Connectivity and Inference Problems for Temporal Networks

Transcription

1 Connectivity and Inference Problems for Temporal Networks (Extended Abstract for STOC 2000) David Kempe Jon Kleinberg Ý Amit Kumar Þ October, 1999 Abstract Many network problems are based on fundamental relationships involving time. Consider, for example, the problems of modeling the flow of information through a distributed network, studying the spread of a disease through a population, or analyzing the reachability properties of an airline timetable. In such settings, a natural model is that of a graph in which each edge is annotated with a time label specifying the time at which its endpoints communicated. We will call such a graph a temporal network. To model the notion that information in such a network flows only on paths whose labels respect the ordering of time, we call a path time-respecting if the time labels on its edges are non-decreasing. A model equivalent to temporal networks was proposed recently by Berman, in work that raised a number of interesting open questions about their basic properties. The central motivation for our work is the following question: how do the basic combinatorial and algorithmic properties of graphs change when we impose this additional temporal condition? The notion of a path is intrinsic to many of the most fundamental algorithmic problems on graphs; spanning trees, connectivity, flows, and cuts are some examples. When we focus on time-respecting paths in place of arbitrary paths, many of these problems acquire a character that is different from the traditional setting, but very rich in its own right. In focusing on these questions, we are additionally guided by issues that arise naturally in the study of broadcasting and gossiping in distributed networks. We provide results on two types of problems for temporal networks. First, we consider connectivity problems, in which we seek disjoint time-respecting paths between pairs of nodes. The natural analogue of Menger s Theorem for node-disjoint paths fails in general for time-respecting paths; we give a non-trivial characterization of those graphs for which the theorem does hold in terms of an excluded subdivision theorem, and provide a polynomial-time algorithm for connectivity on this class of graphs. (The problem on general graphs is NP-complete.) We then define and study the class of inference problems, in which we seek to reconstruct a partially specified time labeling of a network in a manner consistent with an observed history of information flow. Department of Computer Science, Cornell University, Ithaca NY kempe@cs.cornell.edu. Ý Department of Computer Science, Cornell University, Ithaca NY kleinber@cs.cornell.edu. Supported in part by an Alfred P. Sloan Research Fellowship, an ONR Young Investigator Award, and NSF Faculty Early Career Development Award CCR Þ Department of Computer Science, Cornell University, Ithaca NY amitk@cs.cornell.edu.

2 1 Introduction In a variety of settings, one encounters problems that are best modeled using a network with an explicit time-ordering on its edges. Although diverse in motivation, such problems involve a number of common themes, as indicated by the following examples. Communication in a distributed network. As agents in a distributed network communicate over time, information flows in complex ways. Gossip protocols in distributed systems, for example, are based on explicitly scheduling the dissemination of information through a network using node-to-node transmissions [5, 9, 15, 16]. Suppose we are given a network in which nodes have been communicating for a period of time with their neighbors; each edge is labeled by the time(s) at which its endpoints exchanged information. Information flows along a path È in this network only if the time labels on the edges of È are monotonically non-decreasing; thus, such time-respecting paths are crucial structures in understanding the way in which information has disseminated through the network. Epidemiology. The study of epidemics the spread of an infectious disease is a well-developed area of applied mathematics [1]. If we picture a network of individuals coming into contact with one another, there is a natural analogy to the previous setting; and indeed, this analogy has been exploited in the design of protocols for distributed systems [5, 16]. Thus, the spread of a disease (or a computer virus, or a rumor) can be investigated by studying the time-respecting paths in the network. Scheduled Transportation Networks. Finally, many of the same issues arise in the context of scheduled transportation, such as airline travel [3]. We may be given a network of airports, with edges labeled by the time(s) at which flights depart and arrive; the time-respecting paths are those that can be feasibly used by a traveler in this network. Formally, we say that a temporal network is an undirected graph Î µ in which each edge is annotated with a time label µ specifying the time at which its two endpoints communicated. Thus, one can view a temporal network as the pair µ, where is a function from the edge set to the real numbers; we refer to as a time labeling of. A path È in is called time-respecting if the labels on its edges are non-decreasing. Our definition is simple but quite robust; by the use of direct gadget reductions, we can model many other natural types of temporal labelings. For example, in Section 2, we show how to encode a model in which the graph is directed and each edge has separate departure and arrival times. The central motivation for our work is the following question: how do the basic combinatorial and algorithmic properties of graphs change when we impose this additional temporal condition? The notion of a path is intrinsic to many of the most fundamental algorithmic problems on graphs; spanning trees, connectivity, flows, and cuts are some examples. When we focus on time-respecting paths in place of arbitrary paths, many of these problems acquire a character that is different from the traditional setting, but very rich in its own right. In particular, we provide results on two types of problems for temporal networks: connectivity problems, in which we seek disjoint time-respecting paths between pairs of nodes; and inference problems, in which we seek to reconstruct a partially specified time labeling of a network in a manner consistent with an observed history of information flow. We describe these in detail below. Background. There is a large literature on gossiping and broadcasting algorithms in networks; see [9] for a survey. Two paradigmatic problems in this area are (i) the Telephone Problem [2, 4, 8, 18], in which we seek a way for Ò individuals to each transmit a distinct piece of information to everyone else using the minimum number of person-to-person phone calls; and (ii) the Minimum Broadcast Time (see [15] and the references therein), in which we seek a way for a designated source node in a graph to transmit a piece 1

