Deciding feasibility of a booking in the European gas market on a cycle is in P for the case of passive networks

We show that the feasibility of a booking in the European entry‐exit gas market can be decided in polynomial time on single‐cycle networks that are passive, i.e., do not contain controllable elements. The feasibility of a booking can be characterized by solving polynomially many nonlinear potential‐based flow models for computing so‐called potential‐difference maximizing load flow scenarios. We thus analyze the structure of these models and exploit both the cyclic graph structure as well as specific properties of potential‐based flows. This enables us to solve the decision variant of the nonlinear potential‐difference maximization by reducing it to a system of polynomials of constant dimension that is independent of the cycle's size. This system of fixed dimension can be handled with tools from real algebraic geometry to derive a polynomial‐time algorithm. The characterization in terms of potential‐difference maximizing load flow scenarios then leads to a polynomial‐time algorithm for deciding the feasibility of a booking. Our theoretical results extend the existing knowledge about the complexity of deciding the feasibility of bookings from trees to single‐cycle networks.


INTRODUCTION
During the last decades, the European gas market has undergone ongoing liberalization [10][11][12], resulting in the so-called entry-exit market system [22]. The main goal of this market re-organization is the decoupling of trading and actual gas transport. To achieve this goal within the European entry-exit market, gas traders interact with transport system operators (TSOs) via bookings and nominations. A booking is a capacity-right contract in which a trader reserves a maximum injection or withdrawal capacity at an entry or exit node of the TSO's network. On a day-ahead basis, these traders are then allowed to nominate an actual load flow up to the booked capacity. To this end, the traders specify the actual amount of gas to be injected to or withdrawn from the network such that the total injection and withdrawal quantities are balanced. On the other hand, the TSO is responsible for the transport of the nominated amounts of gas. By having signed the booking contract, the TSO guarantees that the nominated amounts can actually be transported through the network. More precisely, the TSO needs to be able to transport every set of nominations that complies with the signed booking contracts. Thus, an infinite number of possible nominations must be anticipated and checked for feasibility when the TSO accepts bookings. As a consequence, the entry-exit market decouples In real-world gas networks, the arc set is typically partitioned into different types of arcs that correspond to different elements of the network, e.g., pipes, compressors, and control valves. However, we restrict our analysis to passive networks that consist of pipes only. We follow the notation and definitions of [32], which we briefly introduce in the following.

Definition.
A load is a vector = ( u ) u∈V ∈ R V ≥0 , with u = 0 for all u ∈ V 0 . The set of load vectors is denoted by L.
A load vector thus corresponds to an actual situation at a single point in time by specifying the amount of gas u that is supplied at u ∈ V + or withdrawn at u ∈ V − . Since we only consider stationary flows, we need to impose that the supplied amount of gas equals the withdrawn amount, which leads to the definition of a nomination.

Definition.
A nomination is a balanced load vector , i.e., ⊤ = 0. The set of nominations is given by A booking, on the other hand, is a load vector defining bounds on the admissible nomination values. More precisely, we have the following definition.

Definition.
A booking is load vector b ∈ L. A nomination is called booking-compliant w.r.t. the booking b if ≤ b holds, where "≤" is meant componentwise throughout this article. The set of booking-compliant (or b-compliant) nominations is given by Next, we introduce the notion of feasibility for nominations and bookings. We model stationary gas flows using an abstract physics model based on the Weymouth pressure drop equation and potential flows; see, e.g., [19] or [25]. It consists of arc flow variables q = (q a ) a∈A ∈ R A and potentials on the nodes = ( u ) u∈V ∈ R V ≥0 . We note that, in this context, potentials are linked to gas pressures at the nodes via u = p 2 u for the case of horizontal pipes. An in-depth explanation for nonhorizontal pipes is given in [19].

Definition.
A nomination ∈ N is feasible if a point (q, ) exists that satisfies ∑ a∈ out (u) where out (u) and in (u) denote the sets of arcs leaving and entering node u ∈ V, Λ a > 0 is an arc-specific constant for any a ∈ A, and 0 < − u ≤ + u are potential bounds for any u ∈ V.
Constraints (1a) ensure that flow is conserved at every node w.r.t. the nomination . For any a = (u, v) ∈ A, Constraint (1b) links the flow q a to the difference u − v of potentials at the endpoints of a. We note that flow can be negative, if it flows in the opposite direction of the orientation of the arc. Finally, due to technical restrictions of the network, the potentials need to satisfy bounds (1c). In a weakly connected network that only consists of pipes, the flow q = q( ) corresponding to a given nomination ∈ N is unique since it is the optimal solution of a strictly convex minimization problem [28]. The potentials = ( ) are the corresponding dual variables and are unique as soon as a reference potential is fixed; see, e.g., [31]. The potentials are therefore unique up to shifts, which in particular implies that potential differences between nodes are unique for a given nomination . The feasibility of a given nomination can be checked using the approach described in [18]. In contrast, verifying the feasibility of a booking is less researched and much more difficult.
Definition. We say that a booking b is feasible if all booking-compliant nominations ∈ N(b) are feasible.
To assess the feasibility of a booking, by definition, a possibly infinite number of nominations need to be checked.
Remark. Deciding the feasibility of a booking can be seen as very special case of deciding the feasibility of an adjustable robust optimization problem with uncertainty set N(b). Let us briefly highlight this relationship in this remark. In principle, for every booking-compliant nomination, we are allowed to adjust the corresponding flow and the corresponding potentials according to the feasibility system (1). However, the decision rule (in terms of adjustable robust optimization) is very special. Note again that, for a given nomination ∈ N(b), the resulting flow is uniquely determined and all potentials are uniquely determined if we fix a certain potential w at an arbitrarily chosen reference node w, e.g., if we set w = for a reference potential . Thus, we face the adjustable robust problem in which the uncertainty set consists of all booking-compliant nominations and that can be formalized as Here, Y corresponds to the -parameterized set of decision rules, which map given nominations to node potentials, i.e., y ∈ Y and y ∶ N(b) → R V . This, in particular, means that for a given nomination, the only choice is the reference pressure since all flows and potentials are uniquely determined afterward by (1). Consequently, deciding the feasibility of a booking is equivalent to finding a specific decision rule in the -parameterized family of functions Y. Note that the uncertainty is not given in a constraintwise way. Additionally, the decision rules must satisfy (1b), which has a nonlinear right-hand side that is nonsmooth in zero. Consequently, the decision rules are also nonlinear and nonsmooth in general. The related adjustable robust problem is thus a very special one that is, in general, not tractable in terms of adjustable robust optimization; see, e.g., [7] or the recent survey [35] as well as the references therein. One particular contribution of this article is that the problem-specific structure at hand is exploited so that the considered problem (which looks highly intractable at a first glance) can be solved efficiently. The further question on whether the developed techniques may be generalized to general adjustable robust flow problems is beyond the scope of this article.
In every network, the zero flow associated with the zero nomination is always feasible. It is achieved by having the same potential at every node. This, in particular, leads to the following assumption on the bounds of the potentials.

