Balanced Min Cost Flow on Skew Symmetric Networks with Convex Costs

We consider the solution of matching problems with a convex cost function via a network flow algorithm. We review the general mapping between matching problems and flow problems on skew symmetric networks and revisit several results on optimality of network flows. We use these results to derive a balanced capacity scaling algorithm for matching problems with a linear cost function. The latter is later generalized to a balanced capacity scaling algorithm also for a convex cost function. We prove the correctness and discuss the complexity of our solution.


Introduction
Skew symmetric networks have become an important tool for the efficient solution of matching problems [1].Going over from a matching problem to a problem of flow optimization often allows for simplification and speed-up of solution algorithms.Typical matching problems do not only include maximizing a matching but often additionally minimizing the costs for the participants involved.Typical examples include minconvexproblems as studied in [2,3].For such minconvex problems a convex function in the number of matchings has to be minimized for each participant.So far the solution of such problems using skew symmetric networks has not been demonstrated.
In this paper we therefore consider the problem of minimizing a separable convex objective function over a skew-symmetric network with a balanced flow.This problem can be mapped on the aforementioned matching problems and allows for an efficient solution of the latter.Specifically, we will derive a balanced capacity scaling algorithm incorporating the additional problem of a convex cost function.
In Section 2 we will shortly review skew symmetric networks and their connection to matchings.Section 3 will be devoted to a short summary of necessary results for optimality.In Section 4 we will present the balanced capacity scaling algorithm for a linear cost function going over in Section 5 to the description of the algorithm for a convex cost function.In Section 6 we will discuss some possible improvements of the aforementioned algorithm.We will conclude and sum up our results in Section 7.

Skew Symmetric Networks and Matchings
 is a pair of disjoint, finite sets E and V corresponding to the edges and vertices of the graph.If the edges between vertices are directed we have a directed graph.This allows for the definition of a network.

 
, G V E  be a directed graph and two functions.We call the triplet a network and the functions v and w the upper and lower capacity bound.
, : with the following properties the vertices of N contain a source s and a drain t and two sets of vertices exist. - the other edges appear pairwise between X and X  meaning if   i j x x exist then so does   i j x x  and vice versa all edges in the network Definition: flownetwork [5, p. 154] Let N be a network and s and t are vertices in   V N such that s is connected to t via edges in .A flow is a mapping .If a flow is defined on N the network is called a flow network.

 
The flow on the network should later allow to map it to a matching.Therefore additional constraints are necessary.The excess is defined as This allows to define an admissible flow.Definition: admissible A flow on a network is admissible if and .
e s e t   Definition: balanced A flow on a skew symmetric network is balanced if and for all loops , x  is even.An admissible flow can always be turned into a circulation, meaning a flow where no excess is found on any vertex.We just have to introduce a vertex from t to s which has a flow value of   e t .Consequently is does not matter whether we consider circulation problems or flows.
Admissible and balanced flows correspond to matchings that we want to define below.
Definition: matching [6, p. 213] Let be a graph, , .A graph with M is called a matching if at least edges and at most edges in M are incident with the vertex i in V and for every   i is incident with j at least This correspondence is illustrated in Figure 1 showing a graph with a matching and the corresponding skew symmetric flow network.This first section summarized the previously known results on the correspondence between matchings and flow optimization which allows for efficient algorithms [5, p. 207].

Optimality of Network Flows
Since we have now reviewed the mapping between matchings and network flows we want to state several important results on the optimality of network flows which can be directly carried over.e.g. a maximal admissible balanced flow on the skew symmetric network corresponds to a maximal matching [7].We start with the following definition.
Definition: restnetwork [7, p. 21] Let N be a flownetwork and x the flow on it.Then the residual capacity corresponding to x is given by:   ij is the backward edge, when the flow is negative.
The edges   ij with together with the vertices that coincide with an edge form the restnetwork So far we have only introduced the correspondence of network flows and matchings.However, we want to compute optimal matchings with respect to a cost function.This means that we have to consider problems on networks N of the type The additional complication compared to maximum balanced flow problems is the cost function.Therefore we want to introduce the necessary framework in order to deal with it.We start by the following Definition: potential, reduced costs [8] The potential function associates with each vertex a number , the potential i.
The reduced costs of an edge are defined as .
 The length of a path is then obviously defined as the sum of the reduced costs of the individual edges.The shortest valid path between two points in a network connects the two points via a valid path and the path has minimum length.
Corollary: [8] For the reduced costs on a network N any path p from a vertex k to a vertex l fulfills and for any circle Obviously we shouldn't try to find a solution by enhancing or decreasing the flow on arbitrary edges but we need a good measure for distance.
Definition: skew symmetric distances Let P i,j be the set of admissible paths in the restnetwork with start-node and endnode .Then we define the skew symmetric distance from i to j as We call the distance to s and the corresponding function is called d.

