The Computational Complexity of Finding Separators in Temporal Graphs

11/02/2017 ∙ by Philipp Zschoche, et al. ∙ Berlin Institute of Technology (Technische Universität Berlin) 0

Vertex separators, that is, vertex sets whose deletion disconnects two distinguished vertices in a graph, play a pivotal role in algorithmic graph theory. For instance, the concept of tree decompositions of graphs is tightly connected to the separator concept. For many realistic models of the real world, however, it is necessary to consider graphs whose edge set changes with time. More specifically, the edges are labeled with time stamps. In the literature, these graphs are referred to as temporal graphs, temporal networks, time-varying networks, edge-scheduled networks, etc. While there is an extensive literature on separators in "static" graphs, much less is known for the temporal setting. Building on previous work (e.g., Kempe et al. [STOC '00]), for the first time we systematically investigate the (parameterized) complexity of finding separators in temporal graphs. Doing so, we discover a rich landscape of computationally (fixed-parameter) tractable and intractable cases. In particular, we shed light on the so far seemingly overlooked fact that two frequently used models of temporal separation may lead to quite significant differences in terms of computational complexity. More specifically, considering paths in temporal graphs one may distinguish between strict paths (the time stamps along a path are strictly increasing) and non-strict paths (the time stamps along a path are monotonically non-decreasing). We observe that the corresponding strict case of temporal separators leads to several computationally much easier to handle cases than the non-strict case does.

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1 Introduction

In complex network analysis, it is nowadays very common to have access to and process graph data where the interactions among the vertices are time-stamped. When using static graphs as a mathematical model, the dynamics of interactions are not reflected and important information of the data might not be captured. Temporal graphs address this issue. A temporal graph is, informally speaking, a graph where the edge set may change over a discrete time interval, while the vertex set remains unchanged. Having the dynamics of interactions represented in the model, it is essential to adapt definitions such as connectivity and paths to respect temporal features. This directly affects the notion of separators in the temporal setting. Vertex separators are a fundamental primitive in static network analysis and it is well-known that they can be computed in polynomial time (see, e.g., proof of [1, Theorem 6.8]). In contrast to the static case, Kempe et al. [27] showed that in temporal graphs it is NP-hard to compute minimum separators.

Temporal graphs are well-established in the literature and are also referred to as time-varying [29] and evolving [17] graphs, temporal networks [26, 27, 32], link streams [28, 38], multidimensional networks [9], and edge-scheduled networks [8]. In this work, we use the well-established model in which each edge has a time stamp [9, 26, 2, 24, 27, 32, 4]. Assuming discrete time steps, this is equivalent to a sequence of static graphs over a fixed set of vertices [33]. Formally, we define a temporal graph as follows.

Definition 1.1 (Temporal Graph).

An (undirected) temporal graph  is an ordered triple consisting of a set  of vertices, a set  of time-edges, and a maximal time label .

See Figure 1 for an example with , that is, a temporal graph with four time steps, also referred to as layers. The static graph obtained from a temporal graph  by removing the time stamps from all time-edges we call the underlying graph of .

(a) A temporal graph .
(b) Layers of .
Figure 1: Subfigure (a) shows a temporal graph  and subfigure (b) shows its four layers . The gray squared vertex forms a strict temporal -separator, but no temporal -separator. The two squared vertices form a temporal -separator.

Many real-world applications have temporal graphs as underlying mathematical model. For instance, it is natural to model connections in public transportation networks with temporal graphs. Other examples include information spreading in social networks, communication in social networks, biological pathways, or spread of diseases [26].

A fundamental question in temporal graphs, addressing issues such as connectivity [6, 32], survivability [29], and robustness [36], is whether there is a “time-respecting” path from a distinguished start vertex  to a distinguished target vertex .111In the literature the sink is usually denoted by . To be consistent with Michail [33] we use instead as we reserve to refer to points in time. We provide a thorough study of the computational complexity of separating from  in a given temporal graph.

Moreover, we study two natural restrictions of temporal graphs: (i) planar temporal graphs and (ii) temporal graphs with a bounded number of vertices incident to edges that are not permanently existing—these vertices form the so-called temporal core. Both restrictions are naturally motivated by settings e.g. occurring in (hierarchical) traffic networks. We also consider two very similar but still significantly differing temporal path models (both used in the literature), leading to two corresponding models of temporal separation.

Two path models.