3 of information to all other nodes in the minimum number of parallel rounds of node-to-node communication. Note the fundamental difference between this body of work and the types of problems we will be considering. In designing a gossiping or broadcasting algorithm, one seeks to schedule the times at which information crosses edges of a network so as to optimize a particular objective function. In the present work, we are given the times at which communication has occurred, and study properties of the full history of this communication. Thus, our analysis has a more diagnostic character. A network model essentially equivalent to the one we study here was proposed recently by Berman [3], in what appears to be the first work on the subject. He considered a model that he termed scheduled networks: directed networks in which each edge has separate departure and arrival times. As mentioned above, this can be encoded within our model of temporal networks. He gave an algorithm to compute all nodes and edges reachable from a given root by time-respecting paths, and he showed that the max-flow mincut theorem holds, with unit capacities, for time-respecting paths. Both of these problems can be reduced to equivalent problems on standard directed graphs (with no time labels), and in particular, the max-flow min-cut result follows from this reduction; the concern in [3] was with finding algorithms that were more efficient than the reduction would allow. The Present Work: Connectivity Problems. A common question in the analysis of gossip protocols is the following. Suppose that a node Ø has learned a piece of information originally possessed by a node. If we are concerned that some nodes may be faulty and corrupt information, we can ask: for some value of, are there (internally) node-disjoint time-respecting paths from to Ø? If this is the case, we can be more confident in the accuracy of the information Ø has learned, since it came along several independent trajectories through the network [14, 16]. In standard graphs, the existence of node-disjoint paths is characterized by Menger s Theorem [13]: the maximum number of node-disjoint -Ø paths is equal to the minimum number of nodes needed to separate from Ø. But there is no analogue of this theorem for arbitrary temporal networks. As observed in [3], the maximum number of node-disjoint time-respecting paths from to Ø can be strictly less than the minimum number of nodes whose deletion leaves no time-respecting -Ø path. Moreover, finding the maximum number of disjoint time-respecting paths is NP-complete. The breakdown of a Menger-like theorem can be seen quite simply on the graph depicted in Figure 1, with the indicated time labeling. There are no two disjoint time-respecting paths from Ú ½ to Ú, but after deleting any one node (other than Ú ½ or Ú ), there still remains a time-respecting Ú ½ -Ú path. Despite this counter-example, we can ask: for which graphs does the analogue of Menger s Theorem hold, under all time labelings? Specifically, let us say that a graph Î µ is Mengerian if for all time labelings Ê and all Ø ¾ Î, the maximum number of node-disjoint time-respecting paths from to Ø equals the minimum number of nodes whose deletion leaves no time-respecting -Ø path. Understanding this property can provide insight into the relation between disjoint paths and node separators in temporal networks. 5 Ú 3 Ú ½ Ú Ú ¾ Ú 4 Figure 1: The graph with the labeling 2

4 We provide a precise and surprisingly clean characterization of Mengerian graphs: an arbitrary graph is Mengerian if and only if it does not contain a subdivision of. Thus, the Mengerian property has a characterization in the spirit of Kuratowski s Theorem expressing planarity in terms of excluded subdivisions; is in a sense the unique obstacle to the property. Although the statement of the theorem is quite direct, the proof arrives at the min-max relation through a non-trivial structural characterization. Moreover, as a consequence of the proof, we develop a polynomial-time algorithm to find the maximum number of disjoint time-respecting paths between a given source and sink in a time-labeled Mengerian graph. We also consider a special case of the general disjoint paths problem in which we seek disjoint paths connecting a constant number of terminal pairs in the setting of temporal networks. The tractability of this problem was raised as an open question by Berman [3]. We resolve this question in two different respects. First, we show that in an (undirected) temporal network, it is NP-complete in general to decide whether there exist two disjoint time-respecting paths between a given source and sink. On the other hand, in a directed acyclic graph with a time labeling, there is a polynomial-time algorithm for the disjoint timerespecting paths problem for any constant number of terminal pairs. This generalizes a result of Fortune, Hopcroft, and Wyllie [6], who provided such an algorithm for standard (unlabeled) directed acyclic graphs. We prove a number of other complexity results on connectivity problems as well; in particular, it is NP-complete to find a minimum cardinality node separator whose removal leaves no time-respecting path from a source to a sink; and it is NP-complete to approximate the maximum number of node-disjoint timerespecting paths between a given source and sink to within a factor of ½. The proofs of these results can be found in the appendix. The Present Work: Inference Problems. A different type of problem arises when we have a temporal network with only partial information about the time labeling, and we seek to infer values of time labels based on additional data. For example, we may believe that communication has taken place in a network in such a way that a certain set Ë of nodes has learned a certain piece of information; we wish to test the feasibility of this hypothesis by reconstructing a history for the communication in which each node in Ë receives the information along a time-respecting path. A general statement of this problem looks as follows. Suppose we are given a graph Î µ, and each edge is labeled with an interval Á ; the meaning is that we know communication took place on edge at some time in the interval Á, but we do not know exactly when. (Note that setting Á to be a single point indicates precise knowledge of the time of communication, and setting Á to be the interval ½ ½µ models no knowledge whatsoever.) We are also given a root node Ö, a set È Î ÒÖ of positive nodes, and a disjoint set Æ Î Ò Ö of negative nodes. We are told that the nodes in È learned a piece of information originating at Ö, and that the nodes in Æ did not learn this information. We are given no information about nodes in Î Ò È Æ µ. The problem is: Does there exist a time labeling so that µ ¾ Á for each (i.e. consistent with our partial data), there is a time-respecting path from Ö to each node in È, and there is no time-respecting path from Ö to any node in Æ? We will call this the problem of Reachability Inference. We provide a polynomial-time algorithm for Reachability Inference on general temporal networks. Roughly, the algorithm works by maintaining vertex labels to all vertices of these labels provide bounds on the earliest time at which any time-respecting path from the root may reach these vertices. Then the algorithm builds a tree rooted at Ö and assigns labels to the edges of the tree such that all paths in the tree originating at Ö are time-respecting. We show that there exists a solution to the inference problem if and only if the tree can be extended to contain all nodes of È and no node of Æ. 3

5 2 Preliminaries Scheduled Networks. We begin by illustrating a reduction of Berman s scheduled network model [3] to our model. Suppose we have a directed graph ¼ in which each arc has two time labels: a departure time µ, and a larger arrival time µ µ. A path È in this model is time-respecting if, for consecutive arcs and ¼ on È, we have µ ¼ µ. We may view this as modeling the schedule of an airline flight along this edge, for example, or the beginning and end of a file transfer from one process to another. We model ¼ with a temporal network as follows. For each arc Ù Úµ of, we construct two undirected edges Ù Û µ and Û Úµ, where Û is a new vertex. We define µ µ and µ µ. It is easy to verify that a path between two nodes in ¼ is time-respecting if and only if the corresponding path between the two nodes in is time-respecting. The construction also preserves the disjointness of paths. A Reduction to Standard Directed Graphs. We also mention a construction that turns certain basic questions about temporal networks into questions about standard directed graphs. In particular, it implies a min-max theorem for edge-disjoint time-respecting paths [3]. We give the full details in the appendix, and only sketch the construction here. Given a temporal network Î µ, we first convert it into an equivalent directed temporal network ¼ in a standard way; by a directed temporal network, we mean simply a directed graph with a single time label on each edge, and we define the notion of a time-respecting directed path in the obvious fashion. We now build an (unlabeled) directed graph ¼¼ as follows. For each arc of ¼, we create two vertices Ù and Û, and a directed edge Ù Û µ. Also, if and are arcs of ¼, we add an edge Û Ù µ if there is a time-respecting directed path that uses immediately followed by. Again, we refer the reader to the appendix for the full details. Now, it is easy to verify that reachability in under time-respecting paths corresponds naturally to reachability in ¼¼ ; and using breadth-first search in ¼¼ we can answer questions like, Starting at node Ù at time Ø, how early can one reach node Ú using a time-respecting path? Moreover, if we add a super-source and super-sink to ¼¼, the max-flow min-cut theorem for (unlabeled) directed graphs implies Proposition 2.1 (Berman [3]) For a temporal network with nodes and Ø, the maximum number of edgedisjoint time-respecting paths from to Ø is equal to the minimum number of edges whose deletion leaves no time-respecting -Ø path. However, this reduction does not imply anything about node-disjoint paths; and it is clear that there is no simple relation between temporal networks and standard directed graphs preserving the notion of nodedisjointness, given that the Mengerian property for temporal networks has a non-trivial characterization. The reduction is also not useful for reasoning about inference problems, since one can only construct the directed graph associated with a temporal network given the labeling of the edges of. 3 Connectivity Problems and Mengerian Graphs Given a temporal network with underlying graph Î µ and time labeling, and two nodes Ø ¾ Î, we say that a collection of time-respecting -Ø paths is internally node-disjoint (or simply node-disjoint, by Ú slight abuse of terminology) if the paths share no nodes other than and Ø. We use Ô Øµ to denote the maximum cardinality of such a collection. A set Ë Î Ò Øµ is called an -Ø separator (or -Ø 4

