On the History of the Transportation

and Maximum Flow Problems

Alexander Schrijver

Abstract. We review two papers that are of historical interest for combinatorial optimization: an article of A.N. Tolstoĭ from 1930, in which the transportation problem is studied, and a negative cycle criterion is developed and applied to solve a (for that time) large-scale (10 × 68) transportation problem to optimality; and an, until recently secret, RAND report of T. E. Harris and F. S. Ross from 1955, that Ford and Fulkerson mention as motivation to study the maximum flow problem. The papers have in common that they both apply their methods to the Soviet railway network.

2010 Mathematics Subject Classification: 01A60, 05-03, 05C21, 05C85, 90C27 Keywords and Phrases: Maximum flow, minimum cut, transportation, algorithm, cycle cancelling, history

1 Transportation

The transportation problem and cycle cancelling methods are classical in optimization. The usual attributions are to the 1940’s and later^{1}. However, as early as 1930, A.N. Tolstoĭ [1930]^{2} published, in a book on transportation planning issued by the National Commissariat of Transportation of the Soviet Union, an article called Methods of finding the minimal total kilometrage in cargo-transportation planning in space, in which he studied the transportation problem and described a number of solution approaches, including the, now well-known, idea that an optimum solution does not have any negative-cost cycle in its residual graph^{3}. He might have been the first to observe that the cycle condition is necessary for optimality. Moreover, he assumed, but did not explicitly state or prove, the fact that checking the cycle condition is also sufficient for optimality.

1The transportation problem was formulated by Hitchcock [1941], and a cycle criterion for optimality was considered by Kantorovich [1942] (Kantorovich and Gavurin [1949]), Koopmans [1948] (Koopmans and Reiter [1951]), Robinson [1949, 1950], Gallai [1957, 1958], Lur’e [1959], Fulkerson [1961], and Klein [1967].

2Later, Tolstoĭ described similar results in an article entitled Methods of removing irrational transportations in planning [1939], in the September 1939 issue of Sotsialisticheskiĭ Transport.

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Tolstoĭ illuminated his approach by applications to the transportation of salt, cement, and other cargo between sources and destinations along the railway network of the Soviet Union. In particular, a, for that time large-scale, instance of the transportation problem was solved to optimality.

We briefly review the article. Tolstoĭ first considered the transportation problem for the case where there are only two sources. He observed that in that case one can order the destinations by the difference between the distances to the two sources. Then one source can provide the destinations starting from the beginning of the list, until the supply of that source has been used up. The other source supplies the remaining demands. Tolstoĭ observed that the list is independent of the supplies and demands, and hence it

is applicable for the whole life-time of factories, or sources of production. Using this table, one can immediately compose an optimal transportation plan every year, given quantities of output produced by these two factories and demands of the destinations.

Next, Tolstoĭ studied the transportation problem in the case when all sources and destinations are along one circular railway line (cf. Figure 1), in which case the optimum solution is readily obtained by considering the difference of two sums of costs. He called this phenomenon circle dependency.

Finally, Tolstoĭ combined the two ideas into a heuristic to solve a concrete transportation problem coming from cargo transportation along the Soviet railway network. The problem has 10 sources and 68 destinations, and 155 links between sources and destinations (all other distances are taken to be infinite), as given in the following table.

Tolstoĭ’s heuristic also makes use of insight in the geography of the Soviet Union. He goes along all sources (starting with the most remote sources), where, for each source X, he lists those destinations for which X is the closest source or the second closest source. Based on the difference of the distances to the closest and second closest sources, he assigns cargo from X to the destinations, until the supply of X has been used up. (This obviously is equivalent to considering cycles of length 4.) In case Tolstoĭ foresees a negative-cost cycle in the residual graph, he deviates from this rule to avoid such a cycle. No backtracking occurs.

In the following quotation, Tolstoĭ considers the cycles Dzerzhinsk-Rostov-Yaroslavl’-Leningrad-Artemovsk-Moscow-Dzerzhinsk and Dzerzhinsk-Nerekhta-Yaroslavl’-Leningrad-Artemovsk-Moscow-Dzerzhinsk. It is the sixth

3The residual graph has arcs from each source to each destination, and moreover an arc from a destination to a source if the transport on that connection is positive; the cost of the ‘backward’ arc is the negative of the cost of the ‘forward’ arc.

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Transportation and Maximum Flow Problems 171

Table 1: Table of distances (in kilometers) between sources and destinations, and of supplies and demands (in kilotons). (Tolstoĭ did not give any distance for Kasimov. We have inserted a distance 0 to Murom, since from Tolstoĭ’s solution it appears that Kasimov is connected only to Murom, by a waterway.)