Assumption 1. The potential bound intervals have a nonempty intersection
Since the zero nomination is always booking compliant, this assumption is required for having a feasible booking at all. Thus, the assumption is also required to allow for a reasonable study of deciding the feasibility of bookings.
It is shown in Theorem 10 of [25] that a feasible booking b can be characterized by constraints on the maximum potential differences between all pairs of nodes. Therefore, the authors introduce, for every fixed pair of nodes (w 1 , w 2 ) ∈ V 2 , the following problem where w 1 w 2 is the corresponding optimal value function (depending on the booking b). Then, the booking b is feasible if and only if holds for every fixed pair of nodes (w 1 , w 2 ) ∈ V 2 . Hence, to verify the feasibility of a booking using this approach, it is necessary to solve the nonlinear and nonconvex global optimization problems (3). For tree-shaped networks, the authors give a polynomial-time dynamic programming algorithm for solving (3). As a consequence, verifying the feasibility of a booking on trees can be done in polynomial time, which can also be obtained by adapting the results of [32]. In this article, we show that (4) can still be decided in polynomial time on a single cycle.

NOTATION AND BASIC OBSERVATIONS
Entry and exit nodes v ∈ V + ∪ V − are called active if v > 0 holds. We denote by V > the set of active entries and exits, respectively.
Using directed graphs to represent gas networks is a modeling choice that allows us to interpret the direction of arc flows. However, the physical flow in a potential-based network is not influenced by the direction of the arcs. Thus, for u, v ∈ V, we introduce the so-called flow-paths P ∶= P(u, v) = (V(P(u, v)), A(P(u, v))) in which V(P(u, v)) ⊆ V contains the nodes of the path from u to v in the undirected version of the graph G and A(P(u, v)) ⊆ A contains the corresponding arcs of this path. Note that these flow-paths are not necessarily unique. For another pair of nodes u ′ , v ′ ∈ V, we say that (P(u, v)) and A(P(u ′ , v ′ )) ⊆ A(P(u, v)), and if P(u ′ , v ′ ) is itself a flow-path. In particular, this allows us to define an order on the nodes of a flow-path. For P = P(u, v) and u ′ , v ′ ∈ P, we define u ′ ≼ P v ′ if and only if a flow-subpath P(u, u ′ ) ⊆ P(u, v ′ ). 1 We now introduce the characteristic function of an arc a = (u, v) ∈ A. For any flow-path P, it is given by Next, we adapt a classical result from linear flow models to construct a flow decomposition in a gas network.
be the set of flow-paths in G with an active entry as start node and an active exit as end node. Then, a decomposition of the given flow q = q( ) into path flows exists, such that where q(P) is the nonnegative flow along the flow-path P ∈  . Furthermore, we require that if q a > 0 for a ∈ A and a (P) = −1 for P ∈  , then q(P) = 0 holds. Similarly, if q a < 0 for a ∈ A and a (P) = 1 for P ∈  , then q(P) = 0 holds.
Proof. If q a < 0 holds, then we replace arc a = (u, v) by (v, u) and set q (v, u) = −q (u, v) . The resulting flow still corresponds to nomination . We now apply Theorem 3.5 of Chapter 3.5 of the book by Ahuja et al. [1]. Given Constraints (1b), the flow q cannot contain any cycle flows. As a consequence, we obtain a flow decomposition that satisfies all the properties. ▪ Observe that, by construction, the flow q and the path flows need to traverse arcs in the same direction. A direct consequence of the flow decomposition is that the nomination can be expressed as a function of the path flows.

Corollary 3. For any u
and for any v ∈ V > − , the condition ∑ is satisfied.
Next, we define the potential-difference function along a given flow-path.

Definition.
Let ∈ N and a flow-path P be given. Then, the potential-difference function along P is given by where q = q( ).
As a consequence of Constraint (1b), for any node pair u, v ∈ V and for any flow-path P ∶= P(u, v), the equation holds. We note that if the path P is directed from u to v, the potential-difference function simplifies to Since, we will mostly use directed paths in what follows, we state some properties that hold in this case.

Lemma 4.
For u, v ∈ V, let P ∶= P(u, v) be a directed path. Then, the following holds: (a) Π P is continuous.

PROBLEM REDUCTION VIA FLOW-MEETING POINTS
In the remainder of this article, we restrict ourselves to a network that is a single cycle. A stylized example of a cyclic gas network is shown in Figure 1. A first observation is that in a potential-based flow model, there cannot be any cycling flow. Thus, flow in a cycle has to "meet" in at least one node. In this section, we show that the set of all feasible flows in Problem (3) can be restricted to flow along two different paths without changing direction along the way.
In a cycle, for every pair of nodes u, v ∈ V, exactly two flow-paths exist. We denote by P l (u, v) = (V l (u, v), A l (u, v)) the left path obtained when v is reached in counterclockwise direction from u. Similarly, P r (u, v) = (V r (u, v), A r (u, v)) is the right path obtained by using the clockwise direction. Moreover, A = A l (u, v) ∪ A r (u, v) holds. If it is clear from the context, we use previously introduced notations indexed by "l" (left) or "r" (right), when they have to be understood w.r.t. P l or P r .
It is not hard to observe that, given Constraints (1a) and (1b), the highest potential in G is attained at an entry node.

Lemma 5.
Let ∈ N∖{0} and o ∈ V + be an entry with highest potential.
Given that no cycle flow is possible in a gas network, flow needs to change the direction along a single cycle. We now specify a node as flow-meeting point if arc flows from different directions "meet" at this node.
Definition. Let ∈ N∖{0} and o ∈ V + be an entry node with highest potential, i.e., o ( ) This definition is illustrated in Figure 2. Note that we choose the node o ∈ V + with highest potential to ensure that there is no flow through node o. If multiple entry nodes with highest potential exist, flow-meeting points are still well defined. In fact, as a direct consequence of Lemma 5, the definition of a flow-meeting point is independent of the choice of node o. By definition, the flow-meeting point w has nonzero flow entering on one arc and possibly zero flow on the other arc. Thus, w is necessarily an exit.
From Constraints (1a) and (1b), it directly follows that for every nonzero nomination at least one flow-meeting point exists. We note that there can be multiple flow-meeting points with different potentials. However, since every flow-meeting point is an exit, it is not hard to observe that the following result holds. In the remainder of this section, we show that for fixed (w 1 , w 2 ) ∈ V 2 there are optimal solutions of (3) with at most one flow-meeting point. More precisely, we prove that an optimal solution exists that has a special entry node o ∈ V + , a special exit node w ∈ V − , and has nonnegative flow from o to w.
Before we prove several auxiliary results, let us first make a notational comment. For (o, w) ∈ V + × V − , we are interested in the partition of the cycle into two flow-paths P l (o, w) and P r (o, w). When discussing the order of nodes along P l (o, w), we therefore simply write u≼ l v instead of u≼ P l (o,w) v. We use an analogous simplification for P r (o, w).
A first observation is that nominations can be modified such that the flow from an entry node with highest potential to an exit node with lowest potential is nonnegative, while preserving particular potential differences.