We start with the introduction of the “non-strict” path model [27]. Given a temporal graph  with two distinct vertices , a temporal -path of length  in  is a sequence of time-edges in , where for all with and  for all . A vertex set  with is a temporal -separator if there is no temporal -path in . We are ready to state the central problem of our paper.

Temporal -Separation Input: A temporal graph , two distinct vertices , and . Question: Does admit a temporal -separator of size at most ?

Our second path model is the “strict” variant. A temporal -path is called strict if for all . In the literature, strict temporal paths are also known as journeys [2, 3, 33, 32].222We also refer to Himmel [23] for a thorough discussion and comparison of temporal path concepts. A vertex set  is a strict temporal -separator if there is no strict temporal -path in . Thus, our second main problem, Strict Temporal -Separation, is defined in complete analogy to Temporal -Separation, just replacing (non-strict) temporal separators by strict ones.

While the strict version of temporal separation immediately appears as natural, the non-strict variant can be viewed as a more conservative version of the problem. For instance, in a disease-spreading scenario the spreading speed might be unclear. To ensure containment of the spreading by separating patient zero () from a certain target (), a temporal -separator might be the safer choice.

Main results.
General Planar  Temporal core
(Section 3) (Section 4) (Section 5)
-Separation  unbounded  constant constant size
Temporal NP-complete NP-c. open  
Strict Temporal   NP-c. NP-c.   NP-complete
Table 1: Overview on our results. Herein, NP-c. abbreviates NP-complete,  and  denote the number of vertices and time-edges, respectively,  refers to the underlying graph of an input temporal graph.  (Thm. 3.1; W[1]-hard wrt. )  (Thm. 3.6)  (Cor. 4.3)  (Prop. 4.5)  (Thm. 5.2)

Table 1 provides an overview on our results.

A central contribution is to prove that both Temporal -Separation and Strict Temporal -Separation are NP-complete for all  and , respectively, strengthening a result by Kempe et al. [27] (they show NP-hardness of both variants for all ). For Temporal -Separation, our hardness result is already tight.333Temporal -Separation with  is equivalent to -Separation on static graphs. For the strict variant, we identify a dichotomy in the computational complexity by proving polynomial-time solvability of Strict Temporal -Separation for . Moreover, we prove that both problems remain NP-complete on temporal graphs that have an underlying graph that is planar.

We introduce the notion of temporal cores in temporal graphs. Informally, the temporal core of a temporal graph is the set of vertices whose edge-incidences change over time. We prove that Temporal -Separation is fixed-parameter tractable (FPT) when parameterized by the size of the temporal core, while Strict Temporal -Separation remains NP-complete even if the temporal core is empty.

A particular aspect of our results is that they demonstrate that the choice of the model (strict versus non-strict) for a problem can have a crucial impact on the computational complexity of said problem. This contrasts with wide parts of the literature where both models were used without discussing the subtle but crucial differences in computational complexity.

Technical contributions.

To show the polynomial-time solvability of Strict Temporal -Separation for , we prove that a classic separator result of Lovász et al. [30] translates to the strict temporal setting. This is surprising since many other results about separators in the static case do not apply in the temporal case. In this context, we also develop a linear-time algorithm for Single-Source Shortest Strict Temporal Paths, improving the running time of the best known algorithm due to Wu et al. [39] by a logarithmic factor.

We settle the complexity of Length-Bounded -Separation on planar graphs by showing its NP-hardness, which was left unanswered by Fluschnik et al. [19] and promises to be a valuable intermediate problem for proving hardness results. In the hardness reduction for Length-Bounded -Separation we introduce a grid-like, planarity-preserving vertex gadget that is generally useful to replace “twin” vertices which in many cases are not planarity-preserving and which are often used to model weights.

While showing that Temporal -Separation is fixed-parameter tractable when parameterized by the size of the temporal core, we employ a case distinction on the size of the temporal core, and show that in the non-trivial case we can reduce the problem to Node Multiway Cut. We identify an “above lower bound parameter” for Node Multiway Cut that is suitable to lower-bound the size of the temporal core, thereby making it possible to exploit a fixed-parameter tractability result due to Cygan et al. [14].

Related work.

Our most important reference is the work of Kempe et al. [27] who proved that Temporal -Separation is NP-hard. In contrast, Berman [8] proved that computing temporal -cuts (edge deletion instead of vertex deletion) is polynomial-time solvable. In the context of survivability of temporal graphs, Liang and Modiano [29] studied cuts where an edge deletion only lasts for  consecutive time stamps. Moreover, they studied a temporal maximum flow defined as the maximum number of sets of journeys where each two journeys in a set do not use a temporal edge within some  time steps. A different notion of temporal flows on temporal graphs was introduced by Akrida et al. [3]. They showed how to compute in polynomial time the maximum amount of flow passing from a source vertex  to a sink vertex  until a given point in time.

