Temporal Reachability Minimization: Delaying vs. Deleting
TTemporal Reachability Minimization:Delaying vs. Deleting
Hendrik Molter ! Technische Universität Berlin, Faculty IV, Algorithmics and Computational Complexity, Germany
Malte Renken ! Technische Universität Berlin, Faculty IV, Algorithmics and Computational Complexity, Germany
Philipp Zschoche ! Technische Universität Berlin, Faculty IV, Algorithmics and Computational Complexity, Germany
Abstract
We study spreading processes in temporal graphs, i. e., graphs whose connections change over time.These processes naturally model real-world phenomena such as infectious diseases or informationflows. More precisely, we investigate how such a spreading process, emerging from a given setof sources, can be contained to a small part of the graph. To this end we consider two ways ofmodifying the graph, which are (1) deleting connections and (2) delaying connections. We show aclose relationship between the two associated problems and give a polynomial time algorithm whenthe graph has tree structure. For the general version, we consider parameterization by the numberof vertices to which the spread is contained. Surprisingly, we prove W[1]-hardness for the deletionvariant but fixed-parameter tractability for the delaying variant.
Mathematics of computing → Graph algorithms; Mathematics ofcomputing → Paths and connectivity problems; Mathematics of computing → Network flows
Keywords and phrases
Temporal Graphs, Temporal Paths, Disease Spreading, Network Flows,Parameterized Algorithms, NP-hard Problems
Funding
H. Molter and M. Renken are supported by the DFG, project MATE (NI 369/17).
Acknowledgements
This work was initiated at the research retreat of the Algorithmics and Compu-tational Complexity group of TU Berlin in September 2020 in Zinnowitz.
Reachability is a fundamental problem in graph theory and algorithmics [17, 33, 38, 39] andquite well-understood. With the emergence of temporal graphs, the concept of reachabilitywas extended to the dimension of time using temporal paths [6, 34]. For a vertex s to reachanother vertex z in a temporal graph, there must not only be a path between them but theedges of this path have to appear in chronological order. This requirement makes temporalreachability non-symmetric and non-transitive, which stands in contrast to reachabilityin normal (static) graphs. Reachability is arguably one of the most central concepts intemporal graph algorithmics and has been studied under various aspects, such as pathfinding [5, 9, 12, 40], vertex separation [28, 34, 41], finding spanning subgraphs [4, 11],temporal graph exploration [2, 7, 23, 24, 25, 26], and others [3, 10, 32, 35].Perhaps the most prominent application of temporal graph reachability is currentlyepidemiology, dealing with effective prevention or containment of disease spreading [1]. Here,minimizing the reachability of vertices in a temporal graph by manipulating the temporalgraph corresponds to minimizing the spread of an infection in various networks by somecountermeasures. Application instances for this scenario may be drawn from physical contacts Temporal graphs are graphs whose edge set changes over discrete time. a r X i v : . [ c s . D S ] F e b Temporal Reachability Minimization: Delaying vs. Deleting [19, 27] or airline flights [8, 13], but also social networks [14, 31], cattle movements [36], orcomputer networks [37].Enright et al. [21] studied the problem of deleting k time-edges such that no singlevertex can reach more than r other vertices and showed its NP-hardness and W[1] -hardnessfor the parameter k , even in very restricted settings. Here, we shift the focus to a set ofmultiple given sources, thus studying the following problem, which has not been consideredfor computational complexity analysis yet (to the best of our knowledge). Minimizing Temporal Reachablity by Deleting (MinReachDelete)
Input:
A temporal graph G , a set of sources vertices S , and integers k, r . Question:
Can we delete at most k time-edges s.t. at most r vertices are reachable from S ? Imaginably, removing edges or vertices is not the most infrastructure friendly approachto restrict reachability. To address this, other operations have been studied. Enright et al.[22] considered restricting the reachability by just changing the relative order in whichedges are active. Deligkas and Potapov [15] considered restricting the reachability by amerging operation of consecutive edge sets of the temporal graph and by a delay operationof time-edges by δ time steps, i.e., moving a time-edge from time t to t + δ . In particular,they introduced a delay variant of MinReachDelete . Minimizing Temporal Reachablity by Delaying (MinReachDelay)
Input:
A temporal graph G , a set of sources vertices S ⊆ V , and integers k, r, δ . Question:
Can we delay at most k time-edges by δ s.t. at most r vertices are reachable from S ? This is the central problem studied in this paper. We remark that technically Deligkas andPotapov [15] formulate the problem slightly differently, allowing delays of up to δ to appear.However, a simple argument can be given to see that this distinction is not significant:Clearly, delaying a time-edge reduces the number of reachable vertices only if the undelayedtime-edge could be reached from some source s ∈ S . But when this is the case, increasingthe delay of that time-edge can never increase the set of vertices reachable from S , eventhough it might increase the set of vertices reachable from some s ′ ̸ = s .Deligkas and Potapov [15] showed that MinReachDelay is NP-hard and
W[1] -hardwhen parameterized by k , even if underlying graph has lifetime τ = 2. A close look into theproof reveals that this also holds for MinReachDelete . Our contribution.
We study how
MinReachDelete and
MinReachDelay relate to eachother. We show that both problems are polynomial-time solvable on trees. Moreover, thereis an intermediate reduction from
MinReachDelete to MinReachDelay indicating that
MinReachDelay seems generally harder than
MinReachDelete . However, surprisingly,this is no longer true when we parameterize the problems by the number r of reachablevertices. Here, we develop a max-flow-based branching strategy and obtain fixed-parametertractability for MinReachDelay while
MinReachDelete remains
W[1] -hard. This makes
MinReachDelay particularly interesting for applications where the number of reachablevertices should be very small, e.g. when trying to contain the spread of dangerous diseases. That is, an edge at a point in time. . Molter, M. Renken, and P. Zschoche 3
We define N as the positive natural numbers, [ a, b ] := { i ∈ Z | a ≤ i ≤ b } , and [ n ] := [1 , n ].For a function f : V → Z and subset X ⊆ V we denote by f ( X ) the sum P x ∈ V f ( x ).We use standard notation from graph theory [16]. We say for a (directed) graph G that G − X := G [ V ( G ) \ X ] is the induced subgraph of G when the vertices in X are removed,and G \ Y := ( V ( G ) , E ( G ) \ Y ) is the subgraph when the edges in Y are removed, where X is a vertex set and Y is an edge set. For any predicate P , the Iverson bracket [ P ] is 1 if P istrue and 0 otherwise. Parameterized complexity.
