Multicommodity Network Flow - Methods and Applications

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1 Multicommodity Network Flow - Methods and Alications Joe Lindström Linköing University joeli89@student.liu.se Abstract This survey describes the Multicommodity Network Flow roblem, a network flow roblem where several commodities must share resources in a common caacitated network. The roblem has many interesting and imortant alications and variations. The survey tries to cover the most imortant solution methods, all of which exloit the underlying network flow substructure. 1. Introduction In the standard Minimum Cost Network Flow (MCF) roblem the object is to send a flow through a network from some set of source nodes to a set of sink nodes, in some otimal fashion. For a solution to be feasible it must comly with the mass balance constraints for each node, i.e. the flow in to a node must equal the flow out of the node (excet for source and sink nodes), and the flow on each arc must not exceed the arc s limit. In the Multicommodity Minimum Cost Network Flow (MMCF) roblem several entities, or commodities, share the network. Each commodity has its own mass balance constraints, but the arc caacity constraints are shared. Note that this is a generalization of MCF, which can be seen as MMCF with only one commodity. This survey is restricted to the standard linear MMCF, where all constraints and costs are linear functions. Even though MMCF can be formulated as a linear rogram, and thus solved in olynomial time with general urose LP solvers, it has been difficult to use common LP techniques to solve it efficiently. In fact, MMCF is one of the few linear roblems where aroximation algorithms are of ractical interest [4]. This combined with its many alications makes MMCF an interesting roblem to study. Most, if not all, solution methods utilize ideas from linear rogramming. This survey does not go in to any advanced details, but some familiarity with linear rogramming and network flows would be favourable, esecially when reading art 3 about solution methods Model formulation Let be a directed grah consisting of a set of nodes and a set of directed arcs. Let denote the node-arc incidence matrix for this grah (i.e. is an matrix where is +1 if edge j goes away from node i, -1 if edge j enters node i, and 0 otherwise). The flow of commodity k on arc is denoted while the whole flow vector for commodity k is denoted, and its corresonding cost vector!. " is the number of commodities. The MMCF roblem is to minimize *),+ -/ :9 ) 5 ) " <;>=.65?7:9 ) 5 ) ". where 3 is the caacity of arc and is a suly vector for commodity k (entry in this vector is the amount of commodity k node sends out, it is negative for sink nodes and zero for nodes that are neither source nor sink nodes). Note that each commodity has its own suly vector, but that the arc caacity constraints (also called bundle constraints) are shared. In some alications there are also individual flow bounds for each commodity-arc air, so that = ) <)>+ -.65?7:9 ) 5 ) " There are many variations of this model that turn out to be equivalent, like allowing for caacitated nodes or lower limits on arcs. They can be reduced to the above model using the same techniques as for MCF (see [1] for an in-deth exlanation of these methods). There are also some interesting variations which result in harder roblems, e.g. imosing integrality constraints on or constraints on aths [6].