The vertex-variant of Menger’s Theorem [31] states that the maximum number of vertex-disjoint paths from  to  equals the size of a minimum-cardinality -separator. In static graphs, Menger’s Theorem allows for finding a minimum-cardinality -separator via maximum flow computations. However, Berman [8] proved that the vertex-variant of an analogue to Menger’s Theorem for temporal graphs, asking for the maximum number of (strict) temporal paths instead, does not hold. Kempe et al. [27] proved that the vertex-variant of the former analogue to Menger’s Theorem holds true if the underlying graph excludes a fixed minor. Mertzios et al. [32] proved another analogue of Menger’s Theorem: the maximum number of strict temporal -path which never leave the same vertex at the same time equals the minimum number of node departure times needed to separate  from , where a node departure time is the vertex at time point .

Michail and Spirakis [34] introduced the time-analogue of the famous Traveling Salesperson problem and studied the problem on temporal graphs of dynamic diameter , that is, informally speaking, on temporal graphs where every two vertices can reach each other in at most  time steps at any time. Erlebach et al. [16] studied the same problem on temporal graphs where the underlying graph has bounded degree, bounded treewidth, or is planar. Additionally, they introduced a class of temporal graphs with regularly present edges, that is, temporal graphs where each edge is associated with two integers upper- and lower-bounding consecutive time steps of edge absence. Axiotis and Fotakis [6] studied the problem of finding the smallest temporal subgraph of a temporal graph such that single-source temporal connectivity is preserved on temporal graphs where the underlying graph has bounded treewidth. In companion work, we recently studied the computational complexity of (non-strict) temporal separation on several other restricted temporal graphs [20].

2 Preliminaries

Let denote the natural numbers without zero. For , we use .

Static graphs. We use basic notations from (static) graph theory [15]. Let  be an undirected, simple graph. We use  and to denote the set of vertices and set of edges of , respectively. We denote by  the graph  without the vertices in . For  denotes the induced subgraph of  by . A path of length  is sequence of edges  where for all with . We set . Path  is an -path if  and . A set  of vertices is an -separator if there is no -path in .

Temporal graphs. Let  be a temporal graph. The graph  is called layer  of the temporal graph  where . The underlying graph  of a temporal graph  is defined as , where . (We write , and  for short if  is clear from the context.) For  we define the induced temporal subgraph of  by . We say that  is connected if its underlying graph is connected. For surveys concerning temporal graphs we refer to [10, 33, 26, 28, 25].

Regarding our two models, we have the following connection:

Lemma 2.1.

There is a linear-time computable many-one reduction from Strict Temporal -Separation to Temporal -Separation that maps any instance to an instance with  and .

Proof.

Let be an instance of Strict Temporal -Separation. We construct an equivalent instance in linear-time. Set , where  is called the set of edge-vertices. Next, let  be initially empty. For each , add the time-edges to . This completes the construction of . Note that this can be done in  time. It holds that  and that .

We claim that is a yes-instance if and only if is a yes-instance.

: Let  be a temporal -separator in  of size at most . We claim that  is also a temporal -separator in . Suppose towards a contradiction that this is not the case. Then there is a temporal -path  in . Note that the vertices on  alternated between vertices in  and . As each vertex in  corresponds to an edge, there is a temporal -path in  induced by the vertices of . This is a contradiction.

: Observe that from any temporal -separator, we can obtain a temporal -separator of not larger size that only contains vertices in . Let  be a temporal -separator in  of size at most  only containing vertices in . We claim that  is also a temporal -separator in . Suppose towards a contradiction that this is not the case. Then there is a temporal  path  in . Note that we can obtain a temporal -path  in  by adding for all consecutive vertices , , where  appears before  at time-step  on , the vertex . This is a contradiction. ∎

Throughout the paper we assume that the underlying graph of the temporal input graph is connected and that there is no time-edge between  and . Furthermore, in accordance with Wu et al. [39] we assume that the time-edge set is ordered by ascending time stamps.

2.1 The Maximum Label is Bounded in the Input Size

In the following, we prove that for every temporal graph in an input to (Strict) Temporal -Separation, we can assume that the number of layers is at most the number of time-edges. Observe that a layer of a temporal graph that contains no edge is irrelevant for Temporal -Separation. This also holds true for the strict case. Hence, we can delete such a layer from the temporal graph. This observation is formalized in the following two data reduction rules.

