On Efficiently Finding Small Separators in Temporal Graphs

# On Efficiently Finding Small Separators in Temporal Graphs

## Abstract

Vertex separators, that is, vertex sets whose deletion disconnects two distinguished vertices in a graph, play a pivotal role in algorithmic graph theory. For many realistic models of the real world 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, we study the problem of finding a small vertex set (the separator) in a temporal graph with two designated terminal vertices such that the removal of the set breaks all temporal paths connecting one terminal to the other. Herein, we consider two models of temporal paths: paths that contain arbitrarily many hops per time step (non-strict) and paths that contain at most one hop per time step (strict). We settle the hardness dichotomy (NP-hardness versus polynomial-time solvability) of both problem variants regarding the number of time steps of a temporal graph. Moreover we prove both problem variants to be NP-complete even on temporal graphs whose underlying graph is planar. We show that on temporal graphs whose underlying graph is planar, if additionally the number of time steps is constant then the problem variant for strict paths is solvable in quasi linear time. Finally, for general temporal graphs we introduce the notion of a temporal core (vertices whose incident edges change over time). We prove that on temporal graphs with constant-sized temporal core, the non-strict variant is solvable in polynomial time, where the degree of the polynomial is independent of the size of the temporal core, while the strict variant remains NP-complete.

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## 1 Introduction

In complex network analysis, nowadays it is very common to have access and process graph data where the interactions in the data set are time-stamped. When using static graphs as a mathematical model for problem instances, the dynamics of interactions are not reflected and important information of the data might not be captured. A straightforward approach to incorporate the dynamics of interactions into the model is to use temporal graphs. A temporal graph is, informally speaking, a graph where the edge set may change over a discrete time interval. Having the dynamics of interactions represented in the model, it is essential to adapt definitions of connectivity and paths to respect the temporal nature of the model. This directly affects the notion of separators in the temporal setting. Vertex separators are a fundamental primitive in static network analysis and it is folklore that they can be computed in polynomial time. In contrast to the static case, Kempe et al. [23] showed that the temporal analogue of this problem is NP-hard.

Temporal graphs are a well-established concept in the literature and are also referred to as evolving [15] and time-varying [24] graphs, temporal networks [22, 23, 27], multidimensional networks [7], link streams [33], and edge-scheduled networks [6]. In this work, we use the well-established model in which each edge has a time stamp [7, 22, 1, 20, 23, 27]. Assuming discrete time steps, this is equivalent to a sequence of static graphs over a fixed set of vertices [28]. 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 .

Many real-world applications have temporal graphs as underlying mathematical models. Consider for instance a public transportation network where the edges correspond to bus or train connections between stations (vertices). It is natural to model these connections with time stamps. Other examples include information spreading in social networks, communication in social networks, biological pathways or spread of diseases [22].

A fundamental question in temporal graphs, addressing issues such as connectivity [4, 27], survivability [24], and robustness [31], is whether there is a “time-respecting” path from a distinguished start vertex  to a distinguished target vertex .1 We thoroughly study the computational complexity of separating from  in a given temporal graph by deleting few vertices with the aim to identify tractable cases. To this end, we consider two very similar but still significantly differing path models (both used in the literature) in temporal graphs, leading to two corresponding models of temporal separation.

### The two models.

We start with the introduction of the “non-strict” model [23]. 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  is a temporal -separator if there is no temporal -path in . We can now 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 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 [1, 2, 28, 27]. 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 variant appears 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.

### Our contributions.

In the following we summarize our results; see Table 1 for a survey.

• In Section 3, we investigate the computational hardness of both problems under restrictions of the number of layers and under the restriction that the underlying graph is planar. We prove that both Temporal -Separation and Strict Temporal -Separation are NP-complete for all  and , respectively, strengthening a result by Kempe et al. [23] (they show NP-hardness for all ). Moreover, we prove that both problems are NP-complete on temporal graphs that have a planar underlying graph. To this end, we prove that Length-Bounded -Separation (for the problem definition, see Section 3) is NP-hard on planar graphs.

• In Section 4, we identify tractable cases of both problems on temporal graphs with few layers. For the non-strict problem, our hardness result is already tight.2 For the strict variant, we identify a dichotomy in the computational complexity by proving polynomial-time solvability of Strict Temporal -Separation for . As a side product, we give a linear-time algorithm for the Single-Source Shortest Strict Temporal Paths problem, improving the best known running time [34] by a -factor. Moreover, we prove that for constantly many layers, Strict Temporal -Separation on planar graphs is solvable in time.

• In Section 5, 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 on temporal graphs with temporal core of constant size, Temporal -Separation is solvable in  time, where  is some fixed constant being independent of the size of the temporal core, while Strict Temporal -Separation remains NP-complete even if the temporal core is empty.