  d i
Definition: symmetric distances Let d be the set of skew symmetric distances on a flow network N then the symmetric distances are given by The corresponding set is denoted by sd.
At this point we state a theorem that has been extensively used for proving optimality Theorem: Reduced Cost Optimality [8] Let x be an admissible flow then x is optimal if a potential π exists such that for all edges in the restnetwork The potential π is often called the dual solution.It also has a practical importance [9].Let us assume we have a logistics company with several warehouses.The transport costs per unit load between the different warehouses correspond to the costs on the edges.Then the potential for an optimal solution corresponds to the costs per unit We need two further lemmas conce load for storing them in a warehouse.rning the optimality balanced flow on a network N that fulfills th st optimality also w of network flows Lemma: Let x be a e reduced cost optimality with respect to a potential π.Furthermore sd denotes the symmetric distances with respect to the reduced costs π ij c then i) the flow x fulfills the reduced co ith respect to the potential π π sd    ii) the reduced costs π ij c  a h re zero on t e shortest valid pa e both statements one after the other: ths p and p'.

Proof:
We prov i) Since x fulfills the reduced cost optimality . Furthermore since sd is ca d paths we know: We use the definition of the reduced costs ii) Let p be an st path and p' its bijection.For every The same holds for the bijection.Therefore we have Lemma: flow x fulfills the reduced cost optimality on a Assume a flow network N. If we change the flow both on the shortest valid path p and its bijection p' by we find a new flow x' which also fulfills the reduced cost optimality with respect to the potential π π sd    .

Proof:
From the lemma above we know that the reduced costs ar usly, an important ingredient is the solution of th t to the combination of the results of Sec-e zero on p and p' with respect to π'.Therefore the reduced cost optimality cannot be violated if we enhance the flow by the maximum allowed capacity as defined above.
Obvio e shortest valid path problem.This has been discussed in detail in [10] and its complexity is   n , where m is the number of edges and n is the tices in the network.

Balanced Capacity Scaling
ed capacity scaling enote this minimum capacity by Δ and we call ea n 2 and 3 being an algorithm to compute optimal flows on skew-symmetric networks.
In this Section we describe the balanc algorithm for a linear cost function as in Equation (3) with the additional assumption that the costs are always positive.We will later generalize our result to a convex cost function.The idea of capacity scaling is to look at subgraphs in the restnetwork with some minimum capacity and to optimize the flow on these subgraphs successsively.
We d ch phase of the algorithm where Δ does not change a Δ-scaling phase.We define two sets We can first calculate the maximum balanced flow through the network using one of the algorithms in [7] and use the result b as     e s e t b    .We now begin the algorithm with flow 0 such that the reduced cost optimality is fulfilled but the flow does not fulfill Equation (3) as far as the excess is concerned.
For correctly prescribed Δ initially 0 and potential We now denote the balanced capacity scaling algorit begin hm and afterwards prove its correctness.We state the algorithm in a form close to typical programming language.

   
0,π : 0, , : The balance aximal flow x on a skew symmetric network N with minimal costs.

Proof:
We pro ases.
In the The n ced cost optimality is fulfi etwork is skew symmetric in the beginning.Now we assume the soluti aling phase and go over to the Δ-scaling phase.New edges added in the Δ-scaling phase may have negative reduced costs.For them   holds and we can enhance the flow on them by   ij since the costs are negative.In this case they are of the Δ-restnetwork.Since the same has to hold for the bijection the network will be skew symmetric again.Consequently the reduced cost optimality is fulfilled after the first part.
In the s rescap not part econd part we enhance the flow on shortest valid paths so that the reduced cost optimality will be fulfilled.If an edge   lk is introduced with costs the costs are so high that the edge the final solution.We only need to show that the a will not be part of lgorithm obtains an admissible flow for Δ < 1.However, for Δ < 1 no vertex can exists with an excess greater than one.However, due to the correspondence theorem the problem can only have integer flow variations so that no vertex with an excess smaller than one can exist.Therefore we have found an optimal solution which ha solve the Balanced M