6 vertex-separator or cut-set) with respect to, if it meets every time-respecting -Ø path in an edge or vertex. 1 Ú The minimum size of any -Ø separator with respect to is denoted by ص. In the introduction, we discussed the temporal network in Figure 1, with underlying graph and the indicated time labeling; it has Ô Ú Ú ½ Ú µ ½, but Ú Ú ½ Ú µ ¾. This is in sharp contrast to the statement of Menger s Theorem, which shows that these two quantities are equal for all unlabeled graphs (or equivalently, for all graphs in which µ ¼ for all ). We can in fact generalize this example to provide a family of temporal networks ¾ with nodes Ø for which Ô Ú Ø µ ½, but Ú Ø µ. The graph has µ nodes, indicating a gap of Å Î ½ µ between the maximum number of disjoint paths and the minimum size of a vertex-separator in the worst case. We give the details of this construction in the appendix. In the notation above, we can say that a graph is Mengerian if for all time labelings Ê and Ú all Ø ¾ Î, we have Ô Øµ Ú Øµ. Our first goal in this section is to prove the following characterization theorem. We first recall two definitions from graph theory. We say that a graph À ¼ is a subdivision of a graph À if À ¼ can be obtained from À by replacing each edge with a chain of degree-2 vertices. We say that contains À as a topological minor if it contains a subgraph isomorphic to a subdivision of À. Theorem 3.1 Î µ is Mengerian if and only if it does not contain as a topological minor. The only if direction is easier: Given a graph containing as a topological minor, we can use the subdivision of to label the edges of so that Ô Ú Øµ Ú Øµ. Lemma 3.2 If contains as a topological minor then is not Mengerian. The harder statement here is the if direction, which we prove by induction on the number of edges of. We begin with some lemmas that facilitate the proof. First, if the maximum degree of is, then a set of paths is internally node-disjoint if and only if it is edge-disjoint; applying Proposition 2.1, we have Lemma 3.3 Let be a graph such that the degree of none of its vertices exceeds. Then is Mengerian. Now, if contains a cut-node Ú (a vertex whose deletion disconnects it), then one can show that is Mengerian if and only if each of its two-connected components is. This provides an easy way to apply induction when is not two-connected. Lemma 3.4 Let be a non-mengerian graph that is not two-connected. Then contains a proper subgraph that is non-mengerian. We now develop a structural property of graphs that do not contain a subdivision of. Given a graph Î µ, two vertices Ú Û ¾ Î, and a natural number ½, we say that is Ú Û µ-separable if (i) Ú and Û each have degree. (ii) Either Ò Ú Û consists of connected components; or Ò Ú Û consists of components, and Ú Ûµ is an edge of. ½ connected Figure 2 depicts a Ú Û µ-separable graph; we will refer to the subgraphs induced on the connected components of Ò Ú Û as the lobes of, and denote them by ½. We use (resp. ) to denote the edge from Ú (resp. Û) into ; note that one of the may be empty, in the case that Ú Ûµ ¾, and then we have. 1 The inclusion of edges is only necessary in the special case that there is an edge ص in. In all other cases, we could simply include one of the endpoints of an edge instead of itself. 5

7 Ú ½ ¾ ½ ¾ ½ ¾ Û Figure 2: The form of a Ú Û µ-separable graph Lemma 3.5 Let be a two-connected graph with at least one vertex of degree at least. Then contains as a topological minor, or there are Ú Û ¾ Î and a number so that is Ú Û µ-separable. Proof. Let Ú be a vertex of degree, and Ú ½ Ú be its neighbors. Consider the graph Ò Ú, the result of deleting Ú and all its incident edges. As was two-connected, Ú ½ Ú are still connected in Ò Ú. Let Ì be any inclusion-wise minimal Steiner-Tree spanning Ú ½ Ú. If no vertex in Ì has degree exceeding ¾, then Ì is a path. Let Ú Ú Ú Ú Ð be the first four vertices from Ú ½ Ú appearing on Ì. Then, contains as a topological minor, with Ú Ú Ú Ú Ð as the images of Ú ½ Ú ¾ Ú Ú. Otherwise, let Û be any vertex of degree Û in Ì. Each of the Û subtrees rooted at Û must contain at least one of Ú ½ Ú Ò Û (Û might be identical to one of the Ú ), because otherwise, that subtree could be deleted and Ì would not be minimal. If any subtree contains two vertices Ú Ú, let Ú Ú Ð Ð be vertices in different subtrees (there must be at least three subtrees). Then contains as a topological minor. For either one of Ú Ú lies on the path from Û to the other say, Ú lies on Û Ú in which case we obtain by mapping Ú Ú ½ Ú ¾ Ú Ú µ to Ú Ú Û Ú Ú µ (and the edge Ú Ú ¾ µ to the path Ú Ú Ð Û). Or there must be a last vertex Û ¼ lying on the paths Û Ú and Û Ú, in which case we can map Ú Ú ½ Ú ¾ Ú Ú µ to Ú Ú Û Û ¼ Ú µ (and the edges Ú Ú ¾ µ and Ú Ú µ to the paths Ú Ú Ð Û and Ú Ú Û ¼, respectively). From now on, we may assume that each subtree rooted at Û contains exactly one vertex Ú, so the subtrees are in fact paths, i.e. Ì consists of disjoint paths È ½ È from Û to Ú ½ Ú, respectively. In the case that Û coincides with one of the Ú, the path È is empty. Consider the graph ¼ Ò Ú Û, obtained by removing from the vertices Ú and Û and all their incident edges. Define the graph to be the connected component of ¼ containing Ú (if Û Ú for some, then the corresponding is empty). The set of all must cover ¼. For assume that there were another component of ¼ that contained none of the Ú. No vertex of can be adjacent to Ú (because then, this vertex would be one of the Ú ), and by construction, there is no edge between and any. So, all paths between and Ú must go through Û, and would not have been two-connected. If for some, there must be a path È between Ú and Ú. Let Û be the last vertex of È that also lies on È, and Û the first vertex appearing after Û on È that lies on some path È for ( and might be identical, but don t have to be). With Ð (and Ú Ð Û), we get as a topological minor by mapping Ú Ú ½ Ú ¾ Ú Ú µ to Ú Û Û Û Ú Ð µ (and the edge Ú Ú µ to the path Ú Ú Ñ Û, for some Ñ Ð). 6