Let Σ denote a finite alphabet. A parameterized problem L ⊆{ ( x, k ) ∈ Σ ∗ × N ∪ { }} is a subset of all instances ( x, k ) from Σ ∗ × N ∪ { } , where k denotesthe parameter . A parameterized problem L is in FPT (is fixed-parameter tractable ) if thereis an algorithm that decides every instance ( x, k ) for L in f ( k ) · | x | O (1) time, where f is anycomputable function only depending on the parameter. If a parameterized problem L isW[1]-hard, then it is presumably not fixed-parameter tractable. We refer to Downey andFellows [18] for details. Temporal graphs. A temporal graph G consists of a set of vertices V (or V ( G )), and asequence of edge sets ( E i ) i ∈ [ τ ] where each E i is a set of unordered pairs from V . The number τ is called the lifetime of G . The elements of E ( G ) := S i ∈ [ τ ] E i × { i } are called the time-edges of G . Furthermore it has a transversal time function γ : E ( G ) → N specifying the time it takesto transverse each edge. The temporal graph G is then written as the tuple ( V, ( E i ) i ∈ [ τ ] , γ ).Often we assume γ to be the constant function γ ≡ G = ( V, ( E i ) i ∈ [ τ ] ).The underlying graph of G is the graph ( V, S τi =1 E i ). For a time-edge set Y and temporal graph G , we denote by G \ Y the temporal graph where V ( G \ Y ) = V ( G ) and E ( G \ Y ) = E ( G ) \ Y .A temporal s - z -path in G is a sequence of time-edges P = ( e i = ( { v i − , v i } , t i )) mi =1 where v = s and v m = z , the sequence of edges ( { v i − , v i } ) mi =1 forms an s - z -path in the underlying graph of G , and t i ≥ t i − + γ ( e i ) for all i ∈ [ m ].The arrival time of P is t m + γ ( e m ). The set of vertices of P is denoted by V ( P ) = { v i | ≤ i ≤ m } . A vertex w is reachable from v in G (at time t ) if there exists a temporal v - w -pathin G (with arrival time at most t ). In particular, every vertex reaches itself via a trivialpath. Furthermore, w is reachable from S ⊆ V if there is a temporal s - w -path for some s ∈ S , and the set of all vertices reachable from S is denoted the reachable set R G ( W ). Wedrop the index G if it is clear form the context. Delaying a time-edge ( { v, w } , t ) by δ refersto replacing it with the time-edge ( { v, w } , t + δ ). For a temporal graph G and a time-edgeset X ⊆ E ( G ) we denote by G ↗ δ X the temporal graph G where the time-edges in X aredelayed by δ . Preliminary observations.
We present an intermediate reduction from the
MinReachDelete to MinReachDelay . ▶ Lemma 1.
Given an instance I = ( G = ( V, ( E i ) i ∈ [ τ ] ) , S, k, r ) of MinReachDelete ,we can compute in linear time, an instance J = ( G ′ = ( V ′ , ( E ′ i ) τi =1 ) , S ′ , k, r ′ , δ = 3 τ ) of MinReachDelay such that the feedback vertex number of the underlying graph of G and That is, the minimum number of vertices needed to hit all cycles in an undirected graph.
Temporal Reachability Minimization: Delaying vs. Deleting G ′ is the same, and I is a yes -instance if and only if J is a yes -instance. Proof.
We construct G ′ = ( V ′ , ( E ′ i ) τi =1 ) in the following way. Set V e := { e uv , e vu | ( { v, u } , t ) ∈ E ( G ) }} , V s := { s vu | e vu ∈ V e } ,V ′ := V ∪ V e ∪ V s , and S ′ := S ∪ V s . Begin with E ′ i = ∅ for all i ∈ [3 τ ]. Then, add for each time-edge ( { v, u } , t ) ∈ E ( G ), the time-edges ( { v, e vu } , t − , ( { v, e vu } , t ) , ( { e vu , e uv } , t − , ( { u, e uv } , t − { u, e uv } , t )to G ′ . Afterwards, add the time-edge ( { s vu , e vu } , τ ) for each e vu ∈ V e . Finally we set r ′ := r + | V ′ \ V | . Since we add for each time-edge of G a constant number of verticesand time-edges to G ′ , we have that |G ′ | ∈ O ( |G| ) and G ′ can be computed in linear time.Moreover, the underlying graph G ′ of G ′ is obtained from the underlying graph G of G bysubdividing edges and adding leaves, thus G and G ′ have the same feedback vertex number.It remains to prove that the two instances are equivalent. ( ⇒ ): Let X be a solution for I . Then, set X ′ := { ( { e vu , e uv } , t − | ( { v, u } , t ) ∈ X } .Pick s ∈ S and v ∈ V arbitrary. Note that for each temporal s - v -path P ′ in G ′ , we canconstruct a temporal s - v -path P in G which uses a time-edge ( { u, w } , t ) if and only if P ′ usesthe time-edge ( { e uw , e wu } , t − s - v -path in G ′ ↗ δ X ′ cannotuse any delayed time-edge. As s and v are arbitrary, this proves R G ′ ↗ δ X ′ ( S ′ ) ∩ V ⊆ R G\ X ( S ).Thus, at most r + | V ′ \ V | = r ′ vertices are reachable from S ′ in G ′ ↗ δ X ′ , thus J is a yes -instance. ( ⇐ ): Let X ′ be a solution for J . It is not difficult to see that we may assume withoutloss of generality that X ′ ⊆ { ( { e vu , e uv } , t − | ( { v, u } , t ) ∈ E ( G ) } . Set X := { ( { v, u } , t ) | ( { e uv , e vu } , t − ∈ X ′ } . Let s ∈ S and v ∈ V be arbitrary. Note that for each temporal s - v -path P in G , we can construct a temporal s - v -path P ′ in G ′ as above. Thus R G\ X ( S ) ⊆ R G ′ ↗ δ X ′ ( S ′ ) ∩ V . We further have R G ′ ↗ δ X ′ ( S ′ ) ⊇ V ′ \ V , which implies that | R G ′ ↗ δ X ′ ( S ′ ) ∩ V | ≤ r ′ − | V ′ \ V | = r . Therefore | R G\ X ( S ) | ≤ r as required and I is a yes -instance. ◀ Note that the reduction in Lemma 1 preserves the size k of the solution and the feedbackvertex number of the underlying graph. We remark that, in exchange for dropping the latterproperty, one can modify the reduction to instead have constant δ and | S ′ | = | S | . In this section the study
MinReachDelay and
MinReachDelete parameterized by thereachable set size r . In particular, our main result in this section is the fixed-parametertractability of MinReachDelay parameterized by r . This is in stark contrast to the W[1] -hardness of
MinReachDelete parameterized by r which we show first. ▶ Theorem 2.
MinReachDelete parameterized by r is W[1] -hard, even if τ = 2 . Proof.