A particular virtue of our work is that our results point out 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 is noticeable since the literature so far used both models without discussing the subtle differences in computational complexity.

### Related work.

Our main reference is the work of Kempe et al. [23] who proved that Temporal -Separation is NP-hard. In contrast, Berman [6] 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 [24] studied cuts where an edge deletion only last 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. [2]. 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 [26] 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 [6] 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. [23] 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. [27] 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 [29] 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. [14] 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 absence of the edge. Axiotis and Fotakis [4] 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.

## 2 Preliminaries

As a convention, denotes the natural numbers without zero. For , we use .

### Static graphs.

We use basic notations from (static) graph theory [13]. 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  if and only if . 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. Let . The vertices visited by  are denoted by . For surveys concerning temporal graphs we refer to [8, 28, 22, 21].

### Strict and non-strict temporal separators.

Throughout the whole 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. [34] we assume that the time-edge set is ordered by ascending labels. Moreover, we can assume that the number of layers is at most the number of time-edges:

###### Lemma 2.1.

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 .

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 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.

###### Proof of Lemma 2.1.

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, . ∎

Regarding our two models, we have the following connection:

###### Lemma 2.3.

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. ∎

## 3 Hardness Results

In this section we establish some preliminary hardness results. We observe that both Temporal -Separation and Strict Temporal -Separation are strongly related to the NP-complete Length-Bounded -Separation (LBS) problem [9, 32]: Given an undirected graph , two distinct vertices , and two integers , is there a subset  such that  and there is no -path in  of length at most ? We get the following straight-forward reduction.

{restatable}

observationlbstostrprob 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.

Baier et al. [5] showed that Length-Bounded -Separation is NP-complete, even if the lower bound  for the path length is five, and hence Strict Temporal -Separation is NP-complete for all . This at hand, Lemma 2.3 implies that Temporal -Separation is NP-complete for all . However, through closer inspection we get that the non-strict variant is already NP-complete for all . This improves a previous result by Kempe et al. [23] who showed NP-completeness of Temporal -Separation and Strict Temporal -Separation for all . We summarize in the following.

###### Proposition 3.1.

Temporal -Separation is NP-complete for all  and Strict Temporal -Separation is NP-complete for all .

###### Proof.

We will make use of the NP-complete Vertex Cover problem.

Vertex Cover Input: A graph  and . Question: Is there a subset  of size at most  such that for all  it holds ?

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. [5, 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

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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.

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 -path 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 . ∎

In the next section we prove that the bound on  is tight in the strict case (note that 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.

Since Length-Bounded -Separation is W[1]-hard with respect to the postulated separator size [19], from Lemmata 2.3 and 3 we obtain the following.

###### Corollary 3.2.

Both Temporal -Separation and Strict Temporal -Separation are W[1]-hard with respect to .

Furthermore, we show NP-completeness of Temporal -Separation and Strict Temporal -Separation for the restricted class of planar temporal graphs, that is, temporal graphs that have a planar underlying graph. To this end, we prove NP-hardness for Length-Bounded -Separation on planar graphs—a result which we consider to be of independent interest; note that NP-completeness on planar graphs was only known for the edge-deletion variant of LBS on undirected graphs [17] and weighted directed graphs [30].

###### Theorem 3.3.

Length-Bounded -Separation on planar graphs is NP-hard.

Roughly, the idea behind Theorem 3.3 is to reduce from an NP-complete planar edge-weighted edge-deletion variant of Length-Bounded -Separation which has constant vertex degree. Since the degree is constant, one can replace a vertex with a grid-like gadget.

###### Proof.

We give a many-one reduction from the NP-complete [17] edge-weighted variant of Length-Bounded -Cut, referred to as Planar Length-Bounded -Cut, where the input graph  is planar, has edge costs , has maximum degree , the degree of  and  is three, and and are incident to the outerface, to Length-Bounded -Separation. Let  be an instance of Planar Length-Bounded -Cut, and we assume  to be even3. We construct an instance  of Length-Bounded -Separation as follows.

### Construction.

For each vertex , we introduce a vertex-gadget  which is a grid of size , that is, a graph with vertex set  and edge set . There are six pairwise disjoint subsets  of size  that we refer to as connector sets. As we fix and orientation of  such that is in the top-left, there are two connector sets are on the top of , two on the bottom of , one on the left of , and one on the right of . Formally, , , , ,, and .

Note that all -paths are of length at most , for all , because there are only  vertices in .

Let  be a plane embedding of . We say that an edge  incident with vertex  is at position  on  if  is th edge incident with  when counted clockwise with respect to .