8 The only step left to establish the lemma is to show that the existence of several edges Û Üµ and Û Ýµ with Ü Ý ¾ would again yield as a topological minor in. Assume that two such edges exist. Then, Ü Ý, because is simple. Consider paths È Ü and È Ý between Ü and Ú (Ý and Ú, respectively), and let Þ be the first vertex of È Ü that lies on È Ý (such a vertex must exist, because both paths meet Ú ). Because Ü Ý, the vertex Þ cannot be equal to both assume without loss of generality that Þ Ý. Let. We then obtain as a topological minor, mapping Ú Ú ½ Ú ¾ Ú Ú µ to Û Ý Þ Ú Ú µ (and the edges Ú Ú ¾ µ and Ú Ú µ to the paths Û Ü Þ and Û Ú Ú, respectively). If none of the above cases yielding applies, then is Ú Û µ-separable, completing the proof. Using the structural property of Lemma 3.5, we can now complete the proof of Theorem 3.1. Proof of theorem 3.1. One direction has already been established as Lemma 3.2. For the converse direction, assume that there exist non-mengerian graphs that do not contain as a topological minor, and let Î µ have the smallest number of edges among all such graphs ( need not be unique). Then, must be two-connected according to Lemma 3.4, and it must contain at least one vertex Ú of degree greater or equal to according to Lemma 3.3. We can therefore apply Lemma 3.5, and because does not contain as a topological minor by assumption, it must be Ú Û µ-separable for vertices Ú Û ¾ Î and some. Ú Let, and Ø be such that ص ÔÚ Øµ, and È a maximum set of vertex-disjoint timerespecting -Ø paths. We distinguish between four cases based on the locations of and Ø. (1) If and Ø both lie in the same lobe, consider the graph ¼, defined as the subgraph induced by Ú Û, with an additional edge Ú Ûµ labeled ¼ Ú Ûµµ µ (if the edge Ú Ûµ existed in, we just relabel it. All other labelings stay the same.) We prove that Ú ¼ ¼ ص Ú Øµ and Ô Ú ¼ ¼ ص ÔÚ Øµ (this obviously yields a contradiction, ¼ being smaller than ). For the first equation, let Ë be a set of vertices separating from Ø in ¼ ; we show that Ë also separates from Ø in. Let È be any time-respecting -Ø path in, with vertices Î È µ. If Î È µ Î ¼ µ, then È is a path in and must meet Ë. Otherwise, È must pass through Ú and Û, and we obtain another timerespecting -Ø path È ¼ by replacing the subpath ÚÈ Û with the edge Ú Ûµ. Because È ¼ ¼, it meets Ë, and because Î È ¼ µ Î È µ, È must also meet Ë. For the second equation, note that È can contain at most one path È not entirely in. Applying the above replacement of ÚÈ Û by Ú Ûµ to È (if it exists) yields a set È ¼ of time-respecting node-disjoint -Ø paths in ¼ with size È ¼ È. (2) If lies in some lobe, and Ø in another lobe, each time-respecting -Ø path must pass through Ú or Û, i.e. there can be at most two disjoint paths, and deleting Ú and Û suffices to disconnect from Ø. For to be non-mengerian, there would have to be exactly one disjoint time-respecting -Ø path, but each -Ø separator would have to have size ¾. In particular, neither Ú nor Û separate from Ø, so there must be time-respecting -Ø paths È Ú È Û that use Ú (resp. Û), but not Û (resp. Ú). Consider the graph ¼ induced by Ú Û, without the edge Ú Ûµ, if it existed in. Because was minimal by assumption, ¼ must have an -Ø separator of size ½, i.e. some vertex Ú. Ú must meet both È Ú and È Û, because both of these paths lie entirely in ¼. Therefore Ú must lie in or by symmetry, we will assume that Ú ¾. Now, let È be a time-respecting -Ø path in that does not meet Ú. Then, È must pass through both Ú and Û. Assume without loss of generality that Ú appears on È before Û. Let È ¼ be the path obtained by replacing ÚÈ Ø with ÚÈ Ú Ø. È ¼ is a time-respecting -Ø path entirely in ¼, and not meeting Ú, contradicting the fact that Ú is an -Ø separator for ¼. (3) If Ú, and Ø lies in some lobe (or one of the three symmetric cases Û, Ø Ú, or Ø Û), we can apply an argument much like the above. All -Ø paths must use either or, so if were non- Mengerian, we would have Ú Øµ ¾ and ÔÚ Øµ ½. Again, there must be paths È and È using exactly one of and each. 7