Lemma 7.
Given ∈ N∖{0} with flow q = q( ), let o ∈ V + be an entry with highest potential and w a flow-meeting point with lowest potential. Furthermore, assume that P l (o, w) and P r (o, w) are directed paths. Then, for a given x ∈ V l (o, w), a nomination ′ ∈ N exists such that the following properties hold (with q ′ = q( ′ )): Proof. We modify nomination and q( ) such that the required properties are satisfied. To this end, we consider a flow decomposition as in Lemma 2.
Since o ∈ V + has highest potential and P l (o, w) and P r (o, w) are directed paths, it follows that q a ≥ 0 for all a ∈ out (o). In analogy, q a ≥ 0 for all a ∈ in (w). From Lemma 2, we then deduce that q(P(u, v)) = 0 if one of the following four conditions holds: In other words, since there is no flow through o or w, there cannot be a flow-path with nonzero flow through them, either. Consequently, for an arc a ∈ P l (o, w), we can simplify Equation (5) to where  l contains all flow-paths on the left side of the cycle, i.e., We note that  l depends on the choice of o and w. However, these are fixed nodes throughout this section. We now modify the flow q such that Property (9b) is satisfied. To this end, for every arc a ∈ P l (o, w) and for every P ∈  l , we set the flow q(P) = 0 if a (P) = −1 holds. We denote the modified flow and the corresponding nomination by q ′ and ′ . Then, for a ∈ A l (o, w), the modified flow is given by and satisfies (9b). Furthermore, by Corollary 3, the corresponding modified nomination ′ satisfies (9a). Additionally, (9c) is satisfied because we have not modified any arc flows q a for a ∈ A r (o, w). Due to Lemma 4(b), the modifications possibly increase the potential difference between o and x, as well as, between o and w. This is the case if and only if the corresponding flow-path contains an arc with negative flow in q, which is now set to zero in the modified flow q ′ . Next, we need to iteratively adapt nomination ′ and flow q ′ to ensure the remaining properties (9d) and (9e).
Step 1: If an arc a ∈ A l (o, x) with q a < 0 exists, then, the potential difference between o and x is increased, i.e., Given the flow decomposition, we then know that we have not modified arc flows on P l (u, x). Consequently, We further note that Π P l (o,u) (0) = 0 and, by Lemma 4(e), Π P l (o,u) (q) ≥ 0 holds because o ≥ u . Consequently, due to Lemma 4(a) and (b), we can decrease path flows q( holds. These flow modifications only decrease the nomination at entries and exits in V l (o, u). Thus, Lemma 4(d) implies the Properties (9a)-(9d). We note that we have not changed an arc flow of A l (x, w) in the modifications of Step 1.
Now it is left to show that we can modify the flow q ′ and the corresponding nomination ′ such that, additionally, Property (9e) is satisfied. To this end, we assume that an arc a ∈ A l (x, w) with q a < 0 exists. Otherwise the claim follows directly from Lemma 4.
Step 2: If an arc a ∈ A l (x, w) with q a < 0 exists, then, be a node such that q a ′ ≥ 0 holds for all a ′ ∈ A l (x, u) and |V l (x, u)| is maximal. Given the flow decomposition, we then know that we have not modified arc flows on A l (x, u). Thus, Π P l (x,u) (q ′ ) = Π P l (x,u) (q) and Π P l (u,w) (q ′ ) > Π P l (u,w) (q) hold. Furthermore, Π P l (u,w) (0) = 0 and Π P l (u,w) (q) ≥ 0 are valid. The latter is satisfied due to w ≤ u and Lemma 4(e). Similar to Step 1, the potential difference Π P l (u,w) (q ′ ) is only determined by positive path flows q(P l (v 1 , v 2 )) with u≼ l v 1 ≼ l v 2 ≼ l w. Due to Lemma 4, we can again decrease path flows q(P l (v 1 , v 2 )) for u≼ l v 1 ≼ l v 2 ≼ l w such that Π P l (u,w) (q ′ ) = Π P l (u,w) (q) holds and Property (9e) is satisfied. Furthermore, this modification does not affect any of Properties (9b)-(9d). Since we only decrease the nomination at entries and exits, Property (9a) is also satisfied.
In total, we can modify nomination and q( ) by repeatedly applying Steps 1 and 2 such that ′ and the corresponding q( ′ ) satisfy Properties (9). ▪ The same result can be established for the symmetric situation.

Corollary 8.
Given ∈ N∖{0} with flow q = q( ), let o ∈ V + be an entry with highest potential and w a flow-meeting point with lowest potential. Furthermore, assume that P l (o, w) and P r (o, w) are directed paths. Then, for a given x ∈ V r (o, w), a nomination ′ ∈ N exists such that the following properties hold (with q ′ = q( ′ )): Lemma 9. Given ∈ N∖{0} with flow q = q( ), let o ∈ V + be an entry with highest potential and w a flowmeeting point with lowest potential. Furthermore, assume that P l (o, w) and P r (o, w) are directed paths. Then, for given o≼ l x≼ l y≼ l w with Π P l (x,y) (q) ≥ 0, a nomination ′ ∈ N with q ′ = q( ′ ) exists such that Properties (9a) and (9b) are satisfied and Π P l (x,y) (q ′ ) = Π P l (x,y) (q) ≥ 0 holds.
Proof. In analogy to the proof of Lemma 7, we consider a flow decomposition of Lemma 2. Furthermore, for every Consequently, the modified flow q ′ , given as in (11), and the corresponding nomination ′ satisfy (9a) and (9b). By this modification, we increase the potential difference only if an arc in P l (o, w) with negative flow in q exists.
If an arc a ∈ A l (o, x) with q a < 0 exists, we apply Step 1 of the proof of Lemma 7, where we do not change the flow on any arc of P l (x, w). On the other hand, if an arc a ∈ A l (y, w) with q a < 0 exists, we apply Step 2, where we do not change the flow on any arc of P l (o, y). If an arc a ∈ A l (x, y) with q a < 0 exists, then, Π P l (x,y) (q ′ ) > Π P l (x,y) (q) ≥ 0 holds. Due to Lemma 4(a) and (b), and Π P l (x,y) (0) = 0, we can decrease path flows q(P l (v 1 , v 2 )) such that Π P l (x,y) (q ′ ) = Π P l (x,y) (q) and Properties (9a) and (9b) are still satisfied. This modification possibly decreases the potential differences Π P l (o,x) (q ′ ) and Π P l (y,w) (q ′ ). As a consequence of Lemma 4, we deduce that However, this can be easily fixed. Since the arc flow of any a ∈ A r (o, w) stays unchanged, Π P r (o,w) (q ′ ) = Π P r (o,w) (q) ≥ 0 holds as a consequence of Lemma 4(e). Since Property (9b) is satisfied for the modified flow q ′ , we deduce that and Properties (9a) and (9b) still hold. ▪ Analogously, we derive the symmetric result.

Corollary 10. Given
∈ N∖{0} with flow q = q( ), let o ∈ V + be an entry with highest potential and w a flowmeeting point with lowest potential. Furthermore, assume that P l (o, w) and P r (o, w) are directed paths. Then, for given o≼ r x≼ r y≼ r w with Π P r (x,y) (q) ≥ 0, a nomination ′ ∈ N with q ′ = q( ′ ) exists such that Properties (12a) and (12b) are satisfied and Π P r (x,y) (q ′ ) = Π P r (x,y) (q) ≥ 0 holds.
As a final auxiliary result, we give a sufficient condition for the existence of a unique flow-meeting point.