We present a parameterized reduction from the
W[1] -hard [18]
Clique problemparameterized by ℓ , where given a graph H = ( U, F ) we are asked whether H contains aclique of size ℓ .Let H = ( U, F ) be a graph, where | F | = m . We construct an instance I = ( G , { s } , k = m − (cid:0) ℓ (cid:1) , r = 1 + ℓ + (cid:0) ℓ (cid:1) ) of MinReachDelete , where G := ( V, ( E i ) i ∈ [2] ) is the temporalgraph given by V := U ∪ { s } ∪ { e f | f ∈ F } ,E := {{ s, e f } | f ∈ F } , and E := (cid:8) { e { u,v } , u } , { e { u,v } , v } (cid:12)(cid:12) { u, v } ∈ F (cid:9) . . Molter, M. Renken, and P. Zschoche 5 Note that I can be constructed in polynomial time. ( ⇒ ): Let C = ( V ′ , F ′ ) be a clique of size ℓ in H . We set X := { ( { s, e f } , | f ∈ F \ F ′ } .Note that | X | ≤ k and that for each edge f ∈ F we can reach e f from s if and only if f ∈ F ′ .Hence, by the construction of G , a vertex u ∈ U is reachable from s in G \ X if and onlyif u ∈ V ′ . Hence, we can reach 1 + (cid:0) ℓ (cid:1) + ℓ many vertices from s in G \ X . Thus, I is a yes -instance. ( ⇐ ): Let X ⊆ E ( G ) be a solution for I . Without loss of generality, we can assumethat X does not contain a time-edge ( { e f , u } , { e f , s } , (cid:0) ℓ (cid:1) vertices from { e f | f ∈ F } are reachable from s in G \ X . Since r = 1 + (cid:0) ℓ (cid:1) + ℓ , we can reach from s at most ℓ vertices from U . Hence, U ∩ R G\ X ( { s } ) mustform a clique of size ℓ in H . ◀ Due to Theorem 2, we know that there is presumably no f ( r ) · |G| O (1) -time algorithm todecide whether we can keep the reachable set of a vertex s small (at most r vertices), bydeleting at most k time-edges. However, this changes when we delay (instead of deleting) atmost k edges. Formally, we show the following. ▶ Theorem 3.
MinReachDelay is solvable in r r · k · |G| time. The proof of Theorem 3 is structured as follows.
Step 1 (reduction to slowing):
We reduce
MinReachDelay to an auxiliary problem whichwe call
MinReachSlow . Here, instead of delaying a time-edge (moving it δ layers forwardin time) we slow it, i. e., increase the time required to transverse it by δ . Step 2 (flow-based techniques):
Our new target now is a fixed-parameter algorithm for
MinReachSlow . Since we do not aim to preserve a specific temporal graph class,we simplify the input by replacing S with a single-source s . Then we transform thetemporal graph G into a (non-temporal) directed graph D in which the deletion of an edgecorresponds to slowing a temporal edge in G . Using this, we derive at a max-flow-basedpolynomial-time algorithm which checks whether the source s can be prevented fromreaching any vertices outside of a given set R by slowing at most k time-edges in G . Step 3 (resulting search-tree):
We are aiming for a search-tree algorithm for
MinReach-Slow . Let R be a set of vertices and our max-flow-based algorithm failed to prevent s from reaching any vertices outside of R . Now, if there exists a solution for the giveninstance of MinReachSlow , then we can identify less than | R | · r vertices such that atleast one of them will be always reached from s . We can then try adding each of themto R , gradually building a search-tree to find the solution.Henceforth the details. Instead of solving MinReachDelay directly, we reduce it toan auxiliary problem introduced next. Let G = ( V, ( E i ) i ∈ [ τ ] , γ ) be a temporal graph. Slowing a time-edge ( { v, w } , t ) by δ refers to increasing γ (( { v, w } , t )) by δ . We define G ↑ δ X := ( V, ( E i ) i ∈ [ τ ] , γ ′ ) where γ ′ ( e ) := γ ( e ) + δ · [ e ∈ X ]. Our auxiliary problem is thefollowing. Minimizing Temporal Reachablity by Slowing ( MinReachSlow ) Input:
A temporal graph G = ( V, ( E i ) i ∈ [ τ ] , γ ), a set of sources S ⊆ V , and integers k, r, δ . Question:
Is there a time-edge set X ⊆ E ( G ) of size at most k such that | R G↑ δ X ( S ) | ≤ r ? By the following, solving an instance of
MinReachSlow also solves
MinReachDelay . ▶ Lemma 4.
An instance I = ( G = ( V, ( E i ) i ∈ [ τ ] , γ ) , S, k, r, δ ) of MinReachDelay is a yes -instance if and only if J = ( G , γ, S, k, r, δ ) is a yes -instance of MinReachSlow . Temporal Reachability Minimization: Delaying vs. Deleting
Proof. ( ⇒ ): Let X ⊆ E ( G ) be a solution for I . Note that for every temporal v - w -path P in G ↑ δ X , we can construct a temporal v - w -path P ′ in G ↗ δ X by replacing a time-edge( e, t ) ∈ P with ( e, t + δ ) whenever ( e, t ) ∈ X . Hence, the reachable set of s ∈ S in G ↗ δ X isa superset of the reachable set of s in G ↑ δ X . Thus, J is a yes -instance. ( ⇐ ): Let X ⊆ E ( G ) be an inclusion-minimal solution for J . We claim that every vertexreachable from S in G ↗ δ X until some time t is also reachable from S in G ↑ δ X untiltime t . Suppose for contradiction that the claim does not hold true for some vertex z andlet t be the time S reaches z in G ↗ δ X . We may assume z to be chosen to minimize t .Clearly t >
0, i. e., z / ∈ S . Let P be a temporal s - z -path in G ↗ δ X with arrival time atmost t and s ∈ S . Let u be the penultimate vertex of P and ( { u, z } , t ′ ) the last time-edgeof P . By minimality of t , u must be reachable from S until time t ′ also in G ↑ δ X . Hence,the last time-edge ( { u, z } , t ′ ) of P is not present in G ↑ δ X and therefore ( { u, z } , t ′ − δ ) ∈ X .By minimality of X , there must be a source s ′ ∈ S and a temporal s ′ - u -path P ′ in G ↑ δ X reaching either u or z at time t ′ − δ . If P ′ reaches z , then this is clearly a contradiction. Butif P ′ reaches u , then appending ( { u, z } , t − δ ) to P ′ produces a temporal s ′ - z -path in G ↑ δ X arriving at time t , thus also a contradiction. ◀ In the reminder of this section, we show that
MinReachSlow is fixed-parameter tractable,when parameterized by r . Formally, we aim for the following theorem, which in turn clearlyimplies Theorem 3 by the means of Lemma 4. ▶ Theorem 5.
MinReachSlow can be solved in r r · k · |G| time. The remainder of this section is dedicated to proving Theorem 5. The advantage of considering
MinReachSlow instead of
MinReachDelay is that we do not have to deal with newtime-edges appearing due to the delay operation. This allows us to translate the reachabilityof a temporal graph to a (non-temporal) directed graph specially tailored to
MinReachSlow .In particular, the removal of some edges in the directed graph corresponds to slowing thecorresponding time-edges by δ in the temporal graph. Before giving the details of theconstruction, we first reduce to the case where S is a singleton. ▶ Lemma 6.
Given an instance I = ( G = ( V, ( E i ) i ∈ [ τ ] ) , S, k, r, δ ) of MinReachSlow , wecan construct in linear time a instance J = ( G ′ , { s } , k, r + 1 , δ ) of MinReachSlow suchthat I is a yes -instance if and only if J is a yes -instance. Proof.