9 Let ¼ be the subgraph induced by Ú Û, with an additional edge Ú Ûµ labeled ¼ Ú Ûµµ µ. Again, ¼ is smaller than, and the rest of the argument is the same as in the previous case. (4) The last remaining case is that Ú and Ø Û (or vice versa). In that case, any time-respecting -Ø path passes through exactly one of the. Let È denote such a path through, if it exists. Because all paths through have to use, there can be at most one disjoint path through each, and for different, the È are disjoint. Because each È can be blocked by deleting, we obtain that completing the proof. Ô Ú Øµ there is a time-respecting -Ø path through Ú Øµ A polynomial-time algorithm The proofs of Lemmas and Theorem 3.1 are implicitly algorithmic; taken together, they can be used to obtain a polynomial-time algorithm for the following problem: Given a Mengerian graph, with vertices Ø ¾ Î, find a maximum set of disjoint time-respecting paths from to Ø. The proof of Lemma 3.4 provides an efficient algorithm to identify either a subdivision of in, or Ú, Û, and such that is Ú Û µ-separable. In the latter case, if Ø Ú Û, then we can find a maximum set of disjoint paths directly. Otherwise we can identify the locations of and Ø relative to the set of lobes ; by strict analogy with the inductive structure of Theorem 3.1, we can then reduce certain parts of the graph and proceed recursively. We thus obtain the following theorem; details of the proof will be given in the full version. Theorem 3.6 There is a polynomial-time algorithm that takes a graph Î µ, and vertices Ø ¾ Î, and returns either a maximum set of disjoint time-respecting paths from to Ø or a subgraph of isomorphic to a subdivision of. Disjoint Paths with a Fixed Number of Terminal Pairs In the general disjoint paths problem, we are given a graph Î µ and pairs of terminals ½ Ø ½ µ Ø µ, where Ø ¾ Î. We seek node-disjoint paths È ½ È so that È joins to Ø. This problem is NPcomplete when is part of the input [10], or when is directed and ¾. When is a fixed constant, the problem is polynomial-time solvable when is undirected, by an algorithm of Robertson and Seymour [17], and when is a directed acyclic graph, by an algorithm of Fortune, Hopcroft, and Wyllie [6]. In [3], Berman asked about the natural extension of these results to the case of time-respecting disjoint paths: can we find node-disjoint paths in polynomial time when is a fixed constant? We first show that the result for undirected graphs does not carry over (assuming È ÆÈ ), even when ¾, all are the same, and all Ø are the same. Theorem 3.7 (a) Given a temporal network µ, with vertices and Ø, it is NP-complete to decide whether there exist two node-disjoint time-respecting paths from to Ø. (b) It is also NP-hard to approximate the maximum number of disjoint time-respecting -Ø paths within a factor of Ç ½ µ for any ¼. However, by generalizing the pebbling game of Fortune, Hopcroft, and Wyllie [6] to the setting of time-respecting paths, we obtain the following result. The details are given in the appendix. Theorem 3.8 There is a polynomial-time algorithm for the time-respecting disjoint paths problem with a fixed number of terminal pairs in a directed acyclic graph. 8

10 4 Inference Problems Thus far, we have always been given a graph and a complete time labeling, and we have sought to decide certain properties of the labeled graph. We now turn to the class of inference problems, in which we are given an incomplete time labeling; the task is then to extend the labeling to ensure that the graph has a certain property, or conclude that such an extension is not possible. In this section, we provide a polynomial-time algorithm for the Reachability Inference Problem discussed in the introduction. Consider an undirected graph Î µ rooted at Ö. Let Ò Î and Ñ. Each edge of the graph is assigned an interval Á. This interval can be closed, open or semiopen however, for the purpose of the following exposition, we will assume without loss of generality that all intervals Á are closed. We are also given two disjoint sets È Æ Î Ò Ö. We now wish to find a time labeling which satisfies the requirements of the reachability inference problem. We will call such a labeling a valid labeling. The algorithm will assign vertex labels Úµ to each vertex of Î this will denote the fact that the labeling should be such that there is no time-respecting Ö-Ú path which arrives at Ú at time Úµ or earlier. For each edge, let ÐÓÛÖ µ and ÙÔÔÖ µ be the lower and upper limits of the interval Á assigned to. The algorithm is shown in Figure 3. Algorithm 1 : 1. Initialize Úµ ½ for all Ú ¾ Æ and Úµ ½ for all Ú ¾ Î Ò Æ. Call an edge Ù Úµ ¾ insufficient if Ùµ ÐÓÛÖ µ and ÙÔÔÖ µ Úµ. 2. While there is an insufficient edge Ù Úµ in do Ùµ ÑÜ Ùµ ÐÓÛÖ µµ. If Öµ ½, there is no valid labeling of. 3. Initialize Ì to be the root Ö (Ì will remain a tree rooted at Ö). Let be a sufficiently small positive constant (specified below). 4. If Ú is a vertex in Ì, define Ø Ú to be the label µ of the last edge on the unique Ö-Ú path in Ì, and set Ø Ö ½. Call an edge Ù Úµ admissible if Ø Ù ÙÔÔÖ µ and Úµ ÙÔÔÖ µ. Let Æ Ì µ denote the set of edges ¾ such that Ì contains exactly one end-point of. 5. While Æ Ì µ contains an admissible edge do Let Ù Úµ, Ù ¾ Ì, be the admissible edge in Æ Ì µ which attains the minimum value of the quantity Õ µ ÑÜ Úµ Ø Ù ÐÓÛÖ µµ. Add to Ì and set µ Õ µ. 6. If È Ì then For all unlabeled edges, set µ ÐÓÛÖ µ. is a valid labeling of. Else There is no valid labeling of. Figure 3: The algorithm for label assignments Let us first analyze the running time of this algorithm. Call an edge sufficient if it is not insufficient. Note that once an insufficient edge becomes sufficient, it remains sufficient (because values can only increase). Therefore, step 2 can be implemented to take Ç Ñ ÐÓ Òµ time. Step 5 can be implemented in Ç Ñ Ò ÐÓ Òµ time, so the running time of the algorithm is Ç Ñ ÐÓ Òµ. Let us now prove the correctness of the algorithm. We maintain the invariant that any valid labeling of cannot contain a time-respecting Ö-Ú path which reaches Ú on or before Úµ (i.e., the label of the last edge on this path must be greater than Úµ). This is clearly true after step 1. Consider an insufficient edge Ù Úµ. If there is a labeling such that a time-respecting Ö-Ù path reaches Ù on or before ÐÓÛÖ µ, 9