Lemma 11. Given
∈ N∖{0} with flow q = q( ), let o ∈ V + be an entry with highest potential and let w ∈ V∖{o} be an arbitrary node. Furthermore, assume that P l (o, w) and Proof. Since q ≥ 0, then w ≤ v holds for all v ∈ V. Let x ∈ V l (o, w) be such that q a = 0 holds for all a ∈ A l (x, w) and |V l (x, w)| is maximal. By construction of x, it is the only flow-meeting point and x = w may hold. ▪ Recall that it is sufficient to solve Problem (3) for each fixed node pair (w 1 , w 2 ) ∈ V 2 and then check Inequality (4) to decide the feasibility of a booking. We now combine the previous results to show that an optimal solution of Problem (3) with at most one flow-meeting point exists.
Theorem 12. Let b be a booking and let (w 1 , w 2 ) ∈ V 2 be a fixed pair of nodes. Then, there is an optimal solution of Problem (3) that has at most one flow-meeting point w.
Proof. Let ( , q, ) be an optimal solution of (3). Choose an entry o ∈ V + with highest potential and a flow-meeting point w with lowest potential. Due to Lemma 6, w ≤ v holds for all v ∈ V. Without loss of generality, we assume that P l (o, w) and P r (o, w) are directed.
The zero nomination corresponds to a feasible point that satisfies the claim and w 1 − w 2 = 0. Thus, we can assume that holds. If there is only one flow-meeting point, we are done. Hence, we now additionally assume that admits at least two different flow-meeting points.
Thus, we can equivalently reformulate (13) as We now apply Lemma 7 with x = w 1 , which does not change Π P l (o,w 1 ) (q) and Π P r (o,w 2 ) (q). Then, we apply Corollary 8 with x = w 2 , which does not change Π P l (o,w 1 ) (q) and Π P r (o,w 2 ) (q). Consequently, the obtained nomination ′ and the corresponding flow q ′ = q( ′ ) are still optimal. Thus, (13) is satisfied by q( ′ ) ≥ 0. The claim then follows by Lemma 11.
The claim follows in analogy to Case 1.
In this case, (13) reads We first apply Corollary 8 with x = w, which does not change Π P l (w 1 ,w 2 ) (q). Thus, (13) is still satisfied and q ′ a ≥ 0 holds for every a ∈ P r (o, w). We now apply Lemma 9 with x = w 1 and y = w 2 , which does not change the objective value w 1 − w 2 = Π P l (w 1 ,w 2 ) (q). Consequently, q ′ ≥ 0 holds and (13) is still satisfied. The claim then again follows from Lemma 11. Case 4: w 1 , w 2 ∈ V r (o, w) and w 1 ≼ r w 2 . The claim follows in analogy to Case 3. Case 5: w 1 , w 2 ∈ V l (o, w) and w 2 ≼ l w 1 . Inequality (13) then reads We first apply Corollary 8 with x = w, which does not change Π P l (w 2 ,w 1 ) (q). Thus, (13) is still satisfied and q ′ a ≥ 0 for every a ∈ P r (o, w).
. Thus, Π P l (w 2 ,w 1 ) (q) ≥ 0 also holds, which contradicts (13). Hence, we conclude that u ∈ V l (w 2 , w)∖{w 2 }. By Lemma 2 and the construction of u, we deduce that for a ∈ A l (u, w) the flow is given by We now set the flow q(P l ) = 0 for P l ⊆ P l (u, w) and a (P l ) = 1. By this modification, we have possibly decreased Π P l (w 2 ,w 1 ) and thus also Π P l (o,w) . In particular, (13) is still satisfied. Lemma 4(d) implies holds. After this modification, (13) is still satisfied and its value is possibly increased, i.e., the objective function value w 1 − w 2 is possibly increased by the modifications. Consequently, the obtained solution is still optimal. Moreover, w is now connected to a flow-meeting point in V l (o, u) because q ′ a = 0 holds for all a ∈ P l (u, w). Consequently, for nomination ′ a flow-meeting point in V l (o, u) with lowest potential exists. We now repeat this procedure until either the claim holds or a new flow-meeting point with lowest potential is an element of V l (o, w 1 ). Then, we apply the respective case of Cases 1-4.
Case 6: w 1 , w 2 ∈ V r (o, w) and w 2 ≼ r w 1 . The claim follows in analogy to Case 5. ▪ As a direct consequence of this result, we deduce the following corollary.
Corollary 13. Let b be a booking and let (w 1 , w 2 ) ∈ V 2 be a fixed pair of nodes. Then, there exist nodes (o, w) ∈ V + × V − and an optimal solution ( , q, ) of Problem (3) with q ≥ 0, if we assume that P l (o, w) and P r (o, w) are directed paths.
The previous result implies that when determining potential-difference maximizing nominations solving Problem (3) for fixed (w 1 , w 2 ) ∈ V 2 , we can additionally restrict the search space by iteratively considering (o, w) ∈ V + × V − and imposing that there is flow from o to w. This is further formalized and exploited in the next section.

STRUCTURE OF POTENTIAL-DIFFERENCE MAXIMIZING NOMINATIONS
In this section, we fix (w 1 , w 2 ) ∈ V 2 and show that there exist optimal solutions of (3) with additional structure that allows us to reduce the dimension of the problem. Based on the results of Section 5, in particular, Corollary 13, we next show that (4) can be decided by considering the following variant of Problem (3) for every (o, w) ∈ V + × V − : where b is a booking and A ′ is obtained from A by orienting all arcs from o to w. Note that in addition to the constraints of (3), we now also impose nonnegative flow from o to w, thus effectively reducing the feasible domain of the problem.

Theorem 14. Let b be a booking. Then
holds. Furthermore, the optimal values are finite and attained.
Let ( , q, ) be an optimal solution corresponding to max (o,w)∈V + ×V − ow w 1 w 2 (b). First, observe that the arc orientation does not play any role in Problem (3). If an arc has a different orientation, we just switch the sign of the corresponding flow variable. Thus, we assume w.l.o.g. that P l (o, w) and P r (o, w) are directed paths in the given instance of (3). Consequently, ( , q, ) is feasible for (3). Thus, The other inequality follows directly from Corollary 13. ▪ As a consequence, the feasibility of a booking can be characterized using Problem (14) as follows.

Corollary 15. A booking b is feasible if and only if for every pair
We now further analyze the structure of optimal solutions of (14) for fixed (o, w) ∈ V + × V − and given (w 1 , w 2 ) ∈ V 2 w.r.t. their respective position in the cycle. Without loss of generality, we assume that P l (o, w) and P r (o, w) are directed paths.

Nodes on different sides of G
Assume that w 1 ∈ P l (o, w) and w 2 ∈ P r (o, w) hold. We show that an optimal solution ( , q, ) of (14) exists that additionally satisfies the following properties: A possible configuration of nodes o, w 1 , s l 1 , s l 2 , t l 1 , w, w 2 , s r 1 , t r 1 , t r 2 is given in Figure 3. To show the existence of such a solution, we introduce a bilevel problem, where the lower level is given by (14) and the upper level chooses, among all lower-level optimal solutions, one with the additional structure. It is given by (14), where M = ∑ u∈V b u and f 1 , … , f 4 are continuous functions that we specify later. By Constraints (16b) and (16c), the variables x ≼ l v and x ≽ l v model the existence of an active entry before and after v on P l . Similarly, Constraints (16d) ensure that y ≼ l v determines the existence of an active exit before v on P l . An analogous interpretation can be given for Constraints (16e)-(16g) and the variables x ≽ r , y ≼ r , y ≽ r . Then, the optimal value function reformulation of (16) is given by Here, Constraint (17b) determines the feasible domain of Problem (14) and Constraint (17d) guarantees feasible points with a potential difference of at least ow w 1 w 2 (b). Thus, we only consider optimal solutions of (14). We denote by a feasible point of (17). In particular, we have the following result.
Lemma 16. Let z be feasible for (17). Then ( , q, ) is an optimal solution of (14). Conversely, every optimal solution of (14) can be extended to a feasible point of (17). Proof.
The first statement follows from the previous discussion. For the converse, let an optimal solution ( , q, ) of (14) be given. We construct a solution z as follows:  We now specify the parts of the objective function of (17) and prove connections between these functions and the stated Properties (a)-(d). We discuss and prove the results for f 1 and f 2 in detail, whereas we only state the results for f 3 and f 4 , since they are very similar. The proofs for the results concerning f 3 and f 4 can be found in Appendix A.
For the following proofs, we make use of structures resulting from the negation of Properties (a)-(d) on Page 11. More precisely, we observe that Consider, for instance, the negation of Property (a). It is always possible to satisfy the first part of the property, i.e., there exist two entries s l 1 , s l To achieve this, we simply choose the first and the last active entry node on the left side of the cycle, i.e., s l 1 ≼ l s l 2 ∈ V l + (o, w) such that s l 1 > 0, s l 2 > 0, and u = 0 for all u ∈ V l + (o, s l 1 ) ∪ V l + (s l 2 , w)∖{o, s l 1 , s l 2 }. Now, if Property (a) is not satisfied, there has to exist another node u ∈ V l + (s l 1 , s l 2 )∖{s l 1 , s l 2 } with u < b u , which shows the claim. Analogously, we can obtain the remaining statements. (17) and