Convex Costs neralization of the problem in (15)
The function is a mapping from s to be maximal since e(s) was assumed to be the maximal flow through the network.
We therefore have an algorithm to in Cost Flow Problem on skew symmetric networks.In the next section we want to discuss the additional complication of a problem with a convex cost functions.
We first define the ge Equation (3) with a convex cost function: know that for all , , , , 1 x y D bers and we For a balanced capacity scaling algorithm for t le ion by lin h step we double the number of po nsidered th he probm in Equation ( 15) we follow [11, pp. 556-560].The idea is to approximate the convex cost funct ear interpolation.This interpolation can be improved step by step until it is exact for all integer values as illustrated in Figure 2.
In the figure in eac ints in between which we assume the function to be constant.The frequency polygon therefore becomes a better approximation of the original function until we approximate the function at each integer value.
Since only integer flow values need to be co e solution will be exact.In every Δ-scaling phase only changes of the flow values by   ,0,   need to be considered.Therefore we define ity and cost function for a Δ-scaling phase as: In the previous algorithm we additionally defined c max w work can have costs of: hich we have to do here as well.No edge in the net-   e s rough is chosen again to be the maxim th the network.We denote now the algorithm and prove i um flow value ts correctness for solving problems of the type as in Equation ( 15) afterwards.begin hen ease flow on t and on   reduce flow on   ij and on   However, from the definition of a convex cost function we know and we have the disagreement.The case i) fulfills the reduced cost optimality and we ii) we resolve this issue by are left with ii) and iii).In case enhancing the flow by Δ flow units.After the 2Δscaling phase we have 23)   and we have From the inequality from the 2Δ-scaling phase follows and the inequality follows since the last line of E (25) is smaller than zero by Equation ( 24) and conse-quation quently Equation ( 23) follows.
The reduced costs have to be identical on   j i   since the network is skew symmetric.We can treat case iii) completely analogous.
Everything we still need to show is that the reduced cost optimality is also fulfilled if we enhance the flow by   balcap p and not by Δ.We consider π 0 ij c  and we obtain for the reduced costs Consequently the reduced cost optimality is still fulfilled.The same can be shown for the edge   j i   .
roblem At the end we will therefore obtain an admissible flow which is the solution to the Min Cost Flow P on a skew symmetric network with a convex cost function.
We will finally get to the complexity of this problem using the above algorithm.

Theorem
The balanced capacity scaling algorithm has a complexity of so that the complexity follows.

Possible Improvements
Every balanced admissible flow on a skew symmetric network corresponds to a matching on the corresponding graph with

Figure 1 .
Figure 1.The left side shows a graph with vertices that are matched once (thick lines) and not matched (thin lines).On the left side the corresponding skew symmetric flow network is shown with thick lines corresponding to flows of 1.
connect the two with an edge by induction in the Δ-scaling ph beginning we have so that the redu on to be optimal in the 2Δsc d;
20) so that the reduced cost optimality is fulfilled.Now let us assume that x is the flow after the 2 ow nction -scaling phase.We have the following cases for the fl in since we woul e the Δ-scaling phase: i):

T
computing a shortest valid path.Proof:e end of each 2Δ-scaling phase we haveS    or   2 S   so that at most there is 2n of flo tionally we w that can be shifted along in the next phase.Addichang t the beginning of the Δscaling phase by at most 2m so that the total excess can be at most e the flow a [3] A. Berger and W. Hochstättler, "Minconvex Graph Factors of Prescribed Size and a Simpler ´Reduction to Weighted f-Factors," Electronic Not gorithm for shortest paths.An implementation with Fi-) ly has to use the more involved shortest valid path algorithm in the last phase.The author has also tried to implement other methods used in networks with a conv n like the Out-of-Kilter algorithm, the Relaxation Algorithm, Cancel-and-Tighten and the primal-dual al Vol. 28, 2007, pp.69-76.doi:10.1016/j.endm.2007.01.011