Reduction Rule 2.1.

Let  be a temporal graph and let  be an interval where for all  the layer  is an edgeless graph. Then for all  where  replace  with  in .

Reduction Rule 2.2.

Let  be a temporal graph. If there is a non-empty interval  where for all  the layer  is an edgeless graph, then set  to .

We prove next that both reduction rules are exhaustively applicable in linear time.

Lemma 2.2.

Reduction Rules 2.2 and 2.1 do not remove or add any temporal -path from/to the temporal graph  and can be exhaustively applied in  time.

Proof.

First we discuss Reduction Rule 2.1. Let  be a temporal graph, , be an interval where for all  the layer  is an edgeless graph. Let  be a temporal -path, and let  be the graph after we applied Reduction Rule 2.1 once on . We distinguish three cases.

  1. If , then no time-edge of  is touched by Reduction Rule 2.1. Hence,  also exists in .

  2. If , then there is a temporal -path  in , because .

  3. If , then there is clearly a temporal -path  in 

The other direction works analogously. We look at a temporal -path in  and compute the corresponding temporal -path in .

Reduction Rule 2.1 is applicable
Reduction Rule 2.1 is not applicable
Figure 2: Figure 2 shows a temporal graph where Reduction Rule 2.1 is applicable. In particular, layers  are edgeless. Figure 2 shows the same temporal graph after Reduction Rule 2.1 was applied exhaustively.

Reduction Rule 2.1 can be exhaustively applied by iterating over the by time-edges  in the time-edge set  ordered by ascending labels until the first  with the given requirement appear. Set . Then we iterate further over  and replace each time-edge  with  until the next  with the given requirement appear. Then we set  and iterate further over  and replace each time-edge  with . We repeat this procedure until the end of  is reached. Since we iterate over  only once, this can be done in  time.

Reduction Rule 2.2 can be executed in linear time by iterating over all edges and taking the maximum label as . Note that the sets  and  remain untouched by Reduction Rule 2.2. Hence, the application of Reduction Rule 2.2 does not add or remove any temporal -path. ∎

A consequence of Lemma 2.2 is that the maximum label  can be upper-bounded by the number of time-edges and hence the input size.

Lemma 2.3.

Let be an instance of (Strict) Temporal -Separation. There is an algorithm which computes in  time an instance  of (Strict) Temporal -Separation which is equivalent to , where .

Proof.

Let  be a temporal graph, where Reduction Rules 2.2 and 2.1 are not applicable. Then for each  there is a time-edge . Thus, . ∎

3 Hardness Dichotomy Regarding the Number of Layers

In this section we settle the complexity dichotomy of both Temporal -Separation and Strict Temporal -Separation regarding the number  of time steps. We observe that both problems are strongly related to the following NP-complete [11, 37] problem:

Length-Bounded -Separation (LBS) Input: An undirected graph , distinct vertices , and . Question: Is there a subset  such that  and there is no -path in  of length at most ?

Length-Bounded -Separation is NP-complete even if the lower bound  for the path length is five [7] and W[1]-hard with respect to the postulated separator size [22]. We obtain the following, improving a result by Kempe et al. [27] who showed NP-completeness of Temporal -Separation and Strict Temporal -Separation for all .

Theorem 3.1.

Temporal -Separation is NP-complete for every maximum label  and Strict Temporal -Separation is NP-complete for every . Moreover, both problems are W[1]-hard when parameterized by the solution size .

We remark that our NP-hardness reduction for Temporal -Separation is inspired by Baier et al. [7, Theorem 3.9].

Proof.

To show NP-completeness of Temporal -Separation for we present a reduction from the Vertex Cover problem where, given a graph and an integer , the task is to determine whether there exists a set of size at most such that does not contain any edge.

Construction.

Let  be an instance of Vertex Cover. We say that  is a vertex cover in  of size  if  and  is a solution to . We refine the gadget of Baier et al. [7, Theorem 3.9] and reduce from Vertex Cover to Temporal -Separation. Let  be a Vertex Cover instance and . We construct a Temporal -Separation instance , where are the vertices and the time-edges are defined as

Note that , and can be computed in polynomial time. For each vertex  there is a vertex gadget which consists of three vertices  and six vertex-edges. In addition, for each edge  there is an edge gadget which consists of two edge-edges  and . See Figure 3 for an example.