holds if and only if satisfies Property (a).
Proof. Let z be feasible for (17).
holds. Assume now that Property (a) does not hold. Consequently, there are u 1 , u 2 , u 3 ∈ V l + (o, w)∖{o} with u 1 ≺ l u 2 ≺ l u 3 such that u 1 > 0, u 2 < b u 2 , and u 3 > 0 hold. Thus, x ≼ l If satisfies Property (a), then we set x ≼ l u = 0 for all u ∈ P l (o, s l 1 )∖{s l 1 }. Otherwise, we set x ≼ l u = 1. Additionally, we set x ≽ l u = 1 for all u ∈ P l (o, s l 2 ) and otherwise we set x holds due to Property (a). Consequently, f 1 ( , x ≼ l , x ≽ l ) = 0 holds. ▪ Lemma 18. Let z be feasible for (17) and Then, there exists y ≼ l such that f 2 ( , y ≼ l ) = 0 holds if and only if satisfies Property (b).
Proof. Let z be feasible for (17).
holds. Assume now that Property (b) does not hold. Consequently, there are u 1 , holds due to Property (b). Consequently, f 2 ( , y ≼ l ) = 0 holds. ▪ Lemma 19. Let z be feasible for (17) and Then, there exists x ≽ r such that f 3 ( , x ≽ r ) = 0 holds if and only if satisfies Property (c).
Lemma 20. Let z be feasible for (17) and Then, there exists (y ≼ r , y ≽ r ) such that f 4 ( , y ≼ r , y ≽ r ) = 0 holds if and only if satisfies Property (d).
In the following, we consider f 1 , … , f 4 as specified in Lemmas 17-20. As a next step, we show that changing the nomination on the boundary nodes of Properties (a)-(d) does not affect the values of f 1 , … , f 4 , since the corresponding products of binary variables are zero.

Lemma 21.
Let z be an optimal solution of (17) and let u 1 ,u 3 ∈ V l + (o, w) with u 1 ≼ l u 3 be nodes such that u 1 > 0,

Proof.
Optimality of z and the choice of u 1 and u 3 imply x Consequently, a change of u 1 or u 3 does not change

Lemma 22. Let z be an optimal solution of (17) and let v
Proof. Optimality of z and the choice of v 1 imply y Thus, a change of v 1 does not change f 2 ( , y ≼ l ). ▪ Lemma 23. Let z be an optimal solution of (17) and let u 1 ∈ V r + (o, w) be a node such that u 1 > 0 and u = 0 for all u ∈ V r + (u 1 , w)∖{u 1 }. Suppose further that z ′ is feasible for (17) with

Lemma 24. Let z be an optimal solution of (17) and let v
The two last proofs can again be found in Appendix A. We next show that there is an optimal solution of (14) that satisfies Properties (a)-(d). More precisely, we prove that the optimal value of (17) is zero by individually treating f 1 , … , f 4 . The final result then easily follows from Lemmas 17-20.

Proof.
Let z be an optimal solution of (17). By contradiction, we assume that f 1 ( , x ≼ l , x ≽ l ) > 0 holds. Lemma 17 implies that does not satisfy Property (a). Consequently, there are entries u 1 , u 2 , u 3 ∈ V l + (o, w)∖{o} with u 1 ≺ l u 2 ≺ l u 3 such that u 1 > 0, u 2 < b u 2 , and u 3 > 0. If q a > 0 for a ∈ out (o) ∩ P l (o, w), we replace u 1 = o. Otherwise, we choose u 1 ≠ o such that u = 0 holds for all u ∈ V l + (o, u 1 )∖{o, u 1 } and we choose u 3 such that u = 0 holds for all u ∈ V l + (u 3 , w)∖{u 3 }. We now consider a flow decomposition as in Lemma 2. Due to q ≥ 0, an exit v 3 ∈ V l − (u 3 , w) with q(P l (u 3 , v 3 )) > 0 exists. Moreover, by the choice of u 1 , there is an (u 1 , v 1 )) > 0. We need to distinguish two cases.
Case 1: v 1 ≺ l u 2 holds. We now decrease q(P l (u 3 , v 3 )) by > 0 and increase q(P l (u 2 , v 3 )) by the same amount . This increases the potential difference Π P l (o,w) (q) due to u 2 ≺ l u 3 . Thus, we decrease q(P l (u 1 , v 1 )) bỹ> 0. Due to Lemma 4, we can choose and̃such that Π P l (o,w) (q) stays the same as before the modification and u 1 > 0, u 2 ≤ b u 2 , u 3 > 0, v 1 > 0 holds. In particular, the binary variables of z stay the same. Due to this and , the values of f 2 , f 3 , and f 4 stay the same. Moreover, the modified solution satisfies Constraints (17b). Furthermore, by this modification we decrease q a for a ∈ P l (u 1 , v 1 ), increase q a for a ∈ P l (u 2 , u 3 ), and the remaining arc flows stay the same. Hence, since u 1 ≺ l v 1 ≺ l u 2 ≺ l u 3 and by Lemma 4(d), we possibly increase the potential difference between w 1 and w 2 and Constraint (17d) is still satisfied. Consequently, z is still feasible for (17). Due to this modification, we decrease u 1 > 0 and u 3 > 0 and increase u 2 . By Lemma 21, considering only the decrease of u 1 and u 3 does not change the objective function value. In contrast, the increase of u 2 decreases f 1 because decreases. Thus, the modification decreases the objective function value, which contradicts the optimality of the original solution.
Case 2: u 2 ≺ l v 1 holds. We now decrease q(P l (u 1 , v 1 )) by > 0 and increase q(P l (u 2 , v 1 )) by the same amount . This decreases the potential difference Π P l (o,w) (q) due to u 1 ≺ l u 2 . Thus, we now decrease q(P l (u 3 , v 3 )) bỹ> 0 and increase q(P l (u 2 , v 3 )) by the same amount̃, which increases the potential difference Π P l (o,w) (q) due to u 2 ≺ l u 3 . Due to Lemma 4, we can choose and̃such that Π P l (o,w) (q) stays the same and u 1 > 0, u 2 ≤ b u 2 , u 3 > 0 holds. In analogy to Case 1, the function values of f 2 , f 3 , and f 4 stay the same and the modified solution satisfies Constraints (17b). Furthermore, the modification only decreases q a for a ∈ P l (u 1 , u 2 ) and increases flow q a for a ∈ P l (u 2 , u 3 ). The remaining arc flows stay the same. Hence, since u 1 ≺ l u 2 ≺ l u 3 and by Lemma 4(d), we possibly increase the potential difference between w 1 and w 2 and Constraint (17d) is still satisfied. Consequently, z is feasible for (17) after modification. In analogy to Case 1, the modification decreases f 1 , which contradicts the optimality of the original solution. ▪ Lemma 26. If z is an optimal solution of (17), then f 2 ( , y ≼ l ) = 0 holds.