2 g 3 Unlike MCF, an otimal solution to MMCF is not necessarily integral, which is required in many alications. This is illustrated in Figure 1. There are three commodities, we wish to shi as much as ossible through the network (so this is a maximum flow roblem). Each arc has caacity 1. The otimal solution shis 0.5 units between each source/sink-air, for a total of 1.5 units, while the otimal integral solution shis 1 unit between one source/sink-air. Converting MMCF to a maximum flow roblem is done by setting all suly and demand to zero, and adding an artificial arc going from each sink to the corresonding source. This arc should have a negative cost, thereby forcing the solution to have a circulation of maximum amount of flow. Figure 1. Maximum total flow is 1.5 units 1.. Comlexity As stated above MMCF can be formulated as a linear rogram, and is consequently solvable in olynomial time [8]. With side constraints on aths the roblem becomes,g -hard. The standard MMCF only has constraints on its nodes and arcs. With the extended model we can also require that every ath in the solution have certain qualities, for more information, see [6]. The same haens when the solution is required to be integral [5]. The LP formulation can still serve a urose in these variations however, either as an aroximation or as a relaxation in a branch-and-bound algorithm Otimality conditions With the hel of LP duality theory, we can deduce some results about otimal feasible solutions. In articular we can deduce formulas for verifying whether or not a feasible solution is otimal. Duality theory can also hel in verifying correctness of algorithms and even give us a hint to how good algorithms may be develoed. The dual of MMCF is to maximize: P RXO ZY U H I KJLNM + POQSR T%(' VUW [ R U \0 ;,=. S1] O^ Note that in this formulation of the dual there are two different kinds of dual variables, O, which is the arc rice, corresonding to the bundle constraints in the rimal, and U (i), the node otential for commodity k on node i, corresonding to the mass balance constraints in the rimal. Each variable in the dual corresonds to a constraint in the rimal. Given a feasible solution for the rimal and dual, the comlementary slackness theorem states that if and only if the solutions are otimal, the roduct of each variable and its corresonding constraint s slack 1 is zero [8]. In the MMCF roblem this becomes O T%(' - Y + a - RbO -cy U [ R U \0K d =. So these two conditions are enough to verify that a feasible solution is otimal. They can also be used to rove the following theorem: Theorem 1 (Partial Dualization) Let e be an otimal solution to MMCF and let O - denote otimal arc rices. Then for each commodity k, the flow variables f solve the following (uncaacitated) MCF roblem: Minimize H I KJWLNM a h3 d *;=. 0S1F g where the cost vector VcRXO For a roof, see [1] This theorem can be used in an iterative method where otimal arc rices are first found, and then otimal node 1 The slack in a constraint of the form ikj?l is lnmoi

3 otentials are searched for with the hel of MCF. Or actually the single source shortest ath roblem since the MCF roblem in question is uncaacitated (uncaacitated MCF equals shortest ath when there is only one source and one sink, but we can easily convert any multi-source/sink MMCF to a single-source/sink MMCF with the hel of suer-sources/sinks as illustrated in [3]). This is encouraging since there are efficient methods for solving the shortest ath roblem.. Alications Since MMCF is a generalization of MCF, which can model a wide variety of combinatorial roblems, it should come as no surrise that MMCF also has many interesting alications. The last decade, articles have been ublished mainly in these areas []: Transortation and logistics This is erhas the most obvious alication, where commodities corresond to goods, messages, vehicles etc. and the network corresonds to road mas, comuter networks, ieline systems and so on. Design of integrated circuits Alications include cell layout and circuit clustering. Scheduling, manufacturing and lanning In these alications time comes in to lay and the network is often artitioned into timestes. Telecommunication network design and management Minimizing the traffic, or distributing it evenly over a telecommunications network can be modeled as a MMCF roblem. In these alications MMCF is however not the main roblem but rather an evaluation for the real roblem, which is to construct a well functioning network..1. A transortation examle Assume that the owner of two factories wish to shi goods to two buyers through a railroad system. Different routes in the railroad system have different amount of traffic, the amount of trains and the size of the cars running between location A and B decides how much the owner can send that way. Also, different trains have different rices er unit of shied goods. Determining how to shi the goods in the cheaest way can easily be seen as a MMCF roblem. Locations in the railroad system corresond to nodes in the grah, railroads connecting the locations corresond to arcs. The number of trains and size of cars running between two locations corresond to arc caacities, and the shiing rices corresond to arc costs. If the goods shied are oil, rice or wheat, the standard LP suffices. If it s comuters, cars or something else indivisible we need an integral