We set G ′ := ( V ∪ { s } , ( E ′ i ) i ∈ [ τ + δ ] ) where s is a new vertex, E ′ := {{ s, s ′ } | s ′ ∈ S } , E ′ i := ∅ for all i ∈ [ δ ] \ { } , and E ′ i + δ := E i for all i ∈ [ τ ]. Observe that slowing an edgein E ′ has no effect. Thus, I is a yes -instance if and only if J is a yes -instance. Clearly, J can be computed in linear time. ◀ A network ( D, c ) consists of a directed graph D = ( V, A ) and a edge capacities c : A → N ∪ {∞} . A function f : A → N ∪ { } is an s - z -flow for two distinct vertices s, z ∈ V if ∀ e ∈ A : f ( e ) ≤ c ( e ) and ∀ v ∈ V \ { s, z } : P ( u,v ) ∈ A f (( u, v )) = P ( v,u ) ∈ A f ( v, u )).The value of f is denoted by | f | := P ( s,v ) ∈ A f (( s, v )). An arc set C ⊆ A is an s - z -cut of annetwork (( V, A ) , c ) if s, z ∈ V and there is no s - z -path in ( V, A \ C ). The capacity of the s - z -cut C is c ( C ) := P e ∈ C c ( e ).Let G = ( V, ( E i ) i ∈ [ τ ] , γ ) be a temporal graph, s ∈ R ⊆ V and δ ∈ N . We definethe temporal neighborhood of a vertex v ∈ V at time point t ∈ [ τ ] as the set N G ( v, t ) := S τi = t N ( V,E i ) ( v ) containing all neighbors of v in the layers t through τ . . Molter, M. Renken, and P. Zschoche 7 v w v v v w w w e e Figure 1
Left : An excerpt of a tempo-ral graph G containing the time-edge e =( { v, w } , Right : An excerpt of the flow net-work F ( G , s, R, δ ) showing the correspondingpart for e , where solid arc have capacity ∞ and the dashed arc has capacity 1. We then define the flow network F ( G , s, R, δ ) := ( D, c ) where D = ( V ′ , A ) is the directedgraph defined by V ′ := { s , z } ∪ (cid:26) e , e , v t , w t , v t + γ ( e ) , w t + γ ( e ) ,v t + γ ( e )+ δ , w t + γ ( e )+ δ (cid:12)(cid:12)(cid:12)(cid:12) v, w ∈ R and e = ( { v, w } , t ) ∈ E ( G ) (cid:27) and A := n ( v t , v t ′ ) (cid:12)(cid:12)(cid:12) v t ∈ V ′ , t ′ = min { i | i > t and v i ∈ V ′ } ̸ = ∞ o (1) ∪ ( v t , e ) , ( w t , e ) , ( e , e ) , ( e , v t + γ ( e ) ) , ( e , w t + γ ( e ) ) , ( v t , w t + γ ( e )+ δ ) , ( w t , v t + γ ( e )+ δ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v, w ∈ R and e = ( { v, w } , t ) ∈ E ( G ) (2) ∪ (cid:8) ( v t , z) (cid:12)(cid:12) v ∈ R, t = max { i | N G ( v, i ) ⊈ R } ̸ = −∞ (cid:9) . (3)and we set c (( e , e )) = 1 for all e ∈ E ( G ) and c ( a ) = ∞ for all other a ∈ A . ConsiderFigure 1 for an illustration. ▶ Lemma 7.
For any given G = ( V, ( E i ) i ∈ [ τ ] , γ ) , s ∈ R ⊆ V , and k, δ ∈ N , one can test in O ( k · |G| ) time whether F ( G , s, R, δ ) has a s - z -flow of value at least k + 1 and compute sucha flow or a maximum flow otherwise. Proof.
Note, that the flow network F ( G , s, R, k, δ, r ) can be computed in O ( |G| ) time byiterating over E ( G ) first forward and then backwards once. Then we can compute a flow ofvalue k + 1 or of maximum value, whichever is smaller, by running at most k + 1 rounds ofthe Ford-Fulkerson algorithm [29]. This gives a overall running time of O ( k · |G| ) time. ◀▶ Lemma 8.
Let G = ( V, ( E i ) i ∈ [ τ ] , γ ) , s ∈ R ⊆ V , δ ∈ N , and (( V ′ , A ) , c ) := F ( G , s, R, δ ) .Let X ⊆ E ( G ) and C := { ( e , e ) ∈ A | e ∈ X } . For any x t ∈ V ′ , there is a s - x t -path in ( V ′ , A \ C ) if and only if G ↑ δ X contains a temporal s - x -path with arrival time at most t . Proof.
Let γ ′ be the transversal time function of G ↑ δ X , i. e., γ ′ ( e ) := γ ( e ) + δ · [ e ∈ X ]. ( ⇐ ): Let P be a temporal s - x -path P in G ↑ δ X for some vertex x ∈ V and let t bethe arrival time of P . Then we construct a s - x t -path ˆ P in ( V ′ , A \ C ) as follows. Startwith ˆ P being just the vertex s and perform the following two steps for every time-edge e = ( { v, w } , b ) of P in order. Note that the currently last vertex of ˆ P is v c for some c ≤ b . As long as c < b , appendto ˆ P the arc ( v c , v d ) where d > c is chosen minimal (cf. (1)). Afterwards, the currentlylast vertex of ˆ P is v b . If e / ∈ X , i. e., ( e , e ) / ∈ C , then append to ˆ P the arcs ( v b , e ), ( e , e ), and ( e , w b + γ ( e ) ) (cf.(2)). Otherwise, if e ∈ X , i. e., γ ′ ( e ) = γ ( e ) + δ , then append to ˆ P the arc ( v b , w b + γ ( e )+ δ ).Note that in both cases the new last vertex of ˆ P is w b + γ ′ ( e ) . ( ⇒ ): Let P be a s - x t -path in ( V ′ , A \ C ). Then we construct a s - x -path P ′ with arrivaltime at most t in G ↑ δ X as follows. Start with P ′ being just the vertex s (with arrival time0) and repeat the following steps until all arcs of P have been processed. Temporal Reachability Minimization: Delaying vs. Deleting Let b be the arrival time of P ′ and v the last vertex. Note that the last vertex of P is v c for some c ≥ b . Ignore all arcs of P up to the last arc containing v c for some c ≥ b . If the next three arcs are ( v c , e ) , ( e , e ) , ( e , w c + γ ( e ) ) for some time-edge e = ( { v, w } , c ) ∈E ( G ), then append to P ′ that time-edge e . Note that, by assumption, ( e , e ) / ∈ C , thus e / ∈ X , and thus the arrival time of e is c + γ ′ ( e ) = c + γ ( e ). Otherwise the next arc must be ( v c , w c + γ ( e )+ δ ) for some time-edge e = ( { v, w } , c ) ∈ E ( G ).Then append to P ′ that time-edge e . Note that the arrival time of e is c + γ ′ ( e ) ≤ c + γ ( e ) + δ . ◀ We now show that we can use Lemma 7 to check whether s can be prevented fromreaching any vertices outside of R by slowing at most k time-edges by δ each. ▶ Lemma 9.