For each edge , we introduce an edge-gadget  that differs on the weight of , as follows. Let  be at position  on  and at position  on . If , then  is constructed as follows. Add a path consisting of  vertices and connect one endpoint with each vertex in  by an edge and connect the other endpoint with each vertex in  by an edge. If , then  is constructed as follows. We introduce a planar matching between the vertices in  and . That is, for instance, we connect vertex  with for each , if , or we connect vertex  with for each , if and (we omit the remaining cases). Then, replace each edge by a path of length at least  where its endpoints are identified with the endpoints of the replaced edge. Hence, a path between two vertex-gadgets has length at least .

Next, we choose connector sets  and  such that no vertex  is adjacent to a vertex from an edge-gadget. Such  and  always exist because the degree of  and  is three. Now, we add two special vertices  and  and edges between  and each vertex in , as well as between  and each vertex in .

Finally, we set Note that can be computed in polynomial time. Moreover, one can observe that  is planar by obtaining an embedding from .

### Correctness.

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

: Let  be a yes-instance. Thus, there is a solution  with  such that there is no -path of length at most  in . We construct a set  of size at most  by taking for each  one arbitrary vertex from the edge-gadget  into . Note that since , each edge in  is of cost one.

Assume towards a contradiction that there is a shortest -path  of length at most  in . Since a path between two vertex-gadgets has length at least , we know that  goes through at most  edge-gadgets. Otherwise  would be of length at least Now, we reconstruct an -path  in  corresponding to  by taking an edge  into  if  goes through the edge-gadget . Hence, the length of  is at most . This contradicts that there is no -path of length at most  in . Consequently, there is no -path of length at most  in  and  is a yes-instance.

: Let  be a yes-instance. Thus, there is a solution  of minimum size (at most ) such that there is no -path of length at most  in . Since  is of minimum size, it follows from the following claim that  for all .

###### Claim 3.4.

Let  be a vertex-gadget and  with . Then, for each vertex set  of size at most  it holds that there are  and  such that there is a -path of length at most  in .

###### Proof of Claim 3.4.

Let  be a vertex-gadget and  two connector sets of , where  and . We add vertices  and and edges  and  to , where  and . There are  different cases in which . It is not difficult to see that in each case there are  vertex-disjoint -paths. The claim then follows by Menger’s Theorem [26]. ∎

Note that by minimality of , it holds that  for all  with . We construct an edge set  of cost at most  by taking  into  if there is a .

Assume towards a contradiction that there is a shortest -path  of length at most  in . We reconstruct an -path  in  which corresponds to  as follows. First, we take an edge  such that . Such a  always exists, because  and . Let  be the first edge of  and at position  on . Then we add a -path  in , such that . Due to Claim 3.4, such a -path  always exists in  and is of length at most .

We take an edge-gadget  into  if  is in . Recall, that an edge-gadget is a path of length . Due to Claim 3.4, we can connect the edge-gadgets  of two consecutive edges  in  by a path of length at most  in . Let  be the last edge in , be at position  on , and . We add a -path of length  in  (Claim 3.4). Note that  visits at most  vertex-gadgets and  edge-gadgets. The length of  is at most This contradicts that  forms a solution for . It follows that there is no -path of length at most  in  and  is a yes-instance. ∎

As Lemmata 2.3 and 3 are planarity preserving, we get the following:

###### Corollary 3.5.

Both Temporal -Separation and Strict Temporal -Separation on planar temporal graphs is NP-complete.

## 4 On Temporal Graphs with Few Layers

Consider the situation where a commuter wants to reach her working place () from home () by public transport. There is a certain time when she needs to be at work and a certain time span that she is willing to spend traveling from her home to work. Hence, considering the available transportation network, only a certain time interval is relevant. If we restrict the transportation network to this time interval, then it is reasonable to assume only few layers to be present. In this scenario, if there is no temporal -separator of size , then the commuter can reach her working place in time, even if transport hubs are blocked.

In the previous section we showed that (Strict) Temporal -Separation is NP-complete already for (five) two layers (Proposition 3.1). In this section we determine for any smaller number of layers whether the respective problems are polynomial-time solvable. It is folklore that a minimum -separator in static graphs can be computed in polynomial time. Thus, Temporal -Separation is polynomial-time solvable for . The strict variant is more interesting: We show in this section that a dichotomy takes place when increases from four to five, that is, if we can solve Strict Temporal -Separation in  time. Furthermore, we show that Strict Temporal -Separation on planar temporal graphs is solvable in time for any arbitrary but fixed number of layers. Studying planar temporal graphs is well-motivated for instance in the context of transportation, since many street networks can be modeled by planar graphs.

A key tool [6,