A Finite Time Combinatorial Algorithm for Instantaneous Dynamic Equilibrium Flows
AA Finite Time Combinatorial Algorithm forInstantaneous Dynamic Equilibrium Flows
Lukas Graf and Tobias Harks ∗Augsburg University, Institute of Mathematics, 86135 Augsburg{ lukas.graf,tobias.harks}@math.uni-augsburg.de
July 17, 2020
Instantaneous dynamic equilibrium (IDE) is a standard game-theoretic con-cept in dynamic traffic assignment in which individual flow particles myopicallyselect en route currently shortest paths towards their destination. We analyzeIDE within the Vickrey bottleneck model, where current travel times along apath consist of the physical travel times plus the sum of waiting times in all thequeues along a path. Although IDE have been studied for decades, several fun-damental questions regarding equilibrium computation and complexity are notwell understood. In particular, all existence results and computational methodsare based on fixed-point theorems and numerical discretization schemes and noexact finite time algorithm for equilibrium computation is known to date. As ourmain result we show that a natural extension algorithm needs only finitely manyphases to converge leading to the first finite time combinatorial algorithm com-puting an IDE. We complement this result by several hardness results showingthat computing IDE with natural properties is NP-hard.
Flows over time or dynamic flows are an important mathematical concept in network flowproblems with many real world applications such as dynamic traffic assignment, productionsystems and communication networks (e. g., the Internet). In such applications, flow parti-cles that are send over an edge require a certain amount of time to travel through each edgeand when routing decisions are being made, the dynamic flow propagation leads to latereffects in other parts of the network. A key characteristic of such applications, especiallyin traffic assignment, is that the network edges have a limited flow capacity which, whenexceeded, leads to congestion. This phenomenon can be captured by the the fluid queueingmodel model due to Vickrey [24]. The model is based on a directed graph G = ( V, E ), whereevery edge e has an associated physical transit time τ e ∈ R + and a maximal rate capacity ∗ The research of the authors was funded by the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) - HA 8041/1-1 and HA 8041/4-1. a r X i v : . [ c s . G T ] J u l wv w q e ( θ ) ν e τ e Figure 1:
An edge e = vw . As the inflow rate at node v exceeds the edge’s capacity, a queue forms atits tail. ν e ∈ R + . If flow enters an edge with higher rate than its capacity the excess particles startto form a queue at the edge’s tail, where they wait until they can be forwarded onto the edge(cf. Figure 1). Thus, the total travel time experienced by a single particle traversing an edge e is the sum of the time spent waiting in the queue of e and the physical transit time τ e .This physical flow model then needs to be enhanced with a behavioral model prescribingthe actions of flow particles. There are two main standard behavioral models in the trafficassignment literature known as dynamic equilibrium (DE) (cf. Ran and Boyce [20, § V-VI]and instantaneous dynamic equilibrium (IDE) ([20, § VII-IX]). Under DE, flow particles havecomplete information on the state of the network for all points in time (including the futureevolution of all flow particles) and based on this information travel along a shortest path.The full information assumption is usually justified by assuming that the game is playedrepeatedly and a DE is then an attractor of a learning process. The behavioral model of IDEis based on the idea that drivers are informed in real-time about the current traffic situationand, if beneficial, reroute instantaneously no matter how good or bad that route will be inhindsight. Thus, at every point in time and at every decision node, flow only enters thoseedges that lie on a currently shortest path towards the respective sink. This concept assumesfar less information (only the network-wide queue lengths which are continuously measured)and leads to a distributed dynamic using only present information that is readily availablevia real-time information. IDE has been proposed already in the late 80’s (cf. Boyce, Ranand LeBlanc [1, 21] and Friesz et al. [7]).A line of fairly recent works starting with Koch and Skutella [16] and Cominetti, Correaand Larré [3] derived very elegant combinatorial characterizations of DE for the fluid queueingmodel of Vickrey. They derived a complementarity description of DE flows via so-called thinflows with resetting which leads to an α -extension property stating that for any equilibriumup to time θ , there exists α > θ + α .An extension that is maximal with respect to α is called a phase in the construction of anequilibrium and the existence of equilibria on the whole R + then follows by a limit argumentover phases using Zorn’s lemma. In the same spirit, Graf, Harks and Sering [9] establisheda similar characterization for IDE flows and also derived an α -extension property.For both models (DE or IDE), it is an open question whether for constant inflow ratesand a finite time horizon, a finite number of phases suffices to construct an equilibrium, see[3, 16] and [9]. This problem remains even unresolved for single-source single-sink series-parallel graphs as explicitly mentioned by Kaiser [15]. Proving finiteness of the number ofphases would imply an exact finite time combinatorial algorithm. Such algorithm is not2nown to date neither for DE nor for IDE. More generally, the computational complexityof equilibrium computation is widely open.
In this paper, we study IDE flows and derive algorithmic and computational complexityresults. As our main result we settle the key question regarding finiteness of the α -extensionalgorithm.Theorem 3.7: For single-sink networks with piecewise constant inflow rates for a fi-nite time horizon, there is an α -extension algorithm computing an IDE after finitelymany extension phases. This implies the first finite time combinatorial exact algorithmcomputing IDE within the Vickrey model.The Vickrey model is arguably one of the most important traffic models (see Li, Huang andYang [18] for a research overview of the past 50 years), and yet, to the best of our knowledge,our algorithm is the first finite time algorithm computing a dynamic instantaneous equilib-rium in the continuous formulation. Let us remark that for dynamic equilibrium, such analgorithm is still not known to date.The proof of our result is based on the following ideas. We first consider the case of acyclic networks and use a topological order of vertices in order to schedule the extensionphases in the algorithm. The key argument for the finiteness of the number of extensionsphases is that for a single node v and any interval with linearly changing distance labelsof nodes closer to the sink and constant inflow rate into v this flow can be redistributed tothe outgoing edges in a finite number of phases of constant outflow rates from v . We showthis using the properties (derivatives) of suitable edge label functions for the outgoing edges(see Figure 4). The overall finiteness of the algorithm follows by induction over the nodesand time. We then generalize to arbitrary single-sink networks by considering dynamically changing topological orders depending on the current set of active edges. Finally, a closerinspection of the proofs also enables us to give an explicit upper bound on the number ofextension steps in the order of O (cid:18) P (cid:16) ∆ +1 (cid:17) L ·| E |· T ·| V | (cid:19) , where P is the number of constant phases of the network inflow rates, ∆ the maximumout degree in the network, T the termination time and L a bound on the derivatives of thedistance labels depending on the network inflow rates and the edge capacities of the givennetwork.We then turn to the computational complexity of IDE flows. Our first result here is alower bound on the output complexity of any algorithm. We construct an instance in whichthe unique IDE flow oscillates with a changing periodicity (see Figure 5). Algorithms for DE or IDE computation used in the transportation science literature are numerical , thatis, only approximate equilibrium flows are computed given a certain numerical precision, see the relatedwork for a more detailed comparison. T : Is there an IDE that terminates before T ?