For any given G = ( V, ( E i ) i ∈ [ τ ] , γ ) , s ∈ R ⊆ V , δ ∈ N , and k ∈ N ∪ { } , themaximum s - z -flow in F ( G , s, R, δ ) has value at most k if and only if there is a set X of atmost k time-edges such that s can not reach any vertices outside of R in G ↑ δ X . Proof.
Write (( V ′ , A ) , c ) := F := F ( G , s, R, δ ). ( ⇒ ): Let the maximum s -z-flow f in F have value at most k . Moreover, let C bea s -z-cut of minimum capacity. From the max-flow min-cut theorem [30], we know that c ( C ) ≤ k . Note that C ⊆ { ( e , e ) ∈ A | e ∈ E ( G ) } , since all other edges have infinitecapacity. Hence, | C | ≤ k . Now set X := { e ∈ E ( G ) | ( e , e ) ∈ C } . Assume towards acontradiction that there is a temporal s - x -path P in G ↑ δ X for some x ∈ V \ R . We maytake P to be minimal, thus the penultimate vertex y of P is contained in R . By Lemma 8there is a s - y t -path ˆ P in ( V ′ , A \ C ) where t is the time P reaches y . The fact that P afterwards proceeds to x and (3) in the definition of F imply that ( V ′ , A \ C ) contains apath from y t to z. This contradicts C being a s -z-cut in ( V ′ , A ). ( ⇐ ): Let X be a time-edge set as assumed. By assumption R G↑ δ X ( { s } ) ⊆ R . We claimthat C = { ( e , e ) | e ∈ X } ⊆ A is a s -z-cut in ( V ′ , A ), which implies that the maximumvalue of an s -z-flow in F is at most c ( C ) ≤ k [30]. So suppose towards a contradiction thatthere is a s -z-path P in D avoiding C . Let x t ∈ V be the penultimate vertex of P . Thenthere is a s - x -path P ′ in G ↑ δ X with arrival time at most t by Lemma 8. The final arcof P is now ( x b , z). Hence, ( x t , z) must be contained in (3), i. e., we can extend P ′ by sometime-edge to end at a vertex in V \ R . This contradicts our assumption R G↑ δ X ( { s } ) ⊆ R . ◀ If F ( G , s, R, δ ) contains a s -z-flow of value k + 1, then we want to find a small set Y ⊆ V ( G ) \ R of vertices such that Y ∩ R G↑ δ X ( s ) ̸ = ∅ for every X ⊆ E ( G ) with | X | ≤ k and | R G↑ δ X ( s ) | ≤ r . ▶ Lemma 10.
Let G = ( V, ( E i ) i ∈ [ τ ] , γ ) , s ∈ R ⊆ V , δ, r ∈ N , and k ∈ N ∪ { } . Assumethat F ( G , s, R, δ ) has a s - z -flow of value k + 1 . We can compute in O ( k · | A | ) time a set Y ⊆ V \ R of size at most | R | · r such that Y ∩ R G↑ δ X ( s ) ̸ = ∅ holds whenever there is a X ⊆ E ( G ) with | X | ≤ k and | R G↑ δ X ( s ) | ≤ r . Proof.
Let f be a s -z-flow of value k + 1 in (( V ′ , A ) , c ) := F ( G , s, R, δ ). We may assume f to never use an arc ( v t , w t ′ ) whenever A contains some arc ( v b , z) with b ≥ t , as we couldotherwise redirect f to use that latter arc instead. Note that performing this modificationcan be done in O ( k · | A | ) time. Now set H := { v t ∈ V ′ | ∃ b ≥ t : ( v b , z) ∈ A, f (( v b , z)) > } ,H ′ := { v t ∈ H | | N G ( v, t ) | < r } , and Y := [ v t ∈ H ′ N G ( v, t ) \ R . . Molter, M. Renken, and P. Zschoche 9
Algorithm 11
Pseudocode of the algorithm behind Theorem 5.
Input:
An instance I = ( G , { s } , k, r, δ ) of MinReachSlow . Output: yes if I is a yes -instance and otherwise no . return g ( { s } ) , where function g ( R ) is Compute a s -z-flow f in F ( G , s, R, δ ) by Lemma 7 if f is of value at most k then return yes if | R | ≥ r then return no Compute a set Y ⊆ V \ R by Lemma 10 foreach v ∈ Y do if g ( R ∪ { v } ) = yes then return yes return noFrom the above definitions and the fact that S v t ∈ H { v } ⊆ R , we see that | Y | < | R | · r .It remains to prove that R G↑ δ X ( s ) ∩ Y ̸ = ∅ , with X ⊆ E ( G ) being an arbitrary solutionfor the MinReachSlow -instance ( G , { s } , k, r, δ ). Define C := { ( e , e ) ∈ A | e ∈ X } . Since f has value k + 1 > c ( C ), there is a s -z-path P in ( V ′ , A \ C ) where each edge e ∈ E ( P )has f ( e ) >
0. Let v t be the penultimate vertex of P . Then clearly v t ∈ H by definitionof H . By Lemma 8, there is a temporal s - v -path P ′ in G ↑ δ X with arrival time at most t .Note that through P ′ , s can reach v as well as all vertices of N G↑ δ X ( v, t ) = N G ( v, t ). Thus1 + | N G ( v, t ) | ≤ | R G↑ δ X ( s ) | ≤ r as X is assumed to be a solution. In particular this alsoimplies v t ∈ H ′ . Thus N G ( v, t ) \ R ⊆ Y ∩ R G↑ δ X ( s ). Further N G ( v, t ) \ R ̸ = ∅ by definitionof H (and also (3) in the definition of F ), thus the claim is proven. ◀ Now we are ready to prove of Theorem 5. The corresponding search-tree algorithm islisted as Algorithm 11.