Figure 3: The Vertex Cover instance  (left) and the corresponding Temporal -Separation instance from the reduction of Theorem 3.1 (right). The edge-edges are dashed (red), the vertex-edges are solid (green), and the vertex gadgets are in dotted boxes.

Correctness.

We prove that is a yes-instance if and only if is a yes-instance.

: Let  be a vertex cover of size  for . We claim that  is a temporal -separator. There are  vertices not in the vertex cover  and for each of them there is exactly one vertex in . For each vertex in the vertex cover  there are two vertices in . Hence, .

First, we consider the vertex-gadget of a vertex . Note that in the vertex-gadget of , there are two distinct temporal -separators and . Hence, every temporal -path in  contains an edge-edge. Second, let  and let  and  be the temporal -paths which contain the edge-edges of edge-gadget of  such that  and . Since  is a vertex cover of  we know that at least one element of  is in . Thus,  or , and hence neither  nor  exist in . It follows that  is a temporal -separator in  of size at most , as there are no other temporal -paths in .

: Let  be a temporal -separator in  of size  and let . Recall that there are two distinct temporal -separators in the vertex gadget of , namely  and , and that all vertices in  are from a vertex gadget. Hence,  is of the form . We start with a preprocessing to ensure that for vertex gadget only one of these two separators are in . Let . We iterate over  for each :

  1. If  or  then we do nothing.

  2. If  then we remove  from  and decrease  by one. One can observe that all temporal -paths which are visiting  are still separated by  or .

  3. If  then we remove  from  and add . One can observe that  is still a temporal -separator of size  in .

  4. If  then we remove  from  and add . One can observe that  is still a temporal -separator of size  in .

That is a complete case distinction because neither  nor  separate all temporal -paths in the vertex gadget in . Now we construct a vertex cover  for  by taking  into  if both  and  are in . Since there are  vertex gadgets in  each containing either one or two vertices from , it follows that ,

Assume towards a contradiction that  is not a vertex cover of . Then there is an edge  where . Hence,  and . This contradicts the fact that  is a temporal -separator in , because is a temporal -path in . It follows that  is a vertex cover of  of size at most . ∎

Observation 3.2.

There is a polynomial-time reduction from LBS to Strict Temporal -Separation that maps any instance  of LBS to an instance with  for all  of Strict Temporal -Separation.

In the remainder of this section we prove that the bound on  is tight in the strict case (for the non-strict case the tightness is obvious). This is the first case where we can observe a significant difference between the strict and the non-strict variant of our separation problem. In order to do so, we have to develop some tools which we need in subroutines. In Section 3.1, we introduce a common tool to study reachability in temporal graphs on directed graphs. This helps us to solve the Single-Source Shortest Strict Temporal Paths efficiently (Proposition 3.4). Note that this might be of independent interest since it improves known algorithms, see Section 3.2. Afterwards, in Section 3.3, we prove that Strict Temporal -Separation can be solved in polynomial time, if the maximum label .

3.1 Strict Static Expansion

A key tool [8, 27, 32, 3, 39] is the time-expanded version of a temporal graph which reduces reachability and other related questions in temporal graphs to similar questions in directed graphs. Here, we introduce a similar tool for strict temporal -paths. Let  be a temporal graph and let . For each , we define the sets  and . The strict static expansion of  is a directed acyclic graph  where and , , , , and (referred to as column-edges of ). Observe that each strict temporal -path in  has a one-to-one correspondence to some -path in . We refer to Figure 4 for an example.

Figure 4: A temporal graph (left) and the strict static expansion for (right). One strict temporal -path in and its corresponding -path in are marked (green).
Lemma 3.3.

Let  be a temporal graph, where  are two distinct vertices. The strict static expansion for  can be computed in  time.

Proof.

Let  be a temporal graph, where  and  are two distinct vertices. Note that because is connected. We construct the strict static expansion for  as in four steps follows: First, we initiate for each an empty linked list. Second, we iterate over the set with non-decreasing labels and for each :

  1. Add to if they do not already exist. If we added a vertex then we push to the -th linked list.

  2. Add to .

This can be done in  time. Observe, that the two consecutive entries in the -th linked list is an entry in . Third, we iterate over each linked list in increasing order to add the column-edges to . Note that the sizes of all linked lists sum up to . Last, we add  and  to  as well as the edges in  and  to .