• Given some k ∈ N : Is there an IDE with of at most k phases?The proof is a a reduction from , wherein for any given -formula we construct anetwork (see Figure 11) with the following properties: If the -formula is satisfiable thereexists a quit simple IDE flow, where all flow particles travel on direct paths towards the sink.If, on the other hand, the -formula is unsatisfiable all IDE flows in the correspondingnetwork lead to congestions diverting a certain amount of flow into a separate part of thenetwork. Placing different gadgets in this part of networks then allows for the reduction tovarious decision problems involving IDE flows. The concept of flows over time was studied by Ford and Fulkerson [5]. Shortly after, Vick-rey [24] introduced a game-theoretic variant using a deterministic queueing model. Sincethen, dynamic equilibria have been studied extensively in the transportation science litera-ture, see Friesz et al. [7]. New interest in this model was raised after Koch and Skutella [16]gave a novel characterization of dynamic equilibria in terms of a family of static flows (thinflows). Cominetti, Correa and Larré [3] refined this characterization and Koch and Sering [23]incorporated spillbacks in the fluid queuing model. In a very recent work, Kaiser [15] showedthat the thin flows needed for the extension step in computing dynamic equilibria can bedetermined in polynomial time for series-parallel networks. The papers [3, 15] explicitlymention the problem of possible non-finiteness of the extension steps.In the traffic assignment literature, the concept of IDE was studied by several papers suchas Ran and Boyce [20, § VII-IX], Boyce, Ran and LeBlanc [1, 21], Friesz et al. [7]. Theseworks develop an optimal control-theoretic formulation and characterize instantaneous userequilibria by Pontryagin’s optimality conditions. For solving the control problem, Boyce,Ran and LeBlanc [1] proposed to discretize the model resulting in finite dimensional NLPwhose optimal solutions correspond to approximative IDE. While this approach only gives anapproximative equilibrium, there are further difficulties. The control-theoretic formulationis actually not compatible with the deterministic queueing model of Vickrey. In Boyce, Ranand LeBlanc [1], a differential equation per edge governing the cumulative edge flow (statevariable) is used. The right-hand side of the differential equation depends on the exit flowfunction which is assumed to be differentiable and strictly positive for any positive inflow.Both assumptions (positivity and differentiability) are not satisfied for the Vickrey model.For example, flow entering an empty edge needs a strictly positive time after which it leaves4he edge again, thus, violating the strict positiveness of the exit flow function. More impor-tantly, differentiability of the exit flow function is not guaranteed for the Vickrey queueingmodel. Non-differentiability (or equivalently discontinuity w.r.t. the state variable) is a well-known obstacle in the convergence analysis of a discretization of the Vickrey model, see forinstance Han et al. [10]. It is a priori not clear how to obtain convergence of a discretizationscheme for an arbitrary flow over time (disregading equilibrium properties) within the Vick-rey model. Indeed, Otsubo and Rapoport [19] report ”significant discrepancies” between thecontinuous and a discretized solution for the Vickrey model. To overcome the discontinu-ity issue, Han et al. [10] reformulated the model using a PDE formulation. They obtaineda discretized model whose limit points correspond to dynamic equilibria of the continuousmodel. The algorithm itself, however, is numerical in the sense that a precision is specifiedand within that precision an approximate equilibrium is computed. The overall discretiza-tion approach mentioned above stands in line with a class of numerical algorithms based onfixed point iterations computing approximate equilibrium flows within a certain numericalprecision, see Friesz and Han [6] for a recent survey.The long term behavior of dynamic equilibria with infinitely lasting constant inflow rateat a single source was studied by Cominetti, Correa and Olver [4]. They introduced theconcept of a steady state and showed that dynamic equilibria always reach a stable stateprovided that the inflow rate is at most the capacity of a minimal s - t cut.Ismaili [13, 14] considered a discrete version of DE and IDE, respectively. He investigatedthe computational complexity of computing best responses for DE showing that the best-response optimization problem is not approximable, and that deciding the existence of aNash equilibrium is complete for the second level of the polynomial hierarchy. In [14] asequential version of a discrete routing game is studied and PSPACE hardness results forcomputing an optimal routing strategy are derived. For further results regarding a discretepacket routing model, we refer to Cao et al. [2], Scarsini et al.[22], Harks et al. [11] andHoefer et al. [12]. Throughout this paper we always consider networks N = ( G, ( ν e ) e ∈ E , ( τ e ) e ∈ E , ( u v ) v ∈ V \{ t } , t )given by a directed graph G = ( V, E ), edge capacities ν e ∈ N ∗ , edge travel times τ e ∈ N ∗ ,and a single sink node t ∈ V which is reachable from anywhere in the graph. Every othernode v ∈ V \ { t } has a corresponding (network) inflow rate u v : R ≥ → Q ≥ indicating forevery time θ ∈ R ≥ the rate u v ( θ ) at which the infinitesimal small agents enter the networkat node v and start traveling through the graph until they leave the network at the commonsink node t . We will assume that these network inflow rates are right-constant step functionswith bounded support and finitely many, rational jump points and denote by P ∈ N ∗ thetotal number of jump points for all network inflow rates.Then, a flow over time is a tuple f = ( f + , f − ) where f + , f − : E × R ≥ → R ≥ areintegrable functions. For any edge e ∈ E and time θ ∈ R ≥ the value f + e ( θ ) describes the (edge) inflow rate into e at time θ and f − e ( θ ) is the (edge) outflow rate from e at time θ . For We restrict ourselves to integer travel times and edge capacities in order to make the statements and proofscleaner. However, all results can be easily applied to instances with rational travel times and capacitiesby simply rescaling the instance appropriately. Note, however, that by upscaling edge travel times shorterthen 1 all explicit bounds will scale accordingly. f we define the cumulative (edge) in- and outflow rates F + and F − by F + e ( θ ) := Z θ f + e ( ζ ) dζ and F − e ( θ ) := Z θ f − e ( ζ ) dζ, respectively. The queue length of edge e at time θ is then defined as q e ( θ ) := F + e ( θ ) − F − e ( θ + τ e ) . (1)Such a flow f is called a feasible flow for the given set of inflow rates u v : R ≥ → R ≥ , if itsatisfies the following constraints (2) to (5). The flow conservation constraints are modeledfor all nodes v = t as X e ∈ δ + v f + e ( θ ) − X e ∈ δ − v f − e ( θ ) = u v ( θ ) for all θ ∈ R ≥ , (2)where δ + v := { vu ∈ E } and δ − v := { uv ∈ E } are the sets of outgoing edges from v andincoming edges into v , respectively. For the sink node t we require X e ∈ δ + t f + e ( θ ) − X e ∈ δ − t f − e ( θ ) ≤ e ∈ E we always assume f − e ( θ ) = 0 f.a. θ < τ e . (4)Finally we assume that the queues operate at capacity which can be modeled by f − e ( θ + τ e ) = ( ν e , if q e ( θ ) > { f + e ( θ ) , ν e } , if q e ( θ ) ≤ e ∈ E, θ ∈ R ≥ . (5)Following the definition in [9] we call a feasible flow an IDE flow if whenever a particlearrives at a node v = t , it can only ever enter an edge that is the first edge on a currentlyshortest v - t path. In order to formally describe this property we first define the current or instantaneous travel time of an edge e at θ by c e ( θ ) := τ e + q e ( θ ) ν e . (6)We then define time dependent node labels ‘ v ( θ ) corresponding to current shortest pathdistances from v to the sink t . For v ∈ V and θ ∈ R ≥ , define ‘ v ( θ ) := , for v = t min e = vw ∈ E { ‘ w ( θ ) + c e ( θ ) } , else. (7)We say that an edge e = vw is active at time θ , if ‘ v ( θ ) = ‘ w ( θ ) + c e ( θ ) and denote the setof active edges by E θ ⊆ E . 