Proof of Theorem 5. et I = ( G , S, k, r, δ ) be the given instance of MinReachSlow . ByLemma 6, we can assume S = { s } after linear time preprocessing. We now prove thatAlgorithm 11 solves ( G , { s } , k, r, δ ) in O ( r r · k · |G| ) time.Let I be a no -instance. Then, by Lemma 9, F ( G , s, R, δ ) will for all { s } ⊆ R ⊆ V have a s -z-flow of value k + 1. Hence, line 4, and thus Algorithm 11 will never return yes .Let I be a yes -instance. Thus there is a set X of at most k time-edges such that | R G↑ δ X ( r ) | ≤ r . We claim that g ( R ′ ) returns yes for all R ′ with s ∈ R ′ ⊆ R G↑ δ X ( s ). Weprove this by reverse induction on | R ′ | . In the base case where R = R G↑ δ X ( s ), g ( R ) returns yes by Lemma 9. Now assume the claim to hold whenever | R | = q and let R be of size q − s ∈ R ⊆ R G↑ δ X ( s ). Assume that F ( G , s, R, k, δ ) has a s -z-flow of value k + 1,otherwise we are done (by line 4). By Lemma 10, the set Y computed in line 6 contains avertex u ∈ R G↑ δ X ( s ) \ R . Thus, g ( R ∪ { u } ) returns yes by induction hypothesis. Hence, byline 8, g ( R ) return yes , completing the induction. In particular, g ( { s } ) returns yes , thereforeAlgorithm 11 is correct.To bound the running time, note that each call g ( R ) makes at most | Y | ≤ | R | · r ≤ r recursive calls by Lemma 10. In each of these recursive calls the cardinality of R increases byone, so the recursion depth is at most r by line 5. By Lemma 7 and Lemma 10, lines 3 and 6take at most O ( k · |G| ) time. Hence, the overall running time is bounded by O ( r r · k · |G| ). ◀ In this section we present an algorithm that solves
MinReachDelete and
MinReachDelay in polynomial time on temporal graphs where the underlying graph is a tree or a forest. Thisis a quite severe yet well-motivated restriction of the input [20, 21, 22]. ▶ Theorem 12.
MinReachDelete and
MinReachDelay are polynomial-time solvable ifthe underlying graph is a forest.
Actually we even provide an polynomial-time algorithm for a generalized version of
Min-ReachDelay . Then, the polynomial-time solvable of
MinReachDelete follows fromLemma 1, since it is forest preserving. We define the generalized problem as follows:
Weighted MinReachDelay on Forests
Input:
A temporal graph G = ( V, ( E i ) i ∈ [ τ ] , γ ) whose underlying graph is a forest, a weightfunction w : V → N ∪ { , ∞} , a set F ⊆ E ( G ) of undelayable time-edges, a set ofsources S ⊆ V , and integers k, r, δ . Question:
Does there exists a time-edge set X ⊆ E ( G ) of size at most k suchthat w ( R G↗ δ X ( S )) ≤ r ? In the remainder of this section, we show how to solve this problem using dynamicprogramming in polynomial time. Informally speaking, our dynamic program works asfollows. As a preprocessing step we unfold vertices of large degree, reducing to an equivalentinstance of maximum degree 3. Then we root each underlying tree at an arbitrary vertexand build a dynamic programming table, starting at the leaves. More precisely, we computea table entry for each combination of a vertex v , a budget k , a time step t , and a flagindicating whether v is first reached from a child or from its parent. This table entry thencontains a minimum reachable subset of the subtree rooted at v that can be achieved byapplying k delay operations to that subtree. ▶ Theorem 13.
Weighted MinReachDelay on Forests is polynomial-time solvable ifthe underlying graph is a forest.
Proof.
Assume for now that the underlying graph of G is a tree, rooted at an arbitrary leaf.We denote by T v the subtree with root v ∈ V . We use the reaching time ∞ to denote “never”.By convention, a vertex can reach itself at time 0. Define N ∗ := N ∪ { , ∞} .We first show how to transform the underlying graph into a binary tree. This will highlysimplify the description of the dynamic programming table. Replace each vertex v of degreedeg( v ) > v ) − v among the new vertices such that each of them has degree 3. Set the weight ofthe path’s first vertex to the weight of v , and the weight of all other path vertices to 0. Notethat this modification produces an equivalent instance of maximum degree at most 3, whileincreasing the number of vertices only by a constant factor.We extend the notion of reachability to vertex-time pairs ( s, t ) ∈ S × [ τ ] by sayingthat ( s, t ) reaches v ∈ V in G if there exists a temporal s - v -path starting at time t or later.For a set A ⊆ V × [ τ ], R G ( A ) is the set of all vertices, reachable from any member of A in G .We say a vertex v is reached through another vertex w if there is a temporal path from asource s ∈ S to v that uses w .Let v ∈ V , k ∈ N . Define T v,k as the set of temporal graphs obtained from T v by applyingup to k delay operations. Partition T v,k into {T v,k,t | t ∈ N ∗ } , where T v,k,t contains those . Molter, M. Renken, and P. Zschoche 11 graphs in which v is reached from S ∩ V ( T v ) exactly at time t . Finally, we set for each t ∈ N ∗ D [ v, k, t, false ] = min (cid:8) w (cid:0) R T ( S ∩ V ( T v )) (cid:1) (cid:12)(cid:12) T ∈ T v,k,t (cid:9) and D [ v, k, t, true ] = min w (cid:0) R T ( S ∩ V ( T v )) ∪ R T ( { ( v, t ) } ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∈ [ t ′ ≥ t T v,k,t ′ . where the minimum of an empty set is ∞ by convention. It is convenient to also define theseentries as ∞ whenever k <
0. Roughly speaking, D [ v, k, t, ι ] contains the minimal weightreached in T v under the assumption that up to k delay operations are applied to T v , that v is first reached at time t , and that v is reached by a source in S ∩ V ( T v ) at time t if ι = false , v is reached by a source in S \ V ( T v ) at time t if ι = true .Note that v might be reached simultaneously from S ∩ V ( T v ) and S \ V ( T v ).We next show how to compute D [ v, k, t, ι ] recursively, starting at the leaf vertices. Observethat D [ v, k, t, ι ] = ∞ whenever v ∈ S is a source and t >
0. Thus, this case shall be excludedin the following. If v has no children. If ι = false and v / ∈ S and t < ∞ , then D [ v, k, t, false ] = ∞ asthere is no way that v can be reached from a source in S ∩ V ( T v ) = ∅ . Otherwise, D [ v, k, t, ι ] = w ( v ) · [ t < ∞ ] . If v has exactly one child v ′ . If ι = false and v / ∈ S , then v must be reached through v ′ at time t . In this case the minimal total weight reached in T v ′ is D := min t ′ ≤ t D [ v ′ , k − κ ( t ′ , t ) , t ′ , false ] , where κ ( t ′ , t ) is the minimal number of delays that need to occur on the edge { v, v ′ } toensure that ( v ′ , t ′ ) reaches v at time t but not earlier. (Set κ ( t ′ , t ) = ∞ if this is impossible.)Consequently, if v / ∈ S , then D [ v, k, t, false ] = w ( v ) · [ t < ∞ ] + D . If ι = true or v ∈ S , then there are two possibilities. If v ′ is reached through v at time t ′ ,with t ′ being the first time v ′ is reached from S , then the minimal total weight reached in T v ′ is D := min t ′ ≥ t D [ v ′ , k − κ ( t, t ′ ) , t ′ , true ] . Otherwise, v ′ must be