Arkhangelsk Yaroslavl’ Murom Balakhonikha Dzerzhinsk Kishert’Sverdlovsk Artemovsk Iledzhk Dekonskaya Demand
Agryz  709  1064  693  2
Aleksandrov  397  1180  4
Almaznaya  81  65  1.5
Alchevskaya  106  114  4
Baku  1554  1563  10
Barybino  985  968  2
Berendeevo  135  430  10
Bilimbai  200  59  1
Bobrinskaya  655  663  10
Bologoe  389  1398  1
Verkhov’e  678  661  1
Volovo  757  740  3
Vologda  634  1236  2
Voskresensk  427  1022  1005  1
V.Volochek  434  1353  1343  5
Galich  815  224  1056  0.5
Goroblagodatskaya  434  196  0.5
Zhlobin  882  890  8
Zverevo  227  235  5
Ivanovo  259  6
Inza  380  735  1272  2
Kagan  2445  2379  0.5
Kasimov  0  1
Kinel’  752  1208  454  1447  2
Kovylkino  355  1213  2
Kyshtym  421  159  3
Leningrad  1237  709  1667  1675  55
Likino  223  328  15
Liski  443  426  1
Lyuberdzhy  268  411  1074  1
Magnitogorskaya  932  678  818  1
Mauk  398  136  5
Moskva  288  378  405  1030  1022  141
Navashino  12  78  2
Nizhegol’  333  316  1
Nerekhta  50  349  5
Nechaevskaya  92  0.5
N.-Novgorod  32  25
Omsk  1159  904  1746  5
Orenburg  76  1.5
Penza  411  1040  883  1023  7
Perm’  1749  121  1
Petrozavodsk  1394  1
Poltoradzhk  1739  3085  1748  4
Pskov  1497  1505  10
Rostov/Don  287  296  20
Rostov/Yarosl  56  454  2
Rtishchevo  880  863  1
Savelovo  325  1206  1196  5
Samara  711  495  1406  7
San-Donato  416  157  1
Saratov  1072  1055  15
Sasovo  504  1096  1079  1
Slavyanoserbsk  119  115  1.1
Sonkovo  193  1337  0.5
Stalingrad  624  607  15.4
St.Russa  558  1507  1515  5
Tambov  783  766  4
Tashkent  3051  1775  3
Tula  840  848  8
Tyumen’  584  329  6
Khar’kov  251  259  60
Chelyabinsk  511  257  949  2
Chishmy  1123  773  889  0.5
Shchigry  566  549  4
Yudino  403  757  999  0.5
Yama  44  52  5
Yasinovataya  85  93  6
Supply  5  11.5  8.5  12  100  12  15  314  10  55  543

Figure 1: Figure from Tolstoĭ [1930] to illustrate a negative cycle

step in his method, after the transports from the factories in Iletsk, Sverdlovsk, Kishert, Balakhonikha, and Murom have been set:

6. The Dzerzhinsk factory produces 100,000 tons. It can forward its production only in the Northeastern direction, where it sets its boundaries in interdependency with the Yaroslavl’ and Artemovsk (or Dekonskaya) factories.

From Dzerzhinsk From Yaroslavl’ Difference to Dzerzhinsk

Berendeevo  430 km  135 km  −295 km
Nerekhta  349 ,,  50 ,,  −299 ,,
Rostov  454 ,,  56 ,,  −398 ,,

From Dzerzhinsk From Artemovsk Difference to Dzerzhinsk

Aleksandrov  397 km  1,180 km  +783 km
Moscow  405 ,,  1,030 ,,  +625 ,,

The method of differences does not help to determine the boundary between the Dzerzhinsk and Yaroslavl’ factories. Only the circle dependency, specified to be an interdependency between the Dzerzhinsk, Yaroslavsl’ and Artemovsk factories, enables us to exactly determine how far the production of the Dzerzhinsk factory should be advanced in the Yaroslavl’ direction.

Suppose we attach point Rostov to the Dzerzhinsk factory; then, by the circle dependency, we get:

Dzerzhinsk-Rostov  454 km −398 km Nerekhta 349 km −299 km
Yaroslavl’- ,, 56 ,, ,, 50 ,,
Yaroslavl’-Leningrad  709 ,, +958 ,, These points remain
Artemovsk- ,, 1,667 ,, unchanged because only the
Artemovsk-Moscow  1,030 ,, −625 ,, quantity of production sent
Dzerzhinsk- ,,  405 ,,  by each factory changes
Total  −65 km  +34 km

Therefore, the attachment of Rostov to the Dzerzhinsk factory causes over-run in 65 km, and only Nerekhta gives a positive sum of differences and hence it is the last point supplied by the Dzerzhinsk factory in this direction.