6 efinition 2.1. A feasible flow over time f is an instantaneous dynamic equilibrium (IDE) ,if for all θ ∈ R ≥ and e ∈ E it satisfies f + e ( θ ) > ⇒ e ∈ E θ . (8)In [9, Section 3] the existence of IDE flows in single-sink networks is proven by the followingalmost constructive argument: An IDE flow up to some time θ can always be extended forsome non-trivial additional time interval on a node by node bases (starting with the nodesclosest to the sink t ). The existence of IDE flows for the whole R ≥ then follows by applyingZorn’s lemma or a limit argument. This leads to a natural algorithm for computing IDEflows in single-sink networks, which we make explicit here as Algorithm 1. Algorithm 1:
IDE-Construction Algorithm from [9]
Input:
A single-sink network N with piecewise constant network inflow rates Output:
An IDE flow f in N Let f be an IDE flow up to time θ ← while not all flow particles have reached the sink t do Let t = v < v < · · · < v n be a topological order w.r.t. G [ E θ ] Choose the largest α > u v and f − e are constant over ( θ, θ + α ) for i = 1 , . . . , n do Compute the (constant) inflow into v i during [ θ, θ + α i − ] and find a constantdistribution to the outgoing edges from v i such that only active edges are used Determine the largest α i ≤ α i − such that the set of active edges does notchange during ( θ, θ + α i ) end for Extend the flow f up to time θ + α n and set θ ← θ + α n end while The extension at a single node can be easily computed in polynomial time using a simplewater filling procedure (see [9, Algorithm 1 (electronic supplementary material)]). It is,however, not obvious whether a finite number of extension phases suffices to construct anIDE flow for all of R ≥ . Since IDE flows always have a finite termination time in single-sinknetworks ([9, Theorem 4.6]) it is actually enough to extend the flow for some finite timehorizon (in an upcoming paper [8] we will even provide a way to explicitly compute such atime horizon). This leaves the possibility of some Zeno-type behavior of the lengths of theextension phases as possible reason for Algorithm 1 not to terminate within finite time. Thus,the question of whether IDE flows can actually be computed was left as an open question in[9]. A first partial answer was found in [17], where finite termination was shown for graphsobtained by series composition of parallel edges. In the following section we give a full answerby showing that the α -extension algorithm terminates for all single-sink networks. In this chapter we will show that IDE flows can be constructed in finite time using Algorithm 1or slight variations thereof. We will first show this only for acyclic networks since there wecan use a single constant order of the nodes for the whole construction. Building on that,7e will then prove the general case by showing that we can always compute IDE flows whilechanging the node order only finitely many times.
For an acyclic network we can use a single static topological order of the nodes with respect tothe whole graph instead of determining a new topological order with respect to the currentlyactive edges for every extension phase. This allows us to rearrange the order of the extensionsteps, leading to Algorithm 2. Note however, that this does not change the the result of thealgorithm.
Algorithm 2:
IDE-Construction Algorithm for acyclic networks
Input:
An acyclic single-sink network N with piecewise constant network inflow rates Output:
An IDE flow f in N Choose T ∈ N large enough such that all IDE flows in N terminate before T Let f be an IDE flow up to time 0 and t = v < · · · < v n a topological order for θ = 0 , . . . , T − do for i = 1 , . . . , n do Compute the inflow function into node v i over the interval [ θ, θ + 1] Distribute this inflow for the whole interval [ θ, θ + 1] to active edges in δ + v i usingmaximal sub-phases of constant flow distribution end for end for Observation . For acyclic networks both variants of the general algorithm (Algorithm 1and Algorithm 2) construct the same IDE flow provided that they use the same tie-breakingrules. Thus, showing that one of them terminates in finite time, also proves the same for theother variant.Using [9, Algorithm 1] we can compute an IDE compliant flow distribution with constantedge-inflow rates at a node v i for any interval wherein the inflow into node v i is constant, thelabels on all the nodes w with v i w ∈ δ + v i change linearly and the set of active edges leaving v i remains constant. We call such an interval a sub-phase . Thus, it suffices to show that inline 6 we can always subdivide the extension interval [ θ, θ + 1] into a finite number of suchsub-phases. We will show this by induction over θ ∈ [ T ] and i ∈ [ n ] using the followingkey-lemma: Lemma 3.2.
Let N be a single-sink network on an acyclic graph with some fixed topologicalorder on the nodes, v some node in N and θ < θ two times. If f is a flow over time in N such that• f is an IDE flow up to time θ for all nodes,• f is an IDE flow up to time θ for all nodes closer to the sink t than v ,• the inflow rate into node v over the interval [ θ , θ ] is constant and• the labels at the nodes reachable via direct edges from v change linearly over this timeinterval,then we can extend f to an IDE flow up to time θ at v using a finite number of sub-phases. w w w w v θ θ ‘ w θ θ ‘ w θ θ ‘ w θ θ ‘ w Figure 2:
The situation in Lemma 3.2: Wehave an acyclic graph with sometopological order on the nodes (herefrom left to right) and an IDE flowup to some time θ for all nodescloser to the sink t than v andup to some earlier time θ for v and all nodes further away than v from t . Additionally, over theinterval [ θ , θ ] the edges leadinginto v have a constant outflow rate(and a physical travel time of atleast θ − θ ) and the nodes w i all have affine label functions ‘ w i .The edges vw i start with some cur-rent queue lengths q vw i ( θ ) ≥ . Proof.
We want to show that a finite number of maximal constant extensions of the flowat node v using the water filling algorithm [9, Algorithm 1] is enough to extend the givenflow for the whole interval [ θ , θ ] at node v . So, let f be the flow after an, a priori, infinitenumber of extension steps getting us to an IDE flow up to some ˆ θ ∈ ( θ , θ ] at node v .Let δ + v = { vw , . . . , vw p } be the set of outgoing edges from v . Then by the lemma’sassumption the label functions ‘ w i : [ θ , θ ] → R ≥ are affine functions and, since we extended f at node v up to ˆ θ , the queue length functions q vw i are well defined on the interval [ θ , ˆ θ ].Thus, for all i ∈ [ p ] we can define functions h i : [ θ , ˆ θ ] → R ≥ , θ τ vw i + q vw i ( θ ) ν vw i + ‘ w i ( θ )such that h i ( θ ) is the shortest current travel time to the sink t for a particle entering edge vw i at time θ and for any edge vw i ∈ δ + v and any time θ ∈ [ θ , ˆ θ ] we have vw i ∈ E θ ⇐⇒ h i ( θ ) = min { h j ( θ ) | j ∈ [ p ] } = ‘ v ( θ ) . (9)We start by showing two key-properties of these functions, which are also visualized inFigures 3 and 4: Claim 1.
If an edge vw i is inactive during some interval [ a, b ] ⊆ [ θ , ˆ θ ] the graph of h i isconvex on this interval. Claim 2.
