Routing with congestion in acyclic digraphs☆
Introduction
The k-disjoint paths problem and related routing problems are among the central problems in combinatorial optimisation. In the most basic variant of the k-disjoint paths problem, a graph G is given with k pairs , …, of vertices and the task is to find k pairwise vertex-disjoint paths linking each to its corresponding target .
The problem is well known to be NP-complete [16]. On undirected graphs with a fixed number k of source/terminal pairs, Robertson and Seymour proved in their monumental graph minor series [23] that the problem is polynomial-time solvable. In fact, they showed that it is fixed-parameter tractable with parameter k: it can be solved in cubic time for every fixed value of k.
For directed graphs, the problem is computationally much harder. Fortune et al. [17] proved that it is already NP-complete for only source/terminal pairs. In particular, this also implies that it is not fixed-parameter tractable on directed graphs. Following this result a lot of work has gone into establishing more efficient algorithms on restricted classes of digraphs.
Fortune et al. [17] showed that the problem can be solved in time on acyclic digraphs, that is, it is polynomial-time for every fixed k. However, as proved by Slivkins [24], the problem is -hard on acyclic digraphs, and therefore unlikely to be fixed-parameter tractable. On the other hand, Cygan et al. [13] proved that the problem is fixed-parameter tractable with parameter k when restricted to planar digraphs. Related to this, Amiri et al. [1] proved that the problem remains NP-complete even in upward planar graphs, but admits a single exponential fixed-parameter algorithm.
Disjoint paths problems have also been studied intensively in the area of approximation algorithms, both on directed and undirected graphs (see, e.g., [4], [6], [7], [8], [9], [10], [11], [12], [20]). The goal is, given an input graph G and demands to route as many pairs as possible in polynomial time. There are many variations what it means for a pair to be routable. In particular, a problem studied intensively in the approximation literature is a relaxed version of disjoint paths where the paths are no longer required to be fully disjoint. Instead, they may intersect but every vertex of the graph is allowed to be contained in at most c paths, for some fixed constant c. This is called congestion c routing. In particular, the well-linked decomposition framework developed in [12] for undirected graphs and later generalised to digraphs in [9] has proved to be very valuable for obtaining good approximation algorithms for disjoint paths problems on planar graphs and digraphs. The problem is also interesting from practical point of view [2].
In this paper, we are interested in exact solutions for high congestion routing on acyclic digraphs. More precisely, we study the following problem.
Definition 1 Let G be a digraph and let be a set of pairs of vertices. Let . A c-routing of I is a set of paths such that, for all , path links to and no vertex appears in more than c paths from . Let . In the -Congestion Routing problem, a digraph G is given in the input together with a set of k pairs of vertices (the demands); the task is to decide whether there is a c-routing of I in G.
We consider -Congestion Routing on acyclic digraphs. First, it is not very difficult to show that, for every fixed , we can generalise the time algorithm of Fortune et al. [17] to -Congestion Routing. By revisiting the W[1]-hardness proof of Slivkins [24] and making appropriate modifications, we can establish that the problem remains W[1]-hard for every fixed congestion . Moreover, by doing the proof in a more modern way (reducing from general subgraph isomorphism instead of maximum clique and invoking a lower bound of Marx [22]), we can show that the time algorithm is essentially best possible with respect to the exponent of n. This lower bound is under the Exponential-Time Hypothesis (ETH), which can be informally stated as n-variable 3Sat cannot be solved in time (see [14], [18], [21] for more background).
Theorem 2 For any fixed integer , -Congestion Routing is W[1]-hard parameterised by k even on DAGs and, assuming ETH, cannot be solved in time for any computable function f.
Intuitively, one can expect the problem to become simpler if c is almost as large as k: after all, the problem is trivial if . Therefore, we study the complexity of the problem in settings close to this extreme case. The main algorithmic result of this paper is to show that for any fixed value of , the problem can be solved in time . That is, the exponent of the polynomial bounding the running time of the algorithm only depends on d but not on the number k.
Theorem 3 For every fixed and for all the -Congestion Routing problem on acyclic digraphs can be solved in time .
A simple corollary of Theorem 2 shows that -Congestion Routing is unlikely to be fixed-parameter tractable and the running time of Theorem 3 essentially cannot be improved (assuming ETH). Observe that if we set , then -Congestion Routing is simply the standard k-disjoint path problem, thus any algorithmic result for -Congestion Routing parameterised by d would imply the essentially same algorithmic result for the fully disjoint version parameterised by k.
Corollary 4 -Congestion Routing is W[1]-hard parameterised by d (if k is part of the input) and, assuming ETH, cannot be solved in time for any computable function f.
Organisation The paper is organised as follows. In Section 3 we fix some notation and prove our main algorithmic result. The corresponding lower bound is then proved in Section 4.
Section snippets
Preliminaries
We review basic notation and concepts of graph theory needed in the paper. We refer to [5], [15] for background.
Let G be a digraph. We write and for its set of vertices and edges, respectively. We assume that there is no edge with the same head and tail, i.e. there are no loops in the digraphs we consider in this paper. If is an edge, then u is its tail and v is its head. G is simple if there are no two distinct edges which have the same tail and the same head. Otherwise we
A polynomial-time algorithm on acyclic digraphs
In this section we prove the first main result of this paper, Theorem 3, which we repeat here for convenience.
Theorem 3 For every fixed , the -Congestion Routing problem on acyclic digraphs can be solved in time .
We first need some additional notation and prove some auxiliary lemmas.
Definition 5 Let G be a digraph and let be a set of paths in G. For every we define the congestion of v with respect to as the number of paths in containing v.
The following lemma provides a simple extension of
Lower bounds
In this section, we prove Theorem 2 by a reduction from Partitioned Subgraph Isomorphism. The input of the Partitioned Subgraph Isomorphism problem consists of a graph H with vertex set and a graph G whose vertex set is partitioned into k classes , …, . The task is to find a mapping such that for every and μ is a subgraph embedding, that is, if and are adjacent in H, then and are adjacent in G. Theorem 10 Assuming ETH, Partitioned Subgraph[22]
Conclusion
In this paper we have studied the -Congestion Routing problem on acyclic digraphs. It is easy to see that the algorithm in [17] for solving the disjoint paths problem on acyclic digraphs can be extended to an algorithm for -Congestion Routing. As we proved in Theorem 2, the time algorithm is essentially best possible with respect to the exponent of n, under the Exponential-Time Hypothesis (ETH). We therefore studied the extreme cases of relatively high congestion
Declaration of Competing Interest
There is no competing interest.
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