solution... A combinatorial examle Given a set A of arbitrary ositive integers, construct a sequence so that each element k in the set occurs twice, k laces aart. This is a generalization of classical skolem sequences [7]. For examle, given the set q, 3, 5, 6r, we can construct the sequence This can be modeled as an MMCF roblem. We construct the network as follows. For each lace in the sequence we have one node. We also have two extra nodes which serve as source and sink nodes for all k commodities, one commodity for each integer in A. For a node corresonding to lace i, we create arcs connecting it to nodes corresonding to laces s i. If there is an integer in A, we connect node i and i- with an arc, so that the commodity corresonding to has cost zero, and all others a ositive cost. We also connect the source node to all nodes corresonding to laces, and all those nodes to the sink node. These arcs can have infinite caacity and zero cost. The idea is to force the flow for integer through two nodes laces aart. To make sure that the flow corresonds to a correct sequence each lace must be used exactly once. This can be done by setting suly and demand to 1 for all commodities, requiring integral solutions and setting a caacity of 1 on all the nodes corresonding to laces. We also have to make sure that all aths have length 3, otherwise the flow could cheat and go straight to the source, only assing one of the laces. So this becomes an integer MMCF with side constraints on aths and caacitated nodes. A solution corresond to a skolem sequence if the cost of the flow is zero, otherwise no skolem sequence exists for the given set. To illustrate this construction, consider the set q 1,3r. The network would have two commodities and 6 nodes. In figure only the arcs between internal nodes are drawn, and their rices for both commodities. The only solution with cost zero is to send commodity 1 through nodes and 1, and commodity through nodes 3 and 0. This corresonds to lacing the integer 1 in lace 1 and, and lacing 3 in lace 0 and 3, resulting in the skolem sequence Solution methods Many suggested methods for solving the MMCF roblem use some kind of decomosition technique. Others are secializations of common linear rogramming methods. The three most common categories are Price-directive decomosition

4 x = 9 Figure. MMCF network for the Skolem set q 1, 3r This means comlicating constraints are removed and rices are introduced in their lace. Resource-directive decomosition These methods give each commodity a certain amount of caacity for each arc, and then solves the resulting single-commodity roblems reeatedly. Partitioning methods Here we use linear rogramming and secialize the simlex algorithm to utilize the secial structure of MMCF. Below is a rice-directive method (Lagrangian relaxation) described in some detail and the other two categories more briefly Lagrangian relaxation The basic idea behind Lagrangian relaxation is to remove comlicating constraints and introduce rices instead which enalizes solutions which aren t feasible in the original roblem. This results in a simler roblem which often can be solved efficiently (such as the shortest ath roblem). The hard art is to enalize correctly, i.e. setting the rices. If the rice is too high we won t find an otimal solution to the original roblem, if they are too low we will find unfeasible solutions. For both otimality and feasibility they need to be just right, this leads to solving the relaxed roblem over and over, each time modifying the rices. For an introduction to Lagrangian relaxation, see [1]. The rest of this section assumes basic knowledge of the technique. A natural choice when alying Lagrangian relaxation to MMCF is to move the bundle constraints to the objective function, i.e. introduce Lagrangian multiliers for each bundle constraint. This will result in a articularly nice roblem. For each arc/commodity-air, the rice now be- comes a RbO^- (t d instead ofu - n -. O^ is the multilier. Since the bundle constraints are gone, this is equivalent to k uncaacitated MCF roblems. More formally, the relaxed roblem is to minimize H I KJvLNM RXO - Y H I KJLNM 43 w.6587:9 ) 5 ) " n - O - + The only thing which formally distinguishes this from k uncaacitated MCF roblems is the last summation in the objective function, but for a given value of Lagrange multiliers this exression is constant and does not influence the otimal solution. So the algorithm roceeds as follows (with subgradient otimization) 1 Choose 4yaz[{K? an arbitrary vector of Lagrange multiliers O - set For each commodity solve the uncaacitated MCF with the multiliers added to the arc rices, the value obtained in this ste is a lower bound on the original MMCF. 3 If the combination of the k subroblems is a feasible solution to the original MMCF roblem, and the comlementary slackness conditions hold, STOP. 4 Set O - F} O - R~x is the combined solution. 5 Set x 9 yaz[{ 6 Set yaz[{ ƒ4yaz[{ R 9 _ 6 7 If yaz[{k s,, go to, otherwise STOP. Y +, where is a limit on the number of iterations to ensure that the algorithm terminates. This will not necessarily converge to an otimal feasible solution. However it is often successful when we want a solution within n of otimum, and it can also be used together with branch-and-bound (serving as a lower-bound) or simlex (sulying it with an advanced start basis).