A new sub-phase starts at time θ if and only if one of of the following two eventshappens:• An edge vw i becomes newly active at time θ , i.e. h i ( θ ) = ‘ v ( θ ) and the left side derivativeof h i is strictly smaller than that of ‘ v at time θ . In this case the right-side derivativeof ‘ v at θ is at most its left-side derivative and at least the left-side derivative of h i , i.e. ∂ − h i ( θ ) ≤ ∂ + ‘ v ( θ ) ≤ ∂ − ‘ v ( θ ) .• The queue of an active edge vw i runs empty at time θ , i.e. h i ( θ ) = ‘ v ( θ ) , q vw i ( θ ) = 0 and the left side derivative of q vw i at time θ is strictly negative. In this case the left-side derivative of ‘ v at θ is at most its right-side derivative and strictly smaller than theright-side derivative of h i , i.e. ∂ − ‘ v ( θ ) ≤ ∂ + ‘ v ( θ ) and ∂ − ‘ v ( θ ) < ∂ + h i ( θ ) . w w w v Start of sub-phase 1 w w w w v Start of sub-phase 2 w w w w v Start of sub-phase 3 w w w w v Start of sub-phase 4 w w w w v Start of sub-phase 5 w w w w v End of sub-phase 5
Figure 3:
A possible flow distribution from the node v in five sub-phases for the situation depicted inFigure 2. The corresponding functions h i are depicted in Figure 4 ‘ v sub-phase 1 sub-phase 2 sub-phase 3 s.-ph. 4 sub-phase 5 h h h h θ θ Figure 4:
The functions h i corresponding to the five sub-phase flow distribution depicted in Figure 3for the situation depicted in Figure 2. The second, third and fifth sub-phase start becausean edge becomes newly active (edges vw , vw and vw again, respectively). The fourthsub-phase starts because the queue on the active edge vw runs empty. By Claim 2 theseare the only two possible events which can trigger the beginning of a new sub-phase. Edge vw is inactive for the whole time interval and – as stated in Claim 1 – has a convex graph.The bold gray line marks the graph of the function ‘ v . Also, note the slope changes of thefunction h i and ‘ v at the beginning of the sub-phases in accordance with Claim 2. roof of Claim 1. By the lemma’s assumption ‘ w i is linear on the whole interval [ θ , θ ]and for an inactive edge e its queue length function consists of at most two linear sections:One where the queue depletes at a constant rate of − ν e and one where it remains constant0. Thus, h i is convex as the sum of two convex functions for any interval, where vw i isinactive. (cid:4) Proof of Claim 2.
The fact that the beginning of a new sub-phase can only be triggered inthe two ways described in the claim is a direct consequence of (9), the definition of h i andthe fact that ‘ w i are linear on the given interval.In the case that an edge vw i becomes newly active we have h i ( θ ) = ‘ v ( θ ) = h j ( θ ) for allpreviously active edges vw j and h i ( θ + ε ) = ‘ v ( θ + ε ) ≤ h j ( θ + ε ) for ε > ∂ + h i ( θ ) = ∂ + ‘ v ( θ ) ≤ ∂ + h j ( θ ) (10)for the right-side derivatives. Since, as an inactive edge, vw i had no inflow before θ we have ∂ − h i ( θ ) ≤ ∂ + h i ( θ ). Similarly, as the previously active edges vw j have smaller or at mostequal inflow rates after θ compared to before, we get ∂ + h j ( θ ) ≤ ∂ − h j ( θ ) = ∂ − ‘ v ( θ ). Togetherwith (10) this yields the first part of the claim.In the case that the queue of an active edge vw i runs empty at time θ the set of activeedges after θ are a subset of the active edges from before θ . The inflow into these edges vw j after θ can only be larger or at least equal to the inflow before. Thus, we get ∂ + ‘ v ( θ ) = ∂ + h j ( θ ) ≥ ∂ − h j ( θ ) and, as all these edge were active before θ as well, ∂ − h j ( θ ) = ∂ − ‘ v ( θ )holds as well. Finally, since vw i had a depleting queue before θ and has a stagnant or evengrowing queue afterwards we have ∂ + h i ( θ ) > ∂ − h i ( θ ) = ∂ − ‘ v ( θ ) (as vw i was active before θ ), which concludes the prove of the second part of the claim. (cid:4) We also need the following observation which is an immediate consequence of the way thewater filling algorithm [9, Algorithm 1] determines the flow distribution combined with thelemma’s assumption that all label functions ‘ w i have constant derivative during the interval[ θ , θ ]. Claim 3.
There are uniquely defined numbers ‘ I,J for all subsets J ⊆ I ⊆ [ p ] such that ‘ v ( θ ) = ‘ I,J within all sub-phases, where { vw i | i ∈ I } is the set of active edges in δ + v and { vw i | i ∈ J } is the subset of such active edges that also have a non-zero queue during thissub-phase. (cid:4) Using these properties we can now first show a claim which implies that the smallest ‘ I,J can only be the derivative of ‘ v for a finite number of intervals. Inductively the samethen holds for all of the finitely many ‘ I,J . The proof of the lemma then concludes by theobservation that an interval with constant derivative of ‘ v can contain only finitely manysub-phases. Claim 4.
Let [ a , b ] , [ a , b ] ⊆ [ θ , ˆ θ ] be two disjoint maximal intervals with constant ‘ v ( θ ) = c for some c ∈ { ‘ I,J | J ⊆ I ⊆ [ p ] } . If b < a and ‘ v ( θ ) ≥ c for all θ ∈ ( b , a ) where thederivative exists, then there exists an edge vw i such that1. the first sub-phase of the interval [ a , b ] begins because vw i becomes newly active attime θ = a and2. this edge is not active between a and a .In particular, the first sub-phase of [ a , b ] is not triggered by vw i becoming newly active. roof of Claim 4. The first part of the claim follows directly from Claim 2 and the fact that ∂ − ‘ v ( a ) > c = ∂ + ‘ v ( a ). For the second part let ˜ θ < a be the last time before the secondinterval, where vw i was active. By Claim 2 we know then that ∂ − h i ( a ) ≤ ∂ + ‘ v ( a ) = c andby Claim 1 this extends to h i ( θ ) ≤ c for the whole interval [˜ θ, a ]. At the same time we have ‘ v ( θ ) ≥ c for all of [ a , a ] and ‘ v ( θ ) > c for at least some proper subinterval all of [ b , a ],since the intervals [ a , b ] and [ a , b ] were chosen to be maximal. Combining these two factswith ‘ v ( θ ) = h i ( θ ) immediately shows ‘ v ( θ ) < h i ( θ ) for all θ ∈ [˜ θ, a ] ∩ [ a , a ]. This gives˜ θ < a and, thus, vw i is inactive for all of [ a , a ], which proves the claim. (cid:4) This claim directly implies that the lowest derivative of ‘ v during [ θ , ˆ θ ] only appears ina finite number of intervals, as each of these intervals has to start with a different edgebecoming newly active. But the,n iteratively applying this claim for the intervals betweenthese intervals shows that any derivative of ‘ v can only appear in a finite number of intervals.Since, by Claim 3, ‘ v can only attain a finite number of values, this implies that [ θ , ˆ θ ] consistsof a finite number of intervals with constant derivative of ‘ v . Claim 5.
Let [ a, b ] ⊆ [ θ , ˆ θ ] be an interval during which ‘ v is constant. Then [ a, b ] containsat most p sub-phases.Proof of Claim 5. By Claim 1 an edge that changes from active to inactive during the interval[ a, b ] will remain inactive for the rest of this interval. Thus, at most p sub-phases can startbecause an edge becomes newly active. By Claim 2 if a sub-phase begins because the queueon an active edge e runs empty, this edge will become inactive. Thus, at most p sub-phasesstart because the queue of an active edge runs empty. Since, also by Claim 2, these are theonly two ways to start a new sub-phase, we can have at most 2 p sub-phases start during[ a, b ]. (cid:4) Combining Claims 4 and 5 we can finally conclude that [ θ , ˆ θ ] only contains a finite numberof sub-phases and, thus, we can achieve ˆ θ = θ with a finite number of extensions.With this lemma the proof of the following theorem is straightforward. Theorem 3.3.