Proof.
Let z be an optimal solution of (17). By contradiction, we assume that f 2 ( , y ≼ l ) > 0 holds. Lemma 18 implies that does not satisfy Property (b). Consequently, there are exits v 1 Next, let an entry u 1 ∈ V l + (o, w) be given so that u 1 > 0, u = 0 for all u ∈ V l + (o, u 1 )∖{o, u 1 }, and in a flow decomposition as by Lemma 2, q(P l (u 1 , v 1 )) > 0 holds. Due to Lemma 4 and v 1 ≺ l v 2 , we can decrease q(P l (u 1 , v 1 )) and increase q(P l (u 1 , v 2 )) such that Π P l (o,w) (q) remains the same and 0 Thus, the binary variables of z stay the same. Furthermore, by Lemmas 21, 23, and 24 the values of f 1 , f 3 , and f 4 stay the same. The modified solution satisfies Constraints (17b) and we only decrease q a for a ∈ P l (u 1 , v 1 ) and increase q a for a ∈ P l (v 1 , v 2 ). The remaining arc flows are unchanged. Then, since u 1 ≺ l v 1 ≺ l v 2 and by Lemma 4(d), Constraint (17d) is still satisfied. Consequently, z is still feasible for (17). Due to this modification, we decrease v 1 > 0 and increase v 2 . By Lemma 22, considering only the decrease of v 1 does not change the objective function value. In contrast, the increase of v 2 decreases f 2 because decreases. Thus, the modification decreases the objective function value, which contradicts the optimality of the original solution. ▪
Again, the proofs for the results concerning f 3 and f 4 can be found in Appendix A. Finally, we obtain the main structural property for nodes w 1 and w 2 on different sides of G by combining the previous lemmas. , w), and w 2 ∈ P r (o, w). Then, an optimal solution ( , q, ) of (14) exists that satisfies Properties (a)-(d).

Proof.
The zero nomination is feasible for Problem (14). Furthermore, the feasible region of the latter problem is compact and thus an optimal solution is attained. Consequently, Problem (17) has an optimal solution, which is attained. Due to Lemmas 25-28 and 17-20, an optimal solution ( , q, , x, y) of Problem (17) exists that satisfies Properties (a)-(d). Additionally, the solution ( , q, ) is also optimal for Problem (14). ▪

Nodes on the same side of G
Assume w 1 , w 2 ∈ P l (o, w) or w 1 , w 2 ∈ P r (o, w) holds. We can w.l.o.g. assume that w 1 , w 2 ∈ P r (o, w) holds. If w 2 ≺ r w 1 holds, then from q ≥ 0 in Problem (14) it follows that Π P r (w 1 ,w 2 ) (q) ≤ 0 is valid. Thus, the zero nomination is an optimal solution for Problem (14). Consequently, we now assume that w 1 ≺ r w 2 holds. We want to show that an optimal solution ( , q, ) of Problem (14) exists such that Properties (a), (b), (d), and (a) w.r.t. P r (o, w) are satisfied, i.e., two entries s r 1 , s r 2 ∈ V r + (o, w) with s r 1 ≼ r s r 2 exists such that Configuration of s and t nodes with o≼ r w 1 ≺ r w 2 ≼ r w. Boxes qualitatively illustrate the amount of the booking that is nominated is satisfied. Figure 4 illustrates a possible node configuration. To this end, we introduce an optimization problem similar to (16), which is given by Note that an analogous variant of Lemma 16 is also valid for Problem (22). We specify the parts of the objective function of (22) as follows: the functions f 1 , f 2 , and f 4 are defined as in Lemmas 17,18,and 20. The function f 3 is defined in analogy to Lemma 17 w.r.t. P r . We note that f i for i = 1, … , 4 also inherit the corresponding properties of Lemmas 17-24. We now prove that the optimal objective value of (22) is zero.

Proof.
The claim follows in analogy to Lemma 25. In doing so, we note that the modifications in the proof of Lemma 25 only affect nodes of P l (o, w). Consequently, we do not change the potential difference between w 1 and w 2 due to w 1 , w 2 ∈ P r (o, w). ▪ Lemma 31. If z is an optimal solution of (22), then f 2 ( , y ≼ l ) = 0 holds.
Proof. The claim follows in analogy to Lemma 26. ▪ To show analogous results for f 3 and f 4 , we make use of an auxiliary lemma.

Lemma 32.
An optimal solution z of (22) exists such that v = 0 for all v ∈ V r − (o, w 1 ) and u = 0 for all u ∈ V r + (w 2 , w) is satisfied.
Proof. We choose an optimal solution z of (22) u is minimal. Note that every addend is nonnegative. By contradiction, we assume that ∑ )∖{v} is satisfied. Consequently, an entry u ∈ V r + (o, v) exists such that u ′ = 0 holds for all u ′ ∈ V r + (o, u)∖{o} and in a flow decomposition, such as in Lemma 2, q(P r (u, v)) > 0 is satisfied. We can now decrease the latter by > 0 such that u > 0 and v > 0 holds. This decreases the potential drop Π P r (o,w) (q). Due to Lemmas 30 and 31, we can assume that q(P l (s l 1 , t l 1 )) > 0 holds. By using Lemma 4, we can now decrease the latter bỹand choose such that Π P l (o,w) (q) = Π P r (o,w) (q) holds and u , v , s l 1 , t l 1 are positive. Moreover, Lemmas 21-24 imply that the solution obtained after the modifications is still feasible and optimal for (22). In doing so, we note that the modifications do not change any flow of P r (w 1 , w 2 ) and thus the potential difference between w 1 and w 2 stays the same. This is a contradiction to the choice of z because ∑ is decreased in the modified solution. Case 2: There is u ∈ V r + (w 2 , w) with u > 0. We now choose u such that u ′ = 0 for all u ′ ∈ V r + (u, w). Due to q ≥ 0, an exit v ∈ V r − (u, w) exists such that v ′ = 0 holds for all v ′ ∈ V r − (v, w)∖{v, w} and q(P r (u, v)) > 0. In analogy to Case 1, the claim follows by decreasing the flow q(P r (u, v)) by > 0 and q(P l (s l 1 , t l 1 )) bỹ> 0. ▪