11 then any label on the edge will result in a time-respecting Ö-Ú path which reaches Ú on or before Úµ, a contradiction. Thus, we can make Ùµ at least ÐÓÛÖ µ. So, the invariant remains true after step 2. Also, all edges are sufficient after step 2. For a sufficiently small constant, we can assume without loss of generality that all valid labelings ¼ satisfy the following property: any time-respecting Ö-Ú path with labeling ¼ does not reach a vertex Ú before Úµ. In steps 3 5, we maintain the following invariant. Let ¼ be a valid labeling of. Then, for any vertex Ú ¾ Ì Ò Ö, the earliest time at which a time-respecting (with respect to ¼ ) Ö-Ú path can arrive at Ú is at least Ø Ú. Clearly, this holds after step 3. We now show that the invariant holds after each iteration of the while-loop in step 5. Let Ù Úµ ¾ Æ Ì µ be an admissible edge which attains the minimum, and suppose that Ì satisfies the invariant before is added to it. First note that the unique path from Ö to Ú in Ì is time-respecting, satisfies the condition that it does not reach Ú before Úµ, and has Õ µ ¾ Á (because is admissible). We need to show that Ì also satisfies the invariant. Suppose not. Then, there exists a valid labeling ¼ and a time-respecting Ö-Ú path È Ú with respect to ¼ which arrives at Ú before Ø Ú Õ µ. This path must contain an edge ¼ Ù ¼ Ú ¼ µ ¾ Æ Ì µ, Ù ¼ ¾ Ì. We claim that ¼ is admissible. By the invariant, È Ú does not reach Ù ¼ before Ø Ù ¼, and because È Ú is time-respecting, Ø Ù ¼ ÙÔÔÖ ¼ µ. Further, the fact that ¼ is valid implies that È Ú cannot reach Ú ¼ on or before Ú ¼ µ, so Ú ¼ µ ÙÔÔÖ ¼ µ. Therefore, ¼ is admissible. Because the algorithm selected, we know that Õ ¼ µ Õ µ. If Ú ¼ µ Õ µ, then clearly, È Ú cannot reach Ú ¼ before Õ µ and hence cannot reach Ú before Õ µ (recall that Ø Ú Õ µ). If Ø Ù ¼ Õ µ or ÐÓÛÖ ¼ µ Õ µ, then È Ú, respecting time, cannot reach Ú before Õ µ (because it reaches Ú ¼ no earlier than Õ µ). Thus, we get a contradiction and the invariant holds after step 5. Claim 4.1 Let ¼ be any valid labeling of. Then, there exists a time-respecting Ö-Ú path with respect to ¼ only if Ú ¾ Ì (here Ì is the tree obtained after step 5). Proof. Suppose the claim is false, and let ¼ be a valid labeling such that there exists a time-respecting Ö-Ú path È Ú with Ú ¾ Ì. Then, there is an edge ¼ ¾ Æ Ì µ È Ú. As we argued above, is an admissible edge and step 5 should not have terminated. Thus, we get a contradiction. So, if È is not a subset of Ì, then there is no valid labeling. If È Ì, then can be extended to all edges as shown in step 6. Indeed, none of the nodes of Æ are in Ì because these nodes have labels ½. Further, if Ù Úµ ¾ such that Ù ¾ Ì Ú ¾ Ì, then the fact that is not admissible implies that Ø Ù ÙÔÔÖ µ ÐÓÛÖ µ or Úµ ÙÔÔÖ µ. In the latter case, the fact that is sufficient implies that Ùµ ÐÓÛÖ µ and so, Ø Ù ÐÓÛÖ µ. Thus, the set of nodes reachable from Ö is exactly equal to those in Ì, and the algorithm is correct. 5 Final Remarks A consequence of the analysis in the preceding section is the following: If µ is a temporal network with a root Ö, such that every node Ú is reachable from Ö by a time-respecting path, then in fact there is a directed arborescence Ì rooted at Ö such that the unique Ö-Ú path in Ì is time-respecting for every node Ú. (We will call such a structure a time-respecting arborescence.) This follows by applying the inference algorithm to the trivial case in which all nodes belong to È, and each interval Á is a single point. This result begins to look a little more unexpected when one considers the failure of natural related reachability properties in temporal networks. For example, suppose we have a temporal network with graph Î µ and labeling, with the property that for every Ù Ú ¾ Î, there is a time-respecting path 10

12 from Ù to Ú. Let us call such a µ a temporally connected network. Given a temporally connected network Î µ on Ò nodes, is there a set ¼ consisting of Ç Òµ edges so that the temporal network on the subgraph Î ¼ µ is also temporally connected? In other words, do all temporal networks have sparse subgraphs preserving this basic connectivity property? The answer to the analogous question is affirmative for standard graphs, simply by taking a spanning tree in the undirected case, or the union of two arborescences in the directed case. We can show, however, that there exist temporally connected networks for which no subgraph of fewer than Å Ò ÐÓ Òµ edges is temporally connected. We ask as an open question: what is the tightest function Òµ for which every temporally connected network on Ò nodes has a subgraph on Òµ edges that is also temporally connected? We can also show that the natural analogue of Edmonds Theorem on disjoint arborescences breaks down for temporal networks, even in the following approximate sense: for every constant, there exists a temporal network with a root Ö, so that Ö has disjoint time-respecting paths to each other node, but there do not exist two disjoint time-respecting arborescences rooted at Ö. We ask: what is the tightest function Òµ so that, in every temporal network on Ò nodes with disjoint time-respecting paths from a root Ö to each other vertex, there are at least Òµ disjoint time-respecting arborescences rooted at Ö? Acknowledgments We thank Yaron Minsky and Robbert van Renesse for valuable discussions on gossip protocols that motivated our investigation of this model. References [1] N. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, Hafner Press, [2] B. Baker, R. Shostak, Gossips and telephones, Discrete Mathematics 2(1972), pp [3] K. Berman, Vulnerability of scheduled networks and a generalization of Menger s Theorem, Networks 28(1996), pp [4] R. Bumby, A problem with telephones, SIAM J. Algebraic and Discrete Methods 2(1981), pp [5] A. Demers, D. Greene, C. Hauser, W. Irish, J. Larson, S. Shenker, H. Stuygis, D. Swinehart, D. Terry, Epidemic algorithms for replicated database maintenance, Proc. ACM Symp. on Operating Systems Principles, [6] S. Fortune, J. Hopcroft, J. Wyllie, The directed subgraph homeomorphism problem, Theoretical Computer Science 10(1980), pp [7] V. Guruswami, S. Khanna, R. Rajaraman, B. Shepherd, M. Yannakakis, Near-Optimal Hardness Results and Approximation Algorithms for Edge-Disjoint Paths and Related Problems, Proc. ACM Symposium on Theory of Computing, 1999, pp [8] A. Hajnal, E. Miller, E. Szemerédi, A cure for the telephone disease, Canadian Math. Bulletin 15(1972), pp [9] S. Hedetniemi, S. Hedetniemi, A. Liestman, A survey of gossiping and broadcasting in communication networks, Networks 18(1988), pp [10] R.M. Karp, On the computational complexity of combinatorial problems, Networks 5(1975), pp