reached from a source in S ∩ V ( T v ′ ) at time t ′ , thus the minimal totalweight reached in T ′ v is D := min t ′ D [ v ′ , k − ˆ κ ( t ′ , t ) , t ′ , false ] , where ˆ κ ( t ′ , t ) is the minimal number of delays that need to occur on { v, v ′ } to ensure that( v, t ) cannot reach v ′ before time t ′ and ( v ′ , t ′ ) cannot reach v before time t . (Again, setˆ κ ( t ′ , t ) = ∞ if this is impossible.) Thus we obtain for the case that ι = true or v ∈ S that D [ v, k, t, ι ] = w ( v ) · [ t < ∞ ] + min { D , D } . If v has two children v ′ , v ′′ . The situation is similar to that of only one child vertex, althoughmore possible cases have to be distinguished. We omit the tedious details. However, it is clearthat D [ v, k, t, ι ] can be computed by simply trying all possible tuples ( t ′ , t ′′ , k ′ , k ′′ , i ′ , i ′′ , ι ′ , ι ′′ )where t ′ , t ′′ are the times at which v ′ , v ′′ are reached; k ′ , k ′′ are the number of delays occurringin T v ′ , T v ′′ ; i ′ , i ′′ are the number of delays occurring on the edges { v, v ′ } , { v, v ′′ } ; and ι ′ , ι ′′ describe whether v ′ , v ′′ are reached from a source in their respective subtrees at time t ′ and t ′′ , respectively. The number of such tuples and the time required to process each ofthem is clearly polynomial in t + k + |G| .After having computed all entries D [ v, k, t, ι ], the solution of the MinReachDelay instance ( G , k ) can be found as the value of R [ r, k ] := min t D [ r, k, t, false ] , where r is the root vertex of G .It remains to consider the case that the underlying graph of G is a disconnected forest.In this case simply apply the above algorithm to each connected component. Afterwards,determining the optimal way to split the overall budget between the connected componentscan be computed by a simple dynamic program. Define X [ i, k ] as the minimum weightreached in the first i trees if up to k time-edges are delayed and use the fact that X [1 , k ] = min k ′ ≤ k ( R [ r , k ′ ])and for all i > X [ i, k ] = min k ′ ≤ k ( R [ r i , k ′ ] + X [ i − , k − k ′ ]) , where r j is the root of the j th tree. ◀ While both problem variants,
MinReachDelete and
MinReachDelay , are polynomial-time solvable on forests and
W[1] -hard when parameterized by k , even if the lifetime τ = 2,their complexities diverge when we parameterize by the number r of reachable vertices. Here, MinReachDelete is W[1] -hard while for
MinReachDelay we found a fixed-parametertractable algorithm. This makes
MinReachDelay particularly interesting for applicationswhere the number of reachable vertices should be very small, e.g. when trying to contain thespread of dangerous diseases.On the practical side we want to point out that our algorithm for
MinReachDelay parameterized by r uses only linear space, and its search-tree-based approach makes it fitfor optimization techniques like further data reduction rules or pruning using lower bounds.Furthermore, our max-flow-based branching technique can be turned into a r -approximationfor MinReachDelay when we minimize the number r of reachable vertices by delaying k time-edges. Refining the presented technique towards better approximation guarantees seemsto be a promising research direction. Moreover, when focusing on specific applications, it isnatural to exploit application-dependent graph properties towards designing more efficientalgorithms. In particular: which well-motivated temporal graph classes beyond trees allowe.g. polynomial-time solvability of MinReachDelete or MinReachDelay ? Finally, fromthe viewpoint of parameterized complexity the parameters k and r are settled, but thelandscape of structural parameters is still waiting to be explored. EFERENCES 13
References Infection prevention and control and preparedness for covid-19 in healthcare settings.Technical report, European Centre for Disease Prevention and Control, 2020. Eleni C. Akrida, George B. Mertzios, and Paul G. Spirakis. The temporal explorer whoreturns to the base. In
Proceedings of the 11th International Conference on Algorithmsand Complexity (CIAC’19) , pages 13–24, 2019. doi:10.1007/978-3-030-17402-6_2. Eleni C. Akrida, George B. Mertzios, Sotiris E. Nikoletseas, Christoforos L. Raptopoulos,Paul G. Spirakis, and Viktor Zamaraev. How fast can we reach a target vertex in stochastictemporal graphs?
J. Comput. Syst. Sci. , 114:65–83, 2020. doi:10.1016/j.jcss.2020.05.005.An extended abstract appeared at ICALP’19. Kyriakos Axiotis and Dimitris Fotakis. On the size and the approximability of min-imum temporally connected subgraphs. In
Proceedings of the 43rd InternationalColloquium on Automata, Languages, and Programming (ICALP’16) , volume 55 of
LIPIcs , pages 149:1–149:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016.doi:10.4230/LIPIcs.ICALP.2016.149. Matthias Bentert, Anne-Sophie Himmel, André Nichterlein, and Rolf Niedermeier. Effi-cient computation of optimal temporal walks under waiting-time constraints.
AppliedNetwork Science , 5(1):73, 2020. doi:10.1007/s41109-020-00311-0. Kenneth A Berman. Vulnerability of scheduled networks and a generalizationof menger’s theorem.
Networks: An International Journal , 28(3):125–134, 1996.doi:10.1002/(SICI)1097-0037(199610)28:3<125::AID-NET1>3.0.CO;2-P. Hans L. Bodlaender and Tom C. van der Zanden. On exploring always-connectedtemporal graphs of small pathwidth.
Information Processing Letters , 142:68–71, 2019.doi:10.1016/j.ipl.2018.10.016. Dirk Brockmann and Dirk Helbing. The hidden geometry of complex, network-drivencontagion phenomena.
Science , 342(6164):1337–1342, 2013. doi:10.1126/science.1245200. Binh-Minh Bui-Xuan, Afonso Ferreira, and Aubin Jarry. Computing shortest, fastest,and foremost journeys in dynamic networks.
International Journal of Foundations ofComputer Science , 14(02):267–285, 2003. doi:10.1142/S0129054103001728. Sebastian Buß, Hendrik Molter, Rolf Niedermeier, and Maciej Rymar. Algorithmicaspects of temporal betweenness. In
Proceedings of the 26th ACM SIGKDD Conferenceon Knowledge Discovery and Data Mining (KDD’20) , pages 2084–2092. ACM, 2020.doi:10.1145/3394486.3403259. Arnaud Casteigts, Joseph G. Peters, and Jason Schoeters. Temporal cliques admitsparse spanners. In
Proceedings of the 46th International Colloquium on Automata, Lan-guages, and Programming (ICALP’19) , volume 132 of
LIPIcs , pages 134:1–134:14. SchlossDagstuhl - Leibniz-Zentrum für Informatik, 2019. doi:10.4230/LIPIcs.ICALP.2019.134. Arnaud Casteigts, Anne-Sophie Himmel, Hendrik Molter, and Philipp Zschoche. Find-ing temporal paths under waiting time constraints. In
Proceedings of the 31st In-ternational Symposium on Algorithms and Computation (ISAAC’20) , volume 181 of
LIPIcs , pages 30:1–30:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.doi:10.4230/LIPIcs.ISAAC.2020.30. Vittoria Colizza, Alain Barrat, Marc Barthélemy, and Alessandro Vespignani. Therole of the airline transportation network in the prediction and predictability of globalepidemics.