Lemma 33. If z is an optimal solution of (22), then f
Proof. Let z be an optimal solution of (22) that satisfies Lemma 32. By contradiction, we assume that f 3 ( , x ≼ r , x ≽ r ) > 0 holds. Lemma 17 implies that does not satisfy Property (a) w.r.t. P r . Consequently, there are entries u 1 , u 1 )∖{o, u 1 } and we choose u 3 such that u = 0 holds for all u ∈ V r + (u 3 , w)∖{u 3 }. We now consider a flow decomposition as in Lemma 2. Due to q ≥ 0, an exit v 3 ∈ V r − (u 3 , w) with q(P r (u 3 , v 3 )) > 0 exists. By the choice of u 1 , there is an exit v 1 ∈ V r − (u 1 , w) with v = 0 for all v ∈ V r − (o, v 1 )∖{v 1 } and q(P r (u 1 , v 1 )) > 0. Consequently, v 1 ≼ r v 3 holds. We now distinguish two cases. Case 1: u 2 ≼ r w 1 . Due to Lemma 32, w 1 ≼ r v 1 holds. Consequently, we can decrease q(P r (u 1 , v 1 )) > 0 by > 0 and we increase q(P r (u 2 , v 1 )) by the same amount such that u 1 > 0 and u 2 ≤ b u 2 holds. Since u 1 ≺ r u 2 holds, this modification decreases the potential difference Π P r (o,w) (q) but the flow on arcs of P r (w 1 , w 2 ) stays the same due to u 2 ≼ r w 1 . Consequently, Π P r (w 1 ,w 2 ) (q) is unchanged. From the proof of Lemma 25, it follows that this modification decreases f 3 . In analogy to Case 1 of Lemma 32, we can now decrease Π P l (o,w) (q) by modifying s l 1 and t l 1 such that Π P r (o,w) (q) = Π P l (o,w) (q) holds without changing the values of f i for i = 1, … , 4. This is a contradiction to the optimality of z because we have decreased f 3 in the first part of the modification.
Case 2: w 1 ≺ r u 2 . Due to u 2 ≺ r u 3 and Lemma 4, we can decrease q(P r (u 3 , v 3 )) by > 0 and increase q(P r (u 2 , v 3 )) by 0 <̃≤ such that Π P r (o,w) (q) = Π P l (o,w) (q), u 3 > 0, v 3 > 0, and u 2 ≤ b u 2 holds. Consequently, the binary variables of z stay the same. By using Lemmas 21,22,and 24, the values f 1 , f 2 , and f 4 stay the same as well. The modified solution satisfies Constraints (17b). Furthermore, the modification only increases q a for a ∈ P r (u 2 , u 3 ) and decreases the flow q a for a ∈ P r (u 3 , v 3 ). The remaining arc flows stay unchanged. Due to w 1 ≺ r u 2 ≺ r u 3 ≺ r w 2 and Lemma 4(d), we possibly increase the potential difference between w 1 and w 2 and thus Constraint (17d) is still satisfied. Case 1 of Lemma 25 implies that the previous modification decreases f 3 , which is a contradiction to the optimality of z. ▪ Lemma 34. If z is an optimal solution of (22), then f 4 ( , y ≼ r , y ≽ r ) = 0 holds.
Proof. Let z be an optimal solution of (22) that satisfies Lemma 32. By contradiction, we assume that f 4 ( , y ≼ r , x ≽ r ) > 0 holds. Lemma 20 implies that does not satisfy Property (d). Consequently, there are 3 , w}. We now consider a flow decomposition as in Lemma 2. Due to q ≥ 0, there is an entry We now distinguish two cases. Case 1: v 2 ≼ r w 2 . Consequently, v 1 ≺ r v 2 ≼ r w 2 holds. We can now decrease q(P r (u 1 , v 1 )) by > 0 and increase q(P r (u 1 , v 2 )) by 0 <̃≤ such that Π P r (o,w) stays the same and u 1 > 0, v 1 > 0, and v 2 ≤ b v 2 holds. In particular, the binary variables of z stay the same after the modification. Due to this and Lemmas 21 and 23, the values of f 1 , f 2 , and f 3 stay unchanged. The modified solution satisfies Constraints (17b). Furthermore, this modification only decreases q a for a ∈ P r (u 1 , v 1 ) and increases arc flows q a for a ∈ P r (v 1 , v 2 ). The remaining arc flows stay the same. Hence, since w 1 ≺ r v 1 ≺ r v 2 ≼ r w 2 and by Lemma 4(d), we possibly increase the potential difference between w 1 and w 2 and Constraint (17d) is still satisfied. Consequently, z is still feasible for (22). In analogy to Case 1 of Lemma 28, it follows that the modification decreases f 4 , which is a contradiction to the optimality of z.
We can now apply Case 2 of Lemma 28. In doing so, we keep in mind that w 1 ≺ r v 1 and w 2 ≺ r v 2 ≺ r v 3 hold which ensures that z still satisfies (17d) after the applied modifications. ▪ Finally, we obtain a result for the present case that is analogous to Theorem 29. Proof. The zero nomination is feasible for Problem (14) and it is optimal if w 2 ≼ r w 1 holds. Furthermore, the feasible region of the latter problem is compact and thus an optimal solution is attained. Consequently, Problem (22) has an optimal solution, which is attained. Due to  and Lemmas 17-20, for w 1 ≺ r w 2 an optimal solution of Problem (22) exists that satisfies Properties (a), (b), (d) and, (a) w.r.t. P r . Additionally, the solution is also optimal for Problem (14). ▪

A POLYNOMIAL-TIME ALGORITHM
Exploiting the special structure of nominations that maximize the potential difference between a pair of nodes, we now show that the feasibility of a booking can be checked in polynomial time on a cycle. First, we obtain an estimate on the number of arithmetic operations necessary to detect the existence of an infeasible nomination, or otherwise certify its nonexistence. In a second step, we then translate this result to the Turing model of computation, resulting in a polynomial-time algorithm for deciding the feasibility of a booking. For doing so, we make the following nonrestrictive assumption on the rationality of the problem data.
Assumption 36. We consider a booking b ∈ Q V and assume that Λ a ∈ Q for all a ∈ A and − u , + u ∈ Q for all u ∈ V. Additionally, we assume that the encoding lengths are bounded from above by .
As a consequence of Corollary 15, a booking b is feasible if and only if, for every (w 1 , w 2 ) ∈ V 2 and (o, w) ∈ V + × V − , admits no solution. We now make several observations. First, recall that A ′ is obtained from A by orienting all arcs from o to w. Then, given (14c), the right-hand sides of (14b) simplify to Λ a q 2 a for all a ∈ A ′ . Second, we eliminate the potentials by aggregating the resulting constraints along P l (o, w) and P r (o, w). We only treat the situation corresponding to Section 5 in which o≼ r w 1 ≺ r w 2 ≼ r w is valid, since it has the highest number of s and t nodes necessary to set up the structural properties and thus represents the worst case in terms of complexity. The situation corresponding to Section 5 with w 1 and w 2 on different paths w.r.t. o and w can however be treated in a similar way. We obtain ∑ It is well known that if the nomination is balanced, the rank of the flow conservation constraints (1a) is |V| − 1, resulting in a single degree of freedom in the case of a cycle. Thus, we introduce w = l w + r w to take into account the supply to the flow-meeting point w along P l and P r separately. Then, for a = (u, v) ∈ A ′ , (1a) leads to solution for (24). Note that this algorithm can in particular handle strict inequalities as required to determine a violation of the potential difference bounds; see, e.g., Notation 11.31 in [5]. We then obtain the following result. We close this section with a short remark on how our results can be applied to other types of utility networks, e.g., to water distribution or power networks.
Remark. The structural properties derived in Sections 2-5 can be applied to potential-based networks if the following assumptions hold: The potentials satisfy (1) where for any arc a ∈ A, the right-hand side of (1b) is a function a ∶ R → R that may depend on the arc flow q a and that is continuous, strictly increasing, and odd, i.e., a (−q a ) = − a (q a ). Consequently, our structural results hold for many different networks such as water, hydrogen, or lossless DC (direct current) power flow networks, if the physics model is chosen appropriately; see [19]. In particular, we can reduce the optimization problem (3), where we replace the right-hand side of (1b) by a , to a fixed inequality system for all these potential-based networks as shown in Section 5. However, the presented complexity result is only valid in the case in which the potential function a (q a ) is a polynomial in the variables |q a | and q a that is strictly increasing and odd.
However, the overall question of deciding the feasibility of a booking discussed in this article is rather specific and tailored to the European gas market system since, e.g., the market design for electricity is different from the one for gas in Europe.