13 [11] K. Kuratowski, Sur le problème des courbes gauches en topologie, Fundam. Math. 15 (1930), pp [12] C. Li, T. McCormick, D. Simchi Levi, The complexity of finding two disjoint paths with min-max objective function, Discrete Appl. Math. 26(1990), pp [13] K. Menger, Zur allgemeinen Kurventheorie, Fundam. Math. 19(1927), pp [14] Y. Minsky, personal communication, April [15] R. Ravi, Rapid rumor ramification: Approximating the minimum broadcast time, Proc. IEEE FOCS, 1994, pp [16] R. van Renesse, Y. Minsky, M. Hayden, A gossip-style failure-detection service, Proc. IFIP [17] N. Robertson, P.D. Seymour, Graph Minors XIII. The disjoint paths problem, J. Combinatorial Theory Ser. B 63(1995), pp [18] R. Tijdeman, On a telephone problem, Nieuw Arch. Wisk. 19(1971), pp Appendix In Section 2, we mentioned a construction which was used in proving the min-max theorem for edge-disjoint time-respecting paths. We give full details of the construction and the proof of the min-max theorem below. Let Î µ be a graph with labeling and nodes Ø ¾ Î. A set Ë is an -Ø edge-separator of with respect to if it meets every time-respecting -Ø path in an edge. Let Ô Øµ denote the maximum number of edge-disjoint time-respecting -Ø paths, and ص the minimum size of any -Ø edge-separator with respect to a labeling. Similarly, Ô Øµ is the maximum number of edge-disjoint -Ø paths (not necessarily time-respecting), and ص the minimum size of any -Ø edge-separator that meets all -Ø paths (not necessarily time-respecting). We can now rephrase Proposition 2.1 as follows: Proposition 6.1 Let Î µ be a graph, a labeling, and Ø ¾ Î. Then, ص Ô Øµ. Proof. We construct a directed unlabeled graph ¼¼ Î ¼¼ ¼¼ µ and vertices ¼¼ Ø ¼¼ from, and show that ¼¼ ¼¼ Ø ¼¼ µ ص and Ô ¼¼ ¼¼ Ø ¼¼ µ Ô Øµ. The claim then follows immediately from the max-flow min-cut theorem for unlabeled edge-disjoint paths. As a first step, we use the standard technique to obtain a directed graph ¼ Î ¼ ¼ µ with labeling ¼ from, such that ¼ ¼ ص ص and Ô ¼ ¼ ص Ô Øµ. µ Ü µ Ú µ Û µ Ý µ Figure 4: The gadget used for edge Ú Ûµ Each undirected edge Ú Ûµ ¾ with label µ is replaced by the gadget depicted in Figure 4, consisting of directed arcs. The new labeling ¼ is defined by giving all arcs of the gadget replacing the labeling µ. It is then straightforward to verify that ¼ ¼ ص ص and Ô ¼ ¼ ص Ô Øµ. 12

14 As the second step of the transformation, we replace every arc ¾ ¼ by two new vertices Ù and Û, and an arc Ù Û µ between them. For two arcs ¾ ¼, we add an arc Û Ù µ to ¼¼ if and only if the head of is the tail of, and ¼ µ ¼ µ. Figure 5 shows a vertex Ú with incident arcs and labels, and the result of the transformation. Ü ½ Ü ¾ Ü ¾ Ú ½ Ý ½ Ý ¾ Ý Ü ½ Ü ¾ Ü Ù ½ Ù ¾ Ù Û ½ Û ¾ Û Ý ½ Ý ¾ Ý Ü Ü Ù Figure 5: A vertex Ú and its replacement. Ù and Û stand short for Ù and Û. The arcs are Ú Úµ or Ú Ú µ To deal with the vertices and Ø in ¼, let ¼¼ and Ø ¼¼ be two new vertices in ¼¼, and add arcs ¼¼ Ù µ and Û Ø ¼¼ µ for all arcs out of and arcs into Ø. First note that every time-respecting directed path È ½ Ø can be mapped in the natural way to the ¼¼ -Ø ¼¼ path È ¼¼ ¼¼ Ù ½ Û ½ Ù Û Ø ¼¼. Furthermore, every ¼¼ -Ø ¼¼ path È ¼¼ must be of this form, and by construction, the corresponding -Ø path must be time-respecting. By a straightforward application of this path mapping, we obtain that Ô ¼¼ ¼¼ Ø ¼¼ µ Ô ¼ ¼ ص. To prove ¼¼ ¼¼ Ø ¼¼ µ ¼ ¼ ص, we map an edge-separator Ë ¼ (with respect to ¼ ) to an edge-separator Ë ¼¼ of ¼¼, setting Ë ¼¼ Ù Ú µ ¾ Ë. The same path mapping argument shows that Ë ¼¼ is indeed an edge-separator of ¼¼. It therefore remains to prove that ¼ ¼ ص ¼¼ ¼¼ Ø ¼¼ µ. Let Ë ¼¼ ¼¼ be an ¼¼ -Ø ¼¼ edge-separator in ¼¼. If Ë ¼¼ contains some arc Û Ù µ, the set Ë ¼¼ Ò Û Ù µµ Ù Û µ has at most the same size, and must also be an ¼¼ -Ø ¼¼ edge-separator, because Ù Û µ is the only arc into Û in ¼¼, and hence every path using Û Ù µ must also use Ù Û µ. Similarly, if Ë ¼¼ contains an arc ¼¼ Ù µ (or Û Ø ¼¼ µ, the symmetric case), we can remove ¼¼ Ù µ (resp. Û Ø ¼¼ µ) from Ë ¼¼, and instead add the arc Ù Û µ (resp. Ù Û µ) to obtain a separator of at most the same size. We can therefore assume that Ë ¼¼ contains only arcs of the form Ù Û µ. Now, define Ë Ù Û µ ¾ Ë ¼¼. Then, Ë Ë ¼¼, and the same path mapping argument as above shows that Ë is an -Ø edge-separator with respect to ¼. We can now apply the max-flow min-cut theorem for unlabeled graphs to the graph ¼¼ with vertices ¼¼ and Ø ¼¼, completing the proof. In Section 3, we showed that the vertex version of Menger s Theorem does not hold for temporal networks. We now construct a family of graphs, ¾, with nodes Ø for which Ô Ú Ø µ ½, but Ú Ø µ, and has µ nodes. For the general idea underlying the following construction, reconsider the three vertices Ú Ú ¾ Ú in Figure 1, which from now on in a slight abuse of terminology shall be called the inner vertices. Every time-respecting Ú ½ -Ú path has to visit at least two inner vertices, ensuring that there can be at most one disjoint path. On the other hand, there is a time-respecting Ú ½ -Ú path through any set of ¾ inner vertices, so that any Ú ½ -Ú separator must have size at least ¾. We generalize this idea to a graph with ¾ ½ inner vertices, and label the edges so that every timerespecting -Ø path visits at least of these vertices (ensuring again that there is exactly one disjoint -Ø 13