For any acyclic single-sink network with piecewise constant network-inflowrates an IDE flow can be constructed in finite time using Algorithm 2.Proof.
We show this by induction over θ ∈ [ T ] and i ∈ [ n ], i.e. we can assume that thecurrently constructed flow f is an IDE flow up to time θ for all nodes v j , j ≥ i and up totime θ + 1 for all nodes v j , j < i with only a finite number of (sub-)phases. In particular,this means that we can partition the interval [ θ, θ + 1] into a finite number of subintervalssuch that within each such subinterval there is a constant inflow rate into node v i and thelabels at all the vertices w with v i w ∈ δ + v i change linearly. Then, by Lemma 3.2, we candistribute the flow at node v i to the outgoing edges using a finite number of sub-phases foreach of these subintervals. Note that, aside from the queue lengths on the edges leaving v i ,the so distributed flow has no influence on the flow distribution in later subintervals and,in particular, does not influence the partition into subintervals or the flow distribution atnodes closer to t than v i . Thus, we can distribute the outflow from v i for the whole interval[ θ, θ + 1] using only a finite number of sub-phases.Closer inspection of the proofs above also allows us to derive a rough but explicit boundon the number of sub-phases the constructed IDE flow can have.12 roposition 3.4. For any acyclic single-sink network with piecewise constant network-inflowrates the number of sub-phases of any IDE flow constructed by Algorithm 2 is bounded by O (cid:18) P (cid:16) ∆ +1 (cid:17) T | V | (cid:19) , where ∆ := max { (cid:12)(cid:12) δ + v (cid:12)(cid:12) | v ∈ V } the maximum out-degree in the given network and P is thenumber of intervals with constant network inflow rates.Proof. First, we look at an interval [ θ , θ ] and a single node v as in Lemma 3.2. Here wecan use Claim 4 to bound the number of intervals of constant derivative of ‘ v by (cid:16)(cid:12)(cid:12)(cid:12) δ + v (cid:12)(cid:12)(cid:12) + 1 (cid:17) | { ( I,J ) | J ⊆ I ⊆ [ | δ + v | ] } | ≤ (cid:16)(cid:12)(cid:12)(cid:12) δ + v (cid:12)(cid:12)(cid:12) + 1 (cid:17) | δ + v | , each of them containing at most 2 (cid:12)(cid:12) δ + v (cid:12)(cid:12) sub-phases by Claim 5. Together this shows that anysuch interval will be subdivided into at most 2(∆ + 1) ∆ +1 sub-phases. Thus, whenever weexecute line 6 of Algorithm 2 every currently existing sub-phase may be subdivided furtherinto at most 2( | ∆ | + 1) ∆ +1 sub-phases. Thus, for every θ ∈ [ T ] the number of sub-phasescan be multiplied by at most Q v ∈ V (cid:16) | ∆ | + 1) ∆ +1 (cid:17) in total during the extension over theinterval [ θ, θ + 1]. Combining this with the at most P sub-phases triggered by changingnetwork inflow rates results in the bound of O (cid:18) P (cid:16) ∆ +1 (cid:17) T | V | (cid:19) . We now want to extend this result to general single-sink networks, i.e. we want to show thatAlgorithm 1 terminates within finite time not only for acyclic graphs, but for all graphs. Wefirst note that the requirement for input-graphs of Algorithm 2 to be acyclic is somewhat tostrong. It is actually enough to have some (static) order on the nodes such that it is always atopological order with respect to the active edges of the network. That is, for a general single-sink network we can still apply Algorithm 2 to determine an IDE-extension with finitely manyphases for any interval during which we have such a static node ordering. Thus, Algorithm 1can also use finitely many extension phases for each interval with such a static ordering.This observation gives rise to Algorithm 3, another slight variant of Algorithm 1.We will prove that this algorithm does indeed construct an IDE flow for arbitrary single-sink networks within finite time by first showing that this algorithm is a special case of theoriginal algorithm. Thus, it is correct and uses only a finite number of phases for any intervalin which the topological order does not change. We can then conclude the proof by showingthat it is enough to change the topological order a finite number of times for any given timehorizon.
Lemma 3.5.
Algorithm 3 is a special case of Algorithm 1. In particular it is correct.Proof.
First note, that ˜ E is clearly always acyclic (except in lines 12 and 13) which guaranteesthat we can always find a topological order with respect to ˜ E . We now only need to showthat such an ordering is also a topological order with respect to the active edges, i.e. thatfor any time θ we have E θ ⊆ ˜ E . For this we will use the following observation13 lgorithm 3: IDE-Construction Algorithm for general single-sink networks
Input:
A single-sink network N with piecewise constant network inflow rates Output:
An IDE flow f in N Choose T large enough such that all IDE flows in N terminate before T Let f be an IDE flow up to time θ ← E ← E Determine a topological order t = v < v < · · · < v n w.r.t. the edges in ˜ E while θ < T do Choose the largest α > u v and f − e are constant over ( θ, θ + α ) for i = 1 , . . . , n do Compute the (constant) inflow into v i during [ θ, θ + α i − ] and find a constantdistribution to the outgoing edges from v i such that only active edges are used Determine the largest α i ≤ α i − such that the set of active edges does notchange during ( θ, θ + α i ) end for Extend the flow f up to time θ + α n and set θ ← θ + α n if E θ \ ˜ E = ∅ then Define ˜ E ← ˜ E ∪ E θ . For each cycle in ˜ E remove the edge e = xy with the largest value ‘ y ( θ ) − ‘ x ( θ )of all edges on this cycle Determine a topological order t = v < v < · · · < v n w.r.t. the edges in ˜ E end if end whileClaim 6. Any edge xy removed from ˜ E in line 13 of Algorithm 3 satisfies ‘ x ( θ ) < ‘ y ( θ ) .Proof. Let C ⊆ ˜ E be a cycle the removed edge xy was part of. Since ˜ E was acyclic before weadded the newly active edges in line 12, this cycle also has to contain some currently activeedge vw . This gives us X e = uz ∈ C \{ vw } ( ‘ z ( θ ) − ‘ u ( θ )) = X e = uz ∈ C ( ‘ z ( θ ) − ‘ u ( θ )) − ( ‘ w ( θ ) − ‘ v ( θ ))= 0 − ‘ w ( θ ) + (cid:16) ‘ w ( θ ) + τ vw + q vw ( θ ) ν vw (cid:17) = τ vw + q vw ( θ ) ν vw ≥ . Thus, C contains at least one edge uz with ‘ z ( θ ) − ‘ u ( θ ) > xy . (cid:4) The claim then immediately implies that in line 13 we only remove inactive edges andthat, afterwards, we still have E θ ⊆ ˜ E . Lemma 3.6.
For any single-sink network there exists some constant
C > such that forany time interval of length C the set ˜ E changes at most | E | times during this interval inAlgorithm 3. roof. The proof of this lemma mainly rest on the following claim stating that for any fixednetwork we can bound the slope of the node labels of any flow in this network by someconstant.
Claim 7.