5 Other methods belonging to the rice-directive decomosition category differ in how they aly rices and how rices are udated. For column-generation aroaches, see [1], [] and [6], for Dantzig-Wolfe decomosition see [1] and [8], and for a Bundle tye algorithm (which remembers old stes in the subgradient calculation) see [4]. 3.. Resource-directive decomosition The idea here is to allocate a certain amount of caacity on each arc to each commodity. Of course we have to make sure that the total amount of caacity allocated does not exceed the arc s caacity in the original MMCF roblem. With this allocation done the resulting roblem is easy to solve, it is simly k standard MCF roblems (not uncaacitated this time). Finding an otimal feasible solution to the original roblem involves adjusting the allocation. So this method also involves solving several easier subroblems reeatedly. The various algorithms differ in how they udate the allocation. A simle aroach is to choose two commodities and one arc, and give some of the first commodity s allocated resource to the other, while keeing all other resources unchanged. A greedy aroach would choose the two commodities which give the largest decrease in the objective function. This method does not guarantee convergence. A method which does guarantee convergence is to use subgradient otimization. For an introduction to this subject, see either [1] or [] Partitioning methods Instead of solving MCF with a general urose LP solver like simlex, a secialized version of simlex has been develoed called network simlex. This algorithm uses sanning trees as basis instead of a general matrix, and ivots with hel of grah oerations such as tree traversal. We can use a similar aroach to seed u simlex for MMCF. So called basis artitioning methods exloit the structure of the basis matrices to erform art of the comutation with network oerations and art of it with general linear rogramming methods. For an introduction, see [1] and []. 4. Current research 5. Conclusion The multicommodity flow roblem forms a challenging art of linear rogramming and combinatorial otimization. It has insired many decomosition methods, which are useful in other domains as well. Because of its structure it is theoretically interesting, and because of its owerful modelling caabilities it has a large number of alications. This makes MMCF an interesting research area. References [1] Ravindra K. Ahuja, Thomas L. Magnanti, James B. Orlin, Network Flows: Theory Algorithms and Alications, 1993 [] P. Chardaire, A. Lisser, Minimum cost multicommodity flow, Handbook on alied otimisation, Resende & Parados éditeurs, 000 [3] Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein, Introduction to Algorithms Second Edition, 001 [4] Frangioni Antonio Gallo Giorgio, A Bundle tye Dualascent Aroach to Linear Multi-Commodity Min Cost Flow Problems, INFORMS Journal of Comuting, 11(4), 1999, [5] Frangioni Antonio Gallo Giorgio Dual Ascent Methods and Multicommodity Flow Problems, Ph.D Dissertation TD 5/97, 1997 [6] Kaj Holmberg, Di Yuan, A Multicommodity Network- Flow Problem with Side Constraints on Paths Solved by Column Generation, INFORMS Journal of Comuting, 15, 003, 4-57, [7] Gustav Nordh, Generalization of Skolem sequences, Master s Thesis LITH-MAT-EX-003/05, Deartment of Mathematics, Linköings universitet, 003 [8] Christos H. Paadimitriou, Kenneth Steiglitz, Combinatorial Otimization: Algorithms and Comlexity, 1998 Most of the current research seem to be focused on the three categories listed in section 3. There is also some work on strictly enumerative aroaches, esecially for integer MMCF, such as genetic algorithms. Much work is done on various variations of the standard MMCF roblem, which are suitable for secific alications in oerations research, such as telecommunications networks or aircraft routing and scheduling.