CONCLUSION
In this work, we prove that deciding the feasibility of a booking in the European entry-exit gas market model is in P for the special case of cycle networks. To the best of our knowledge, this is the first in-depth complexity analysis in this context that considers a nonlinear flow model and a network topology that is not a tree. Our approach requires the combination of both the cyclic structure of the network and properties of the underlying nonlinear potential-based flow model with a general decision algorithm from real algebraic geometry. We show that the size of a polynomial equality and inequality system for deciding the feasibility of a booking is constant and, in particular, does not depend on the size of the cycle. Thus, a general algorithm for solving this system can serve as a constant-time oracle used in an enumeration of polynomial complexity.
Although our theoretical result moves the frontier of knowledge about the hardness of deciding the feasibility of bookings in the European entry-exit gas market, it still remains an open question to exactly determine the frontier between easy and hard cases if a nonlinear and potential-based flow model is considered. Although we believe that the problem is hard on general networks, no hardness results are known so far. Since both trees and single cycle networks are now well understood, a possibility is to consider more general classes of networks. Thus, a reasonable next step could be networks consisting of a single cycle with trees on it or, even more generally, cactus graphs. In our opinion, it is promising to combine the techniques used on trees and cycles in order to solve this larger graph class.
Finally, although the present article is a very specific one, we hope that the structural insights gained can be later put together with other insights to obtain more general techniques for (adjustable) robust and nonlinear flow problems.
holds. Assume now that Property (d) does not hold. Consequently, there are u 1 , u 2 , u 3 ∈ V r − (o, w) ∖ {w} with u 1 ≺ r u 2 ≺ r u 3 such that u 1 > 0, u 2 < b u 2 , and u 3 > 0 hold. Thus, y ≼ r u 1 = y ≽ r u 3 = 1 and ∑ v∈V r − (u 1 ,u 3 ))∖{u 1 ,u 3 } holds. Thus, f 4 ( , y ≼ r , y ≽ r ) > 0. If satisfies Property (d), then we set y ≼ r v = 0 for all v ∈ V r − (o, t r 1 )∖{t r 1 }, otherwise we set y ≼ r v = 1. Additionally, we set y ≽ r v = 1 for all v ∈ V r − (o, t r 2 ) and otherwise we set y ≽ r v = 0. Consequently, for i ∈ V r − (o, t r 1 )∖{t r 1 } or j ∈ V r − (t r 2 , w)∖{t r 2 }, the equality y whenever v 1 or v 3 are in V r − (i, j)∖{i, j}. Consequently, a change of v 1 or v 3 does not change f 4 ( , y ≼ r , y ≽ r ). ▪ Proof of Lemma 27. Let z be an optimal solution of (17). By contradiction, we assume that f 3 ( , x ≽ r ) > 0 holds. Lemma 19 implies that does not satisfy Property (c). Consequently, there are entries u 1 , u 2 ∈ V r + (o, w) with u 1 ≺ r u 2 , u 1 < b u 1 , and u 2 > 0. We now choose u 1 such that u = b u holds for all u ∈ V r + (o, u 1 )∖{u 1 } and u 2 such that u = 0 holds for all u ∈ V r + (u 2 , w)∖{u 2 }. Due to the latter, there is an exit v 2 ∈ V r − (u 2 , w) with v 2 > 0 and v = 0 for all v ∈ V r − (v 2 , w)∖{v 2 , w}. Furthermore, we can assume w.l.o.g. that in a flow decomposition, see Lemma 2, q(P r (u 2 , v 2 )) > 0 holds. Due to Lemma 4 and u 1 ≺ r u 2 , we can decrease q(P r (u 2 , v 2 )) and increase q(P r (u 1 , v 2 )) such that Π P r (o,w) (q) stays the same as before the modification and 0 < u 1 ≤ b u 1 , u 2 > 0, v 2 > 0 hold. Thus, the binary variables of z stay the same. Furthermore, by Lemmas 21,22, and 24 the values of f 1 , f 2 , and f 4 stay the same. The modified solution satisfies Constraints (17b). The modification only decreases q a for a ∈ P r (u 2 , v 2 ), increases q a for a ∈ P r (u 1 , u 2 ), and the remaining arc flows stay the same. Hence, since u 1 ≺ r u 2 ≺ r v 2 and by Lemma 4(d), Constraint (17d) is still satisfied. Consequently, z is still feasible for (17). Due to this modification, we increase u 1 > 0 and decrease u 2 . By Lemma 23, considering only the decrease of u 2 does not change the objective value. In contrast, the increase of u 1 decreases f 3 because decreases. Thus, the modification decreases the objective value, which is a contradiction to the optimality of the original solution. ▪ Proof of Lemma 28. Let z be an optimal solution of (17). By contradiction, we assume that f 4 ( , y ≼ r , y ≽ r ) > 0 holds. Lemma 20 implies that does not satisfy Property (d). Consequently, there are exits v 1 , v 2 , v 3 ∈ V r − (o, w)∖{w} with v 1 ≺ r v 2 ≺ r v 3 , v 1 > 0, v 2 < b v 2 , and v 3 > 0. Furthermore, we choose v 1 such that v = 0 holds for all v ∈ V r − (o, v 1 )∖{v 1 }. If q a > 0 for a ∈ in (w) ∩ P r (o, w), we replace v 3 = w. Otherwise, we choose v 3 ≠ w such that v = 0 holds for all v ∈ V r − (v 3 , w)∖{v 3 , w}. We now consider a flow decomposition such as in Lemma 2. Due to q ≥ 0, there is an entry u 3 ∈ V r + (o, v 3 ) with u = 0 for all u ∈ V r + (u 3 , w)∖{u 3 } and q(P r (u 3 , v 3 )) > 0. Furthermore, an entry u 1 ∈ V r + (o, w) with u = 0 for all u ∈ V r + (o, u 1 )∖{o, u 1 } exists which satisfies q(P r (u 1 , v 1 )) > 0. We now distinguish two cases. Case 1: v 2 ≺ r u 3 holds. We now decrease q(P r (u 1 , v 1 )) by > 0 and increase q(P r (u 1 , v 2 )) by the same amount . This increases the potential difference Π P(o,w) (q). Thus, we decrease q(P r (u 3 , v 3 )) bỹ> 0. Due to Lemma 4, we can choose and̃such that Π P(o,w) (q) stays the same and v 1 > 0, 0 < v 2 ≤ b v 2 , u 3 > 0, v 3 > 0 hold. Thus, the binary variables of z stay the same. Furthermore, by Lemmas 21-23, the values of f 1 , f 2 , and f 3 stay the same. The modified solution satisfies