Again we study two different versions of the problem under consideration. These are the abstract single commodity k-splittable flow problem (k-SFP) and the abstract multicommodity k-SFP. The relation between the stan- dard single commodity k-SFP and the abstract one is that we assume in the abstract setting that (PR) is fulfilled for the paths in P. In the multicom- modity case, we are again given a partition of P and do not require that (PR) is obeyed.
In addition to the set of edges E with its capacity function u and the family of pathsP with its partition P1, . . . ,PK and the demands d1, . . . dK, an
instance of the abstract multicommodity k-SFP consists of integral numbers ki, for i = 1, . . . , K, that specify how many paths may be used to route
commodity i. The task is to find paths Pi
1, . . . , Pkii, for each commodity i,
with corresponding nonnegative flow values fi
1, . . . , fkii such that we meet the
usual requirements of the k-SFP:
ki X j=1 fji = di for i = 1, . . . , K, and K X i=1 X j=1,...,ki: e∈Pi j
We do not require that the paths Pi
1, . . . , Pkii are distinct, for i = 1, . . . , K.
Furthermore, we allow a path to have flow value 0. In other words, we may use less than ki paths for commodity i.
The single commodity case of the k-SFP is the one in which we have only one commodity, i.e., K = 1. Further, we require that P obeys (PR). In this case, we omit the “i-index”, when denoting the input.
Since all other notions are also used as defined in Section 3.2, we desist from further explanations here and refer to that section.
5.3.1 The Abstract Maximum Single Commodity k-SFP
Since in this section we only study the case that we have exactly one com- modity, we omit the term “single commodity” for brevity of presentation. Further, we neglect the given demand, because our objective here is to find a k-splittable flow of maximum value.
Realize that in the abstract setting we can still assume without loss of generality that k ≤ m. This holds, because for each abstract flow x there is (another) flow of the same value that uses at most m paths and sends no more than x(e) units of flow along e∈ E. This follows immediately from the theory of linear programming, because we have only m constraints in the LP formulation of the abstract maximum flow problem—besides nonnegativity constraints—or, in other words, each basic solution to the problem has at most m non-zero entries.
We start considering the problem to find a maximum abstract uniform exactly-k-splittable flow. Since the results by Baier, K¨ohler, and Skutella [11] are mainly based on standard integral maximum flow computations, we can transmit them to the abstract setting by involving McCormick’s algorithm. If the maximum flow value D was known, our problem would reduce to finding a (D/k)-integral flow of value D. This problem can simply be solved by McCormick’s algorithm. Unfortunately, we do not know the maximum flow value. But realize that in a maximum uniform exactly-k-splittable flow at least one edge e ∈ E must be saturated. Let us say that e is used by i ∈ {1, . . . , k} paths, then D/k = u(e)/i. Thus, at most |E|k values are possible for D/k and we need to compute just as many integral maximum flows using McCormick’s algorithm.
To find a maximum abstract uniform k-splittable flow, i.e., one that spreads flow evenly among at most (not necessarily exactly) k paths, one simply has to find the maximum flow of all uniform exactly-i-splittable ones, for i = 1, . . . , k.
flow can be computed in polynomial time.
We can also transmit the Max-Flow-Min-Cut Theorem given in [11] to the abstract setting. As already defined, a cut is given by a set of edges C such that P ∩ C 6= ∅, for all P ∈ P. Baier et al. [11] define the k-uniform cut capacity uk(C) as the maximum volume of a packing of k identically sized
packages into bins having sizes equal to the capacities of the edges in C. This definition can exactly be adopted to abstract networks.
Theorem 5.4. The minimum k-uniform cut capacity equals the value of a maximum abstract uniform single commodity exactly-k-splittable flow. Proof. The proof is analogous to the one given in [11]. Let c∗be the minimum
k-uniform cut capacity of an underlying instance for the considered problem. It is obvious that c∗ yields an upper bound for the value v∗ of a maximum
abstract uniform exactly k-splittable flow.
We have to prove that v∗ ≥ c∗. To do so, we round all edge capacities
down to the nearest multiple of c∗/k. This can only decrease v∗. The capacity
of a standard minimum cut is now at least c∗and by McCormick’s algorithm
we can find a (c∗/k)-integral flow of value c∗. This flow is routed on at most k paths. Thus, the theorem is proven.
Baier et al. [11] prove that the value of a maximum uniform exactly-k- splittable flow is at least half of the value of a maximum k-splittable flow. Since the proof can exactly be transmitted to the abstract setting, we obtain the following result.
Theorem 5.5. There exists a 2-approximation algorithm for the maximum abstract single commodity k-splittable flow problem.
5.3.2 The Abstract Multicommodity k-SFP
Again, we consider the optimization problem to find a solution of minimum congestion. To approximate optimal solutions to the standard multicommod- ity k-SFP, Baier et al. [11] use approximation algorithms for the UFP. We still assume without loss of generality that ki ≤ m, for all i ∈ {1, . . . , K}. It
follows that we can generalize the result in [11], which has already been men- tioned in Section 3.1, that any ρ-approximation algorithm for the UFP yields a ρ-approximation algorithm for the uniform exactly-k-SFP to the abstract model. Thus, we obtain the following theorem by applying the fact that in the multicommodity case the minimum congestion of an optimal uniform exactly-k-splittable flow is at most twice as large as the minimum congestion of an optimal k-splittable flow (see also [11]).
Theorem 5.6. Any ρ-approximation algorithm for the minimum congestion version of the abstract unsplittable flow problem yields a 2ρ-approximation algorithm for the minimum congestion version of the abstract k-spittable flow problem.
Since our results in Section 3.3 are also based on solving the UFP, they can be transmitted to the abstract setting without any changes. Thus, we immediately obtain approximation algorithms for the minimum congestion version of the abstract k-SFP with path capacities. These approximate the underlying problems within a factor 2ρ where ρ is any approximation factor for the minimum congestion version of the abstract UFP.