (V, E) be undirected. For U ⊆ V, define δ(U ) := {{u, v} ∈ E : u ∈ U and v ∈ V \ U }. Similarly dem(U ) is the sum of all demands over commodities which are separated by the cut δ(U ). A necessary condition for the existence of a feasible fractional solution to (LP-MCF) that satisfies A, b, c are without loss of generality; arbitrary nonnegative values can be scaled appropriately . When A ∈ {0, 1} M ×N , we say that we have a (0, 1)-PIP. The best guarantees known for PIPs are due to Srinivasan; those for (0, 1)-PIPs are better than those known for general PIPs by as much as an Ω( √ M ) factor [98, 96]. As witnessed by the (LP-MCF) relaxation, Ufp is a packing problem, albeit one with an exponential number of variables. Motivated by Ufp,  defined the class of column-restricted PIPs (CPIPs): these are the PIPs in which all nonzero entries of column j of A have the same value ρ j , for all j. Observe that a CPIP generalizes Knapsack. If one obtains the fractional solution f to the (LP-MCF) relaxation, one can formulate the rounding problem as a polynomial-size CPIP where the columns of A correspond to the paths used in the fractional solution and the rows correspond to edges in the graph, hence to capacity constraints. The column value ρ j equals the demand d j of the commodity corresponding to the path represented by the column. A preprocessing step requires to transform first the fractional solution to a fractional single-path solution. This is a fractional solution in which (i) at most one path per commodity is used and (ii) if a commodity is routed at least a Ω(1/ log m) fraction of the demand is sent to the sink . In combination with improved bounds for CPIPs this approach yielded the O( √ m log m)-approximation for Ufp mentioned above. The fractional single-path solution concept resurfaced in the algorithm for Edp on DAGs in  (cf. Par. 2.3). A result of independent interest in  shows that any family of column-restricted PIPs can be approximated asymptotically as well as the corresponding family of (0, 1)-PIPs. This result is obtained constructively via the grouping-and-scaling technique which first appeared in  in the context of single-source Ufp (see Par. 3.4 below). Let z * be the fractional optimum. For a general CPIP the result of  translates to the existence of an integral solution of value Ω max z * M 1/( B +1) , z * ζ 1/ B , z * z * M log log M 1/ B . Baveja and Srinivasan  improved the dilation bound for column-restricted PIPs to Ω( z * t 1/ B ) where t ≤ ζ is the maximum column sum of A. 3.3 Combinatorial algorithms and other results. For extended Ufp with polynomially bounded demands,  gave a simple randomized algorithm that achieves an O( √ m log 3/2 m)-approximation and generalized the greedy algorithm for Edp  (cf. Par. 2.2) to Ufp, to obtain an O( √ m log 2 m)-approximation. Azar and Regev  provided the first strongly-polynomial algorithm for weighted Ufp that achieves an O(M )-approximation. This is because, in the notation of Par. 2.2, O 2 is empty. On the other hand there are at most √ m edge-disjoint paths of length more than √ m. See  for other algorithmic results. In transportation logistics a commodity may be splittable in different containers, each of them to be 19 routed along a single path. One wishes to bound the number of containers used. This motivates the bsplittable flow problem, a relaxed version of Ufp where a commodity can be split along at most b ≥ 1 paths, b an input parameter. This problem was introduced and first studied by Baier, Köhler and Skutella . Clearly for b = m, it reduces to solving the fractional relaxation; it is N P -complete for b = 2. See [73, 56] for a continuation of the work in  . The author observes in  that the single-source 2-splittable flow problem admits a simultaneous (2, 1)-approximation for congestion and cost. Finally, a problem in a sense complementary to b-splittable flow and with more history is the multiroute flow where for reliability purposes the flow has to be split along a given number of edge-disjoint paths. See  for definitions and background.