Note that the  as well as  can be upper-bounded by . We employed a constant number of procedures each running in  time. Thus, can be computed in  time. ∎

As a subroutine hidden in several of our algorithms, we need to solve the Single-Source Shortest Strict Temporal Paths problem on temporal graphs: find shortest strict paths from a source vertex to all other vertices in the temporal graph. Herein, we say that a strict temporal -path is shortest if there is no strict temporal -path of length . Indeed, we provide a linear-time algorithm for this. We believe this to be of independent interest; it improves (with few adaptations to the model; for details see Section 3.2) previous results by Wu et al. [39], but in contrast to the algorithm of Wu et al. [39] our subroutine cannot be adjusted to the non-strict case.

Proposition 3.4.

Single-Source Shortest Strict Temporal Paths is solvable in  time.

The following proof makes use of a strict static expansion of a temporal graph. See Sec. 3.1 for more details.

Proof.

By Lemma 3.3, we compute the strict static expansion  of  in  time and define a weight function

Observe that  with  is a weighted directed acyclic graph and that the weight of an -path in  with  is equal to the length of the corresponding strict temporal -path in . Hence, we can use an algorithm, which makes use of the topological order of  on , to compute for all  a shortest -path in  in  time (cf. Cormen et al. [12, Section 24.2]).

Now we iterate over and construct the shortest strict temporal -path in from the shortest -path in , where , and . This can be done in time because . Consequently, the overall running time is . Since the shortest strict temporal -path in can have length , this algorithm is asymptotically optimal. ∎

3.2 Adaptation of Proposition 3.4 for the Model of Wu et al. [39]

Wu et al. [39] considered a model where the temporal graph is directed and a time-edge has a traversal time . In the context of strict temporal path is always one. They excluded the case where , but pointed out that their algorithms can be adjusted to allow . However, this is not possible for our algorithm, because then the strict static expansion can contain cycles. Hence, we assume that for all directed time-edges .

Let be a directed temporal graph. We denote a directed time-edge from  to  in layer by . First, we initiate many linked lists. Without loss of generality we assume that , see Lemma 2.3. Second, we construct a directed temporal graph , where and is empty in the beginning. Then we iterate over the time-edge set by ascending labels. If has then we add  to . If has then we add a new vertex  to  and add time-edge  to and  to the -th linked list, where . We call the original edge of and the connector edge of . If we reach a directed time-edge with label  for the first time, then we add all directed time-edges from the -th linked list to . Observe that for each strict temporal -path in there is a corresponding strict temporal -path in , additionally we have that  is ordered by ascending labels and that  can be constructed in  time.

To construct a strict static expansion for a directed temporal graph , we modify the edge set , where  . Finally, we adjust the weight function from the algorithm of Proposition 3.4 such that if is a column-edges of correspond to a connector-edges, and otherwise. Observe that for a strict temporal -path of traversal time the corresponding -path in the strict static expansion is of weight of the traversal time of .

Our algorithm behind Theorem 3.6 executes the following steps:

  1. As a preprocessing step, remove unnecessary time-edges and vertices from the graph.

  2. Compute an auxiliary graph called directed path cover graph of the temporal graph.

  3. Compute a separator for the directed path cover graph.

In the following, we explain each of the steps in more detail.

The preprocessing reduces the temporal graph such that it has the following properties. A temporal graph with two distinct vertices is reduced if (i) the underlying graph  is connected, (ii) for each time-edge there is a strict temporal -path which contains , and (iii) there is no strict temporal -path of length at most two in . This preprocessing step can be performed in polynomial time:

Lemma 3.5.

Let be an instance of Strict Temporal -Separation. In time, one can either decide  or construct an instance  of Strict Temporal -Separation such that is equivalent to , is reduced, , , and .

The following proof makes use of a strict static expansion of a temporal graph. See Sec. 3.1 for more details.

Proof.

First, we remove all time-edges which are not used by a strict temporal -path. Let  be an instance of Strict Temporal -Separation. We execute the following procedure.

  1. Construct the strict static expansion  of .

  2. Perform a breadth-first search in  from  and mark all vertices in the search tree as reachable. Let  be the reachable vertices from .

  3. Construct , where . Observe that  is the reachable part of  from , where all directed arcs change their direction.

  4. If , then our instance  is a yes-instance.

  5. Perform a breadth-first search from  in  and mark all vertices in the search tree as reachable. Let  be the reachable set of vertices from . In the graph , all vertices are reachable from  and from each vertex the vertex  is reachable.

  6. Output the temporal graph , where   and .

One can observe that  is a temporal subgraph of  and that  is connected. Note that all subroutines are computable in time (see Lemma 3.3). Consequently,