For any given network there exists some constant
L > such that for all flows,all nodes v and all times θ we have | ‘ v ( θ ) | ≤ L .Proof. First note that for any node v we can bound the maximal inflow rate into this nodeby some constant L v as follows: X e ∈ δ − v f − e ( θ ) + u v ( θ ) (5) ≤ X e ∈ δ − v ν e + max { u v ( θ ) | θ ∈ R ≥ } =: L v . Using flow conservation (2) this, in turn, allows us to bound the inflow rates into all edges e ∈ δ + v and, thus, the rate at which the queue length and the current travel time on theseedges can change: − ≤ c e ( θ ) (1),(6) ≤ f + e ( θ ) ν e ≤ L v ν e =: L e . Finally, setting L := P e ∈ E max { , L e } proves the claim, as for all nodes v and times θ wethen have (cid:12)(cid:12) ‘ v ( θ ) (cid:12)(cid:12) ≤ X e ∈ E (cid:12)(cid:12) c e ( θ ) (cid:12)(cid:12) ≤ X e ∈ E L e = L. (cid:4) Now, from Claim 6 we know that, when we remove an edge xy from ˜ E at time θ we musthave ‘ x ( θ ) < ‘ y ( θ ). But at time θ where we last added this edge to ˜ E it must have beenactive (since we only ever add active edges to ˜ E ) and, thus, we had ‘ x ( θ ) = ‘ y ( θ ) + c xy ( θ ) ≥ ‘ y ( θ ) + 1. Therefore, the difference between the labels at x and y has changed at least by1 between θ and θ . Claim 7 then directly implies θ − θ ≥ L . So, for any time interval oflength at most L each edge can be added at most once to ˜ E . Since ˜ E only ever changeswhen we add at least one new edge to it, setting C := L proves the lemma. Theorem 3.7.
For any single-sink network with piecewise constant network-inflow rates anIDE flow can be constructed in finite time using Algorithm 3.Proof.
By Lemma 3.5 Algorithm 3 is a special case of Algorithm 1. Thus, for any intervalwith static ˜ E it produces the same flow as Algorithm 2. In particular, by Theorem 3.3, for anysuch interval the constructed flow consists of finitely many phases. Finally, Lemma 3.6 showsthat the whole relevant interval [0 , T ] can be partitioned into a finite number of intervalswith static set ˜ E . Consequently, Algorithm 3 constructs an IDE flow with finitely manyphases and, thus, terminates within finite time.As in the acyclic case we can again also extract an explicit upper bound on the numberof sub-phases. Proposition 3.8.
For any single-sink network with piecewise constant network inflow ratesthe number of sub-phases of any IDE flow constructed by Algorithm 3 is bounded by O (cid:18) P (cid:16) ∆ +1 (cid:17) L ·| E |· T ·| V | (cid:19) , here, again, ∆ := max { (cid:12)(cid:12) δ + v (cid:12)(cid:12) | v ∈ V } is the maximum out-degree in the given network, P is the number of intervals with constant network inflow rates and L the bound on the slopesof the label functions from Claim 7.Proof. For any time interval with fixed node order Algorithm 3 is equivalent to Algorithm 2and, thus, the bound from Proposition 3.4 applies. Also note, that in Algorithm 2 we couldchange the node order after every unit time step without any impact on correctness orthe bound on the number of sub-phases (as long as we always choose an order which is atopological order with respect to the active edges). As, by Lemma 3.6, the node order inAlgorithm 3 changes at most 2 L · | E | times during any unit time interval, replacing T by2 L · | E | · T in the bound for Algorithm 2 yields a valid bound for the number of sub-phasesof Algorithm 3. Remark . If presented with rational input data (i.e. rational capacities, node inflow rate,current queue lengths, current distance labels and slopes of distance labels of neighbouringnodes) the water filling procedure from [9] again produces a rational output (i.e. rationaledge inflow rates and rational maximal extension length α ). Thus, Algorithm 3 can beimplemented as an exact combinatorial algorithm. While Theorem 3.7 shows that IDE flows can be constructed in finite time, the boundprovided in Proposition 3.8 is clearly superpolynomial. We now want to show that in somesense this is to be expected. Namely, we first look at the output complexity of any suchalgorithm, i.e. how complex the structure of IDE flows can be. Then we show that manynatural decision problems involving IDE flows are actually NP-hard.
As before we call an open interval ( a, b ) ⊆ R ≥ a phase of a feasible flow f , if all edge in- andoutflow rates remain constant during this interval. Then the output of any algorithm com-puting feasible flows certainly has to contain in some way a set of phases and correspondingin- and outflow rates. In particular, the number of phases of a flow is a lower bound forthe runtime of any algorithm determining that flow. This observation will allow us to givean exponential lower bound for the output complexity and therefore also for the worst caseruntime of any algorithm determining IDE flows. This remains true even if we only look atacyclic graphs and we allow for our algorithm to recognize periodic behavior and abbreviatethe output accordingly. Theorem 4.1.
The worst case output complexity of calculating IDE flows is not polynomialin the encoding size of the instance, even if we are allowed to use periodicity to reduce theencoding size of the determined flow. This is true even for series parallel graphs.Proof.
For any given U ∈ N ∗ consider the network pictured in Figure 5 with a constantinflow rate of 2 at s over the interval [0 , U ]. This network can clearly be encoded in O (log U )space. The unique (up to changes on a set of measure zero) IDE flow is displayed up totime θ = 6 . θ = U , it exhibits Ω( U ) distinct phases. This provesthe theorem. 16 vw t (1 , ,
2) (1 , τ s t , ν s t ) = (2 , θ = 0: u ≡ s vw t θ = 1: u ≡ q vt (2) = 1 s vw t θ = 2: u ≡ q vt (3) = 2 s vw t θ = 3: u ≡ q vt (3 .
5) = 1 . q wt (3 .
5) = 0 . s vw t θ = 3 . u ≡ q vt (4 .
5) = 0 . q wt (4 .
5) = 1 . s vw t θ = 4 . u ≡ q vt (5 .
5) = 1 . q wt (5 .
5) = 0 . s vw t θ = 5 . u ≡ q vt (6 .
5) = 2 . s vw t θ = 6 . u ≡ Figure 5:
A network (top left picture) where constant inflow rate of over [0 , U ] leads to an IDEflow with Ω( U ) different phases. The following pictures show the first states of the network,which are described in general in Table 1. θ = f + vt ( θ ) f + wt ( θ ) q vt ( θ ) q wt ( θ ) f + sv ( θ ) f + st ( θ )4 k + 2 − k − − − k & − − k % k + 2 − k − − k % − − k & k + 2 − k + 1 2 0 2 − − k % − − k & k + 2 − k + 2 0 2 3 − − k & % Table 1:
Phases of the (unique) IDE flow in the instance of Figure 5. For all k ∈ N the tableincludes the (constant) inflow rates into all edges on the intervals (4 k + 2 − k − , k + 2 − k ) , (4 k + 2 − k , k + 2 − k + 1) , (4 k + 2 − k + 1 , k + 2 − k + 2) and (4 k + 2 − k + 2 , k + 2 − ( k +1) + 3) aswell as the queue lengths on the edges vt and wt at the beginning of these intervals and therate of change for the queue lengths over the following interval ( % stands for an increase atrate , & for a decrease at rate − ). Remark . In [4, Example 2] Cominetti et al. sketch an instance with O ( d ) verticeswhere a dynamic equilibrium flow exhibits an exponential number of phases (of order Ω(2 d ))before it reaches a stable state. From this, we can conclude that a similar result holdsfor dynamic equilibria. However, since the details of this example are not yet publishedwe do not know whether the phases do exhibit periodic behavior and how complicated theconstructed instance is.The network constructed in the above proof can also be used to gain some insights intothe long term behavior of IDE flows, i.e. how such flows behave if the inflow rates continueforever. In order to analyze this long term behavior of dynamic equilibrium flows Cominettiet al. define in [4, Section 3] the concept of a steady state: Definition 4.3.