15 path). On the other hand, we make sure that there is a time-respecting -Ø path through any set of inner vertices, so every -Ø separator must have size at least. The graph with ¾ ½ inner vertices will be denoted by, and produce a gap of. The skeleton of the graph, consisting of the vertices, Ø, and the inner vertices, is depicted in Figure 6. ¾ ½ ½ Ø 1 3 2k-3 2k-1 k k+2 3k-4 ¾ ¾ 3k-2 ½ ¾ ½ Figure 6: Skeleton of the graph with the labeling The inner vertices are labeled ½ ¾ ½. Vertices ½ are connected to with edges labeled µµ ¾ ½, vertices ¾ ½ to Ø with edges labeled صµ ¾. We will add further labeled edges and vertices to provide time-respecting paths through every set of inner vertices, at the same time maintaining the following key property: If a time-respecting path arrives at vertex Ú at time Ø, it must have visited at least ¾Ú Ø inner vertices up to this point (including Ú). That is, the earlier a path arrives at a point, the longer it must be (contrary to our everyday observations). To make the idea underlying the additional edges more clear, we use the concept of a delay edge. A delay edge is labeled with two times Ø ½ Ø ¾ µ. Ø ½ denotes the time at which the edge can be entered, and it is then left at time Ø ¾, i.e. it produces a delay of Ø ¾ Ø ½. Thus, delay edges are exactly the kind of edges that were considered in [3]. As mentioned in Section 2, a delay edge between vertices Ú and Û can be simply modeled by adding a new vertex Ù, and edges Ú Ùµ, labeled Ú Ùµµ Ø ½, and Ù Ûµ, labeled Ù Ûµµ Ø ¾. It is important that the auxiliary vertex Ù introduced through a delay edge will not affect the size of a minimum separator. For if Ù were part of a separator Ë, we could replace it by Ú to yield a separator of at most the same size. Furthermore, all our delay edges will satisfy Ø ½ Ø ¾, so they can only be traversed in one direction. Between any pair of inner vertices Ú Û, we add parallel delay edges with labels Ú ¾Û Ú ½µ Ú ½ ¾Û Úµ Ú ½ ¾Û Ú ¾µ. Because delay edges are implemented by auxiliary vertices, the resulting graph is still simple. The important properties of the resulting graph and labeling are summarized in the following lemma: Lemma 6.2 Ú Øµ, and Ô Ú Øµ ½. Proof. (1) For the proof of the first equation, first note that deleting the inner vertices ½ obviously disconnects from Ø, so Ú Øµ. To prove that Ú Øµ, let Ë ½ ¾ ½ be any set of inner vertices of size Ë (as argued above, we need not consider auxiliary vertices for -Ø separators). Let Ú ½ Ú ¾ Ú be inner vertices that are not in Ë. Consider the path È Ú ½ Ú Ø which uses the earliest possible delay edge between any pair Ú Ú ½ µ. È does not meet Ë by construction. 14

16 We establish by induction on that È arrives at vertex Ú at time Ø ¾Ú. In the base case ½, this is obvious, because È can use the edge Ú ½ µ, which is labeled Ú ½ µµ ¾Ú ½ ½ (and the edge exists, because Ú ½ ). For the induction step, assume that È arrives at Ú at time ¾Ú. Then, È next uses the delay edge between Ú and Ú ½ labeled ¾Ú ¾Ú ½ Ú Ú ½µ, which we can rewrite as ¾Ú ¾Ú ½ ½µµ, i.e. È reaches Ú ½ at time ¾Ú ½ ½µ. Therefore, È reaches Ú at time ¾Ú, and using the edge Ú Øµ with label Ú Øµµ ¾Ú makes È a time-respecting -Ø path. (2) For the second equation, it is obvious that Ô Ú Øµ ½ (take for instance the path visiting all inner vertices). Therefore, it remains to prove that there can be at most one disjoint time-respecting -Ø path. We will prove that every time-respecting -Ø path must visit at least inner vertices. We establish the following invariant by induction on Ú: If È is a simple time-respecting path from to Ú reaching Ú at time Ø, then È must visit at least ¾Ú Ø inner vertices (including Ú). In the base case Ú ½, the only time-respecting -½ path is the edge ½µ labeled ½µµ ½, so the invariant holds trivially (remember that the delay edges can be thought of as directed, so ½ cannot be reached through any delay edges). Assume now that the property holds for all Û Ú for some Ú ½, and let È be a time-respecting - Ú ½µ path reaching Ú ½ at time Ø. Let Û be the last inner vertex visited before Ú ½ on È, and Ø Û the time at which it was visited. Then, Ø Û Ø, and Û Ú. For if Û Ú ½, the path from Û to Ú ½ cannot be time-respecting. It would either have to lead through or Ø (and both ¾Û ½ ¾Ú ½ and ¾Û ¾Ú ¾ ), or use a delay edge, which we know to be directed the opposite way through the labeling. Therefore, we can apply the induction hypothesis to Û, to obtain that È must visit at least ¾Û Ø Û inner vertices between and Û, i.e. È visits at least ¾Û Ø Û ½ inner vertices including Ú ½. Because È was simple, it must use a delay edge to get from Û to Ú ½, and from the labeling of the delay edges, we obtain that Ø ¾ Ú ½µ Û ½ Ø Û Û ¾Ú ¾Û ½ Ø Û. Solving for Ø Û yields Ø Û ¾Û Ø ¾Ú ½, and È must visit at least ¾Û Ø Û ½ ¾Û ¾Û ¾Ú ½ Ø ½ ¾ Ú ½µ Ø inner vertices including Ú ½, completing the inductive proof. Now, let È be a simple time-respecting -Ø path. Then, È must enter Ø through one of the edges Ú Øµ for Ú, i.e. it must have reached vertex Ú at time Ø ¾Ú. From the invariant established above, we conclude that it must have visited at least ¾Ú Ø inner vertices. We may assume for any set of disjoint time-respecting -Ø paths that they are simple, and since every one of these paths must visit at least inner vertices, there is exactly one disjoint time-respecting -Ø path, completing the proof. Corollary 6.3 The graphs have an Ô Òµ gap between the maximum number of disjoint time-respecting -Ø paths and the minimum size of an -Ø separator. Proof. With Lemma 6.2, it only remains to show that Ò Î µ µ. But this is obvious, because we inserted delay edges (i.e. auxiliary vertices) between any pair of the ¾ ½ inner vertices. Proof of theorem 3.7. The theorem is proved by a direct reduction from the hardness (resp. approximation hardness) of the bounded length vertex-disjoint paths (BLVDP) problem, which is defined as follows: Given a directed graph Î µ with edge lengths Ð Æ Ò ¼, a length bound, and two vertices Ø ¾ Î. What is the maximum number of vertex-disjoint -Ø paths whose lengths are bounded by? The NP-hardness of deciding whether there are two vertex-disjoint paths with lengths bounded by was proved in [12]. [7] proved that it is NP-hard to approximate within a factor of Ñ ½¾ the maximum number of edge-disjoint -Ø paths whose lengths are bounded by some Ç Ñ ½¾ µ. Their proof also works for 15