Proceedings of the National Academy of Sciences , 103(7):2015–2020, 2006.doi:10.1073/pnas.0510525103. Daryl J Daley and David G Kendall. Epidemics and rumours.
Nature , 204(4963):1118–1118, 1964. doi:10.1038/2041118a0. Argyrios Deligkas and Igor Potapov. Optimizing reachability sets in temporal graphsby delaying. In
Proceedings of the 34th AAAI Conference on Artificial Intelligence(AAAI’20) , pages 9810–9817. AAAI Press, 2020. doi:10.1609/aaai.v34i06.6533. Reinhard Diestel.
Graph Theory, 5th Edition , volume 173 of
Graduate Texts in Mathe-matics . Springer, 2016. doi:10.1007/978-3-662-53622-3. Edsger W Dijkstra et al. A note on two problems in connexion with graphs.
NumerischeMathematik , 1(1):269–271, 1959. doi:10.1007/BF01386390. Rodney G. Downey and Michael R. Fellows.
Fundamentals of Parameterized Complexity .Texts in Computer Science. Springer, 2013. ISBN 978-1-4471-5558-4. doi:10.1007/978-1-4471-5559-1. Ken TD Eames and Matt J Keeling. Contact tracing and disease control.
Proceedings ofthe Royal Society of London. Series B: Biological Sciences , 270(1533):2565–2571, 2003. Jessica Enright and Kitty Meeks. Deleting edges to restrict the size of an epidemic: anew application for treewidth.
Algorithmica , 80(6):1857–1889, 2018. doi:10.1007/s00453-017-0311-7. Jessica Enright, Kitty Meeks, George B. Mertzios, and Viktor Zamaraev. Deleting edgesto restrict the size of an epidemic in temporal networks.
Journal of Computer and SystemSciences , 2021. doi:10.1016/j.jcss.2021.01.007. To appear. Jessica Enright, Kitty Meeks, and Fiona Skerman. Assigning times to minimise reacha-bility in temporal graphs.
Journal of Computer and System Sciences , 115:169–186, 2021.doi:10.1016/j.jcss.2020.08.001. Thomas Erlebach and Jakob T. Spooner. Faster exploration of degree-bounded temporalgraphs. In
Proceedings of the 43rd International Symposium on Mathematical Founda-tions of Computer Science (MFCS’18) , volume 117 of
LIPIcs , pages 36:1–36:13. SchlossDagstuhl - Leibniz-Zentrum für Informatik, 2018. doi:10.4230/LIPIcs.MFCS.2018.36. Thomas Erlebach and Jakob T. Spooner. Non-strict temporal exploration. In
Proceedingsof the 27th International Colloquium on Structural Information and CommunicationComplexity (SIROCCO’20) , volume 12156 of
Lecture Notes in Computer Science , pages129–145. Springer, 2020. doi:10.1007/978-3-030-54921-3_8. Thomas Erlebach, Michael Hoffmann, and Frank Kammer. On temporal graph exploration.In
Proceedings of the 42nd International Colloquium on Automata, Languages, andProgramming (ICALP’15) , volume 9134 of
Lecture Notes in Computer Science , pages444–455. Springer, 2015. doi:10.1007/978-3-662-47672-7_36. Thomas Erlebach, Frank Kammer, Kelin Luo, Andrej Sajenko, and Jakob T. Spooner.Two moves per time step make a difference. In
Proceedings of the 46th InternationalColloquium on Automata, Languages, and Programming (ICALP’19) , volume 132 of
LIPIcs , pages 141:1–141:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.doi:10.4230/LIPIcs.ICALP.2019.141. Luca Ferretti, Chris Wymant, Michelle Kendall, Lele Zhao, Anel Nurtay, Lucie Abeler-Dörner, Michael Parker, David Bonsall, and Christophe Fraser. Quantifying SARS-CoV-2transmission suggests epidemic control with digital contact tracing.
Science , 2020.doi:10.1126/science.abb6936. Till Fluschnik, Hendrik Molter, Rolf Niedermeier, Malte Renken, and Philipp Zschoche.Temporal graph classes: A view through temporal separators.
Theoretical ComputerScience , 806:197–218, 2020. doi:10.1016/j.tcs.2019.03.031. L. R. Ford and D. R. Fulkerson. Maximal flow through a network.
Canadian Journal ofMathematics , 8:399–404, 1956. doi:10.4153/CJM-1956-045-5.
EFERENCES 15 L. R. Ford, Jr. and D. R. Fulkerson.
Flows in networks . Princeton University Press,Princeton, N.J., 1962. doi:10.1515/9781400875184. William Goffman and V Newill. Generalization of epidemic theory.
Nature , 204(4955):225–228, 1964. doi:10.1038/204225a0. Roman Haag, Hendrik Molter, Rolf Niedermeier, and Malte Renken. Feedback edge sets intemporal graphs. In
Proceedings of the 46th International Workshop on Graph-TheoreticConcepts in Computer Science (WG’20) , volume 12301 of
Lecture Notes in ComputerScience , pages 200–212. Springer, 2020. doi:10.1007/978-3-030-60440-0_16. John Hopcroft and Robert Tarjan. Algorithm 447: efficient algorithms for graph manipu-lation.
Communications of the ACM , 16(6):372–378, 1973. doi:10.1145/362248.362272. David Kempe, Jon Kleinberg, and Amit Kumar. Connectivity and inference problemsfor temporal networks.
Journal of Computer and System Sciences , 64(4):820–842, 2002.doi:10.1006/jcss.2002.1829. George B Mertzios, Othon Michail, and Paul G Spirakis. Temporal network opti-mization subject to connectivity constraints.
Algorithmica , 81(4):1416–1449, 2019.doi:10.1007/s00453-018-0478-6. Andrew Mitchell, David Bourn, J Mawdsley, William Wint, Richard Clifton-Hadley,and Marius Gilbert. Characteristics of cattle movements in britain–an analysisof records from the cattle tracing system.
Animal Science , 80(3):265–273, 2005.doi:10.1079/ASC50020265. Romualdo Pastor-Satorras and Alessandro Vespignani. Epidemic spreading in scale-freenetworks.
Physical review letters , 86(14):3200, 2001. doi:10.1103/PhysRevLett.86.3200. Omer Reingold. Undirected connectivity in log-space.
Journal of the ACM , 55(4):1–24,2008. doi:10.1145/1391289.1391291. Walter J Savitch. Relationships between nondeterministic and deterministic tape complex-ities.
Journal of Computer and System Sciences , 4(2):177–192, 1970. doi:10.1016/S0022-0000(70)80006-X. Huanhuan Wu, James Cheng, Yiping Ke, Silu Huang, Yuzhen Huang, and Hejun Wu.Efficient algorithms for temporal path computation.
IEEE Transactions on Knowledgeand Data Engineering , 28(11):2927–2942, 2016. doi:10.1109/TKDE.2016.2594065. Philipp Zschoche, Till Fluschnik, Hendrik Molter, and Rolf Niedermeier. The complexityof finding small separators in temporal graphs.