A feasible flow f with forever lasting constant inflow rate reaches a steadystate if there exists a time ˜ θ such that after this time all queue lengths stay the same foreveri.e. q e ( θ ) = q e (˜ θ ) f.a. e ∈ E, θ ≥ ˜ θ. s - t cut is alsoa sufficient condition for any dynamic equilibrium in such a network to eventually reach asteady state ([4, Theorem 4]). We will show that this is not true for IDE flows - even if weconsider a weaker variant of steady states: Definition 4.4.
A feasible flow f reaches a periodic state if there exists a time ˜ θ and aperiodicity p ∈ R ≥ such that after time ˜ θ all queue lengths change in a periodic manner,i.e. q e ( θ + kp ) = q e ( θ ) f.a. e ∈ E, θ ≥ ˜ θ, k ∈ N ∗ . Note that, in particular, every flow reaching a stable state also reaches a periodic state(with arbitrary periodicity).
Theorem 4.5.
There exists a series parallel network with a forever lasting constant inflowrate u at a single node s , satisfying u ≤ P e ∈ δ + X ν e for all s - t cuts X , where no IDE flow everreaches a periodic state.Proof. Consider the network constructed in the proof of Theorem 4.1, i.e the one picturedin Figure 5, but with a constant inflow rate of 2 at s for all of R ≥ . A minimal cut is X = { s, v, w } with P e ∈ δ + X ν e = 2. The unique IDE flow is still the one described in Table 1and, thus, never reaches a periodic state. Remark . In contrast the (again unique) dynamic equilibrium for the network from Fig-ure 5 is displayed in Figure 6 and does indeed reach a steady state at time θ = 4. s vw t (1 , ,
2) (1 , τ s t ,ν s t ) = (2 , θ = 0: u ≡ s vw t θ = 1: u ≡ q vt (2) = 1 s vw t θ = 2: u ≡ q vt (3) = 1 s vw t θ = 3: u ≡ q vt (4) = 1 s vw t θ = 4: u ≡ Figure 6:
The dynamic equilibrium flow for the network constructed in the proof of Theorem 4.5.
We will now show that the decision problem whether in a given network there exists an IDEwith certain properties is often NP-hard – even if we restrict ourselves to only single-sourcesingle-sink networks on acyclic graphs. Note, however, that due to the non-uniqueness ofIDE flows this does not automatically imply that computing any
IDE flow must be hard.We first show that the restriction to a single source can be made without loss of generality.
Lemma 4.7.
For any multi-source single-sink network N with piecewise constant inflowrates with finitely many jump points there exists a (larger) single-source single-sink network N with constant inflow rate such thata) the encoding size of N is linearly bounded in that of N ,b) if N is acyclic, so is N ,c) N is a subnetwork of N (except for the sources), cu s ≡ · [0 , ‘ ‘ ‘ t Figure 7:
The clause gadget C consists of a source node andthree edges leaving it, each with capacity and traveltime . If embedded in a larger network in such a waythat the shortest paths from ‘ , ‘ and ‘ to t all havethe same length (and no queues during the interval [0 , ), the inflow at node s can be distributed in anyway among the three edges. In particular, it is possibleto send all flow over only one of the three edges. Inany distribution there has to be at least one edge whichcarries a flow volume of at least . d) the restriction map composed with some constant translation is a one-to-one correspon-dence between the IDE-flows in N and those in N : { IDE-flows in N } → { IDE-flows in
N } , f f | N ( _ − c ) . Proof.
This can be accomplished by using the construction from the proof of [9, Theorem6.3], which clearly satisfies all four properties.
Theorem 4.8.
The following decision problems are NP-hard:• Given a network and a specific edge: Is there an IDE not using this edge?• Given a network and a specific edge: Is there an IDE using this edge?• Given a network and a time horizon T : Is there an IDE that terminates before T ?• Given a network and some k ∈ N : Is there an IDE consisting of at most k phases?All these decision problems remain NP-hard even if we restrict them to single-source instanceswith constant inflow rate on acyclic graphs. The last problem becomes NP-complete if werestrict k by some polynomial in the encoding size of the whole instance.Proof. We will show this theorem by reducing the NP-complete problem to the aboveproblems. The main idea of the reduction is as follows: For any given instance of we construct a network which contains a source node for each clause with three outgoingedges corresponding to the three literals of the clause. Any satisfying interpretation of the -formula translates to a distribution of the network inflow to the literal edges, whichleads to an IDE flow that passes through the whole network in a straightforward manner. If,on the other hand, the formula is unsatisfiable every IDE flow will cause a specific type ofcongestion which will divert a certain amount of flow into a different part of the graph. Thispart of the graph may contain an otherwise unused edge the diverted flow is then forced touse or a gadget which produces many phases (e.g. the graph constructed for the proof ofTheorem 4.1) or a long travel time (e.g. an edge with very small capacity).We start by providing two types of gadgets: One for the clauses and one for the variablesof a -formula. The clause gadget C (see Figure 7) consists of a source node c witha constant network inflow rate of 12 over some interval of length 1 and three edges withcapacity 12 and travel time 1 connecting c to the nodes ‘ , ‘ and ‘ , respectively. Thisgadget will later be embedded into a larger network in such a way that the shortest pathsfrom the nodes ‘ , ‘ and ‘ to the sink t all have the same length. Thus, the flow entering the19 y ¬ xzz s t V Figure 8:
The variable gadget V . The edges xy and zz have capacity , all other edges have infinite ca-pacity. The travel times on all (solid) edges are while the dashed lines represent paths with alength such that the travel time from s to t isthe same as from y over z and z to t . If flowenters this gadget at any rate over a time inter-val of length one at either x or ¬ x all flow willtravel over the edge zz to the sink t . If, on theother hand, at both x and ¬ x a flow of volume atleast enters the gadget over an interval of length a flow volume of more than will be divertedtowards s . gadget at the source node c can be distributed in any way over the three outgoing edges. Wewill have a copy of this gadget for any clause of the given -formula with the three nodes ‘ , ‘ and ‘ corresponding to the three literals of the respective clause. Setting a literal totrue will than correspond to sending a flow volume of at least 4 towards the respective node.The variable gadget V (see Figure 8) has two nodes x and ¬ x over which flow can enterthe gadget. From both of these nodes there is a path consisting of two edges of length 1leading towards a common node z , from where another edge of length and capacity 1 leadsto node z . From there the gadget will be connected to the sink node t somewhere outsidethe gadget. The path from ¬ x to z has infinite capacity , while the path from x to z consistsof one edge with capacity 1 followed by one edge of infinite capacity with a node y betweenthe two edges. The first edge can be bypassed by a path of length 3 and infinite capacity.From the middle node y there is also a path leaving the gadget towards t via some node s outside the gadget. This path has a total length of one more than the path via z and z to t .We will have a copy of this gadget for every variable of the given -formula. Similarlyto the clause gadget we will interpret the variable x to be set to true if a flow of volume atleast 4 traverses node x and the variable to be set to false if a flow volume of at least 4 passesthrough node ¬ x . Now, if flow travels through only one of these two nodes over the courseof an interval of length 1 (i.e. the variable x is set consistently) than all this flow will travelto t via z . If, on the other hand, both x and ¬ x each are traversed by a flow of volume atleast 4 over the span of a time interval of length 1 a flow of volume more than 1 will leavethe gadget via the edge ys during the unit length time interval three time steps later. Throughout this construction whenever we say that an edge has “infinite capacity” by that we mean somearbitrary capacity high enough such that no queues will ever form on this edge. Since the network weconstruct will be acyclic such capacities can be constructed inductively similarly to the constant L e in theproof of Claim 7
20o verify this, assume that the flow enters at nodes x and ¬ x during [0 , ¬ x will start to form a queue on edge zz two time steps later. This queuewill have reached a length of at least 2 at time 3 and, thus, still has a length of at least 1at time 4. The flow entering through x at first only uses edge xy until a queue of length 2has build up there. After that, flow will only enter this edge at a rate of 1 to keep the queuelength constant, while the rest of the flow travels through the longer path towards y . Thisflow (of volume at least 1) as well as some non-zero amount of flow from the queue on edge xy will arrive at node y during the interval [3 , zz all ofthis flow (of volume more than 1) will be diverted towards s .We can now transform a -formula into a network as follows: Take one copy of theclause gadget c for every clause of the formula, one copy of the variable gadget V for everyvariable and connect them in the obvious way (e.g. if the first literal of some clause is ¬ x connect the node ‘ of this clause’s copy of C with the node ¬ x of the variable x ’s copyof V and so on). Then add a sink node t and connect the nodes z of all variable gadgetsto t via edges of travel time 1 and infinite capacity. Finally, connect the node s (which isthe same for all variable gadgets) to t by first an edge s v of travel time 1 and then anotheredge of travel time 2 and infinite capacity. The resulting network (see Figure 9) has an IDEflow not using edge s v if and only if the -formula is satisfiable. If, on the other hand,the -formula is unsatisfiable every IDE flow will sent a flow volume of more than 1 overedge s v during the interval [4 , s v tV V . . . V n C . . . C k Figure 9:
Schematic representation of the whole network corresponding to a -formula with clauses C , . . . , C k in variables x , . . . , x n . The triangles are clause gadgets (cf. fig. 7), the rectan-gles are variable gadgets (cf. fig. 8). u s ≡ u N · [5 , θ ] s u s ≡ [4 , θ ] vt s N t N N I Figure 10:
The indicator gadget I for a single-sourcesingle-sink network N with network inflow rate u N [0 ,θ ] . All bold edges have infinite capacity,the edge s v has capacity . The edges s s N and s s both have travel time θ , edge s v hasa travel time of and the edges t N t and vt canhave any travel time such that the shortest s - t path through N has a length of exactly one morethan the s - t path using edge s v . If within theinterval [4 , a flow of volume more than ar-rives at s over the dashed edge, all flow enter-ing the network at s will travel trough N (it willarrive at that sub-networks source node s N at arate of u N during the interval [5 + θ , θ ] ).If, on the other hand, a flow volume of at most reaches s via the dashed edge up to time θ all flow originating at s will bypass N usingedge s v and N will forever remain empty. In order to show that the other problems are NP-hard as well, we will introduce a thirdtype of gadget: The indicator gadget I (see Figure 10). We can construct such a gadgetfor any given single-source single-sink network N with constant inflow rate over the interval[0 , θ ] at its source node. It consists of a new source node s with the same inflow rate as N ’ssource node shifted by 5 time steps. The node s is connected to the sink node t (outside thegadget) by two paths: One through the network N (entering it at its original source node s N and leaving it from its sink node t N ) and one through two additional nodes s and v andan edge of capacity and travel time 1 between them. All other edges outside N have infinitecapacity. The two outgoing edge from s both have a length of θ . The path through thegadget has length one more than the path via s and v . The node s has a constant networkinflow rate of 1 starting at time 4 and ending at time 5 + θ . When embedding this gadgetinto a larger network (with sink t ) the gadget is connected to the larger network by one ormore incoming edges into s .If no flow ever enters the gadget via this edge, all flow generated at s will travel throughthe path containing s v . If, on the other hand, a flow of volume more than 1 comes throughthis edge before the inflow at node s starts, all the flow generated there will travel throughthe subnetwork N . Adding this gadget to the network constructed from the -formulaas described above results in a network with the following properties (see Figure 11 for anexample):• If the -formula is satisfiable there exists an IDE flow where the subnetwork N insidegadget I is never used but edge s s is used.• If the -formula is unsatisfiable every IDE flow will be such that its restriction to thesubnetwork N inside I is a (time shifted) IDE flow in the original stand alone network N and the edge s s is never used.Accordingly, if for example we use the network from Figure 5 as sub-network we have areduction from to the fourth problem from Theorem 4.8. Any network N gives us areduction to the second problem (with edge s s as the special edge). And just an edge with22 very small capacity allows a reduction to the third problem. Alternatively, one could alsouse a network wherein flow gets caught in cycles for a long time before it reaches the sinkas, for example, the network constructed to prove the lower bound on the termination timeof IDE in [8]. c ‘ ‘ ‘ c ‘ ‘ ‘ c ‘ ‘ ‘ x y ¬ x x y ¬ x x y ¬ x x y ¬ x ts s s N t N N Figure 11:
The whole network for the -formula ( x ∨ x ∨ ¬ x ) ∧ ( x ∨ ¬ x ∨ x ) ∧ ( ¬ x ∨ x ∨ x ) .The bold edges have infinite capacity, while all other edges have capacity . The solid edgeshave a travel time of , the dashdotted edges may have variable travel time (depending onthe subnetwork N ). Remark . Combining a construction similar to the one above with the single-source multi-sink network constructed in the proof of [9, Theorem 6.3] to show that multi-commodity IDEflows may cycle forever, shows that the problem to decide whether a given multi-sink networkhas an IDE terminating in finite time is NP-hard as well.
Remark . The above construction also shows the following aspect of IDE flows: Whilea network may trivially contain edges that are never used in any IDE, edges that are onlyused in some IDE flows an edges that are used in every IDE, there can also be edges thatare either not used at all or used for some flow volume of at least c , but never with any flowvolume strictly between 0 and c . We showed that Instantaneous Dynamic Equilibria can be computed in finite time for single-sink networks by applying the natural α -extension algorithm. The obtained explicit boundson the required number of extension steps are quite large and we do not think that they aretight. A further analysis is needed.We then turned to the computational complexity of IDE flows. We gave an example of asmall instance which only allows for IDE flows with rather complex structure, thus, implying23hat the worst case output complexity of any algorithm computing IDE flows has to beexponential in the encoding size of the input instances. Furthermore, we showed that severalnatural decision problems involving IDE flows are NP-hard by describing a reduction from .One common observation that can be drawn from many proofs involving IDE flows (inthis paper as well as in [9] and [8]) is that they often allow for some kind of local analysis oftheir structure – something which seems out of reach for Dynamic Equilibrium flows. Thislocal argumentation allowed us to analyse the behavior of IDE flows in the rather complexinstance from section 4.2 by looking at the local behavior inside the much simpler gadgetsfrom which the larger instance is constructed. At the same time, this was also a key aspectof the positive result in section 3 where it allowed us to use inductive reasoning over thesingle nodes of the given network. We think that this local approach to the analysis of IDEflows might also help to answer further open questions about IDE flows in the future. Onesuch topic might be a further investigation of the computational complexity of IDE flows.While both our upper bound on the number of extension steps as well as our lower boundfor the worst case computational complexity are superpolynomial bounds, the latter is atleast still polynomial in the termination time of the constructed flow, which is not the casefor the former. Thus, there might still be room for improvement on either bound. Acknowledgments:
We thank the Deutsche Forschungsgemeinschaft (DFG) for their fi-nancial support.
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