As a result, the following points are attached to the Dzerzhinsk factory:

N. Novgorod  25,000 tons
Ivanova  6,000 ,,
Nerekhta  5,000 ,,
Aleksandrov  4,000 ,,
Berendeevo  10,000 ,,
Likino  15,000 ,,
Moscow 35,000 ,, (remainder of factory’s production)
Total 100,000 tons

After 10 steps, when the transports from all 10 factories have been set, Tolstoĭ “verifies” the solution by considering a number of cycles in the network, and he concludes that his solution is optimum:

Thus, by use of successive applications of the method of differences, followed by a verification of the results by the circle dependency, we managed to compose the transportation plan which results in the minimum total kilometrage.

The objective value of Tolstoĭ’s solution is 395,052 kiloton-kilometers. Solving the problem with modern linear programming tools (CPLEX) shows that Tolstoĭ’s solution indeed is optimum. But it is unclear how sure Tolstoĭ could have been about his claim that his solution is optimum. Geographical insight probably has helped him in growing convinced of the optimality of his solution. On the other hand, it can be checked that there exist feasible solutions that have none of the negative-cost cycles considered by Tolstoĭ in their residual graph, but that are yet not optimum^{4}.

4The maximum objective value of a feasible solution, whose residual graph does not contain any nonnegative-cost cycle of length 4, and not any of the seven longer nonnegativelength cycles considered by Tolstoĭ (of lengths 6 and 8), is equal to 397,226.

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2 Max-Flow Min-Cut

The Soviet rail system also roused the interest of the Americans, and again it inspired fundamental research in optimization.

In their basic paper Maximal Flow through a Network (published first as a RAND Report of November 19, 1954), Ford and Fulkerson [1954] mention that the maximum flow problem was formulated by T. E Harris as follows:

Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, find a maximal flow from one given city to the other.

In their 1962 book Flows in Networks, Ford and Fulkerson [1962] give a more precise reference to the origin of the problem:^{5}

It was posed to the authors in the spring of 1955 by T. E. Harris, who, in conjunction with General F. S. Ross (Ret.), had formulated a simplified model of railway traffic flow, and pinpointed this particular problem as the central one suggested by the model [11].

Ford-Fulkerson’s reference 11 is a secret report by Harris and Ross [1955] entitled Fundamentals of a Method for Evaluating Rail Net Capacities, dated October 24, 1955^{6} and written for the US Air Force. At our request, the Pentagon downgraded it to “unclassified” on May 21, 1999.

As is known (Billera and Lucas [1976]), the motivation for the maximum flow problem came from the Soviet railway system. In fact, the Harris-Ross report solves a relatively large-scale maximum flow problem coming from the railway network in the Western Soviet Union and Eastern Europe (‘satellite countries’). Unlike what Ford and Fulkerson say, the interest of Harris and Ross was not to find a maximum flow, but rather a minimum cut (‘interdiction’) of the Soviet railway system. We quote:

Air power is an effective means of interdicting an enemy’s rail system, and such usage is a logical and important mission for this Arm.

As in many military operations, however, the success of interdiction depends largely on how complete, accurate, and timely is the commander’s information, particularly concerning the effect of his interdiction-program efforts on the enemy’s capability to move men and supplies. This information should be available at the time the results are being achieved.

5There seems to be some discrepancy between the date of the RAND Report of Ford and Fulkerson (November 19, 1954) and the date mentioned in the quotation (spring of 1955).

6In their book, Ford and Fulkerson incorrectly date the Harris-Ross report October 24, 1956.

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The present paper describes the fundamentals of a method intended to help the specialist who is engaged in estimating railway capabilities, so that he might more readily accomplish this purpose and thus assist the commander and his staff with greater efficiency than is possible at present.

First, much attention is given in the report to modeling a railway network: taking each railway junction as a vertex would give a too refined network (for their purposes). Therefore, Harris and Ross propose to take ‘railway divisions’ (organizational units based on geographical areas) as vertices, and to estimate the capacity of the connections between any two adjacent railway divisions. In 1996, Ted Harris remembered (Alexander [1996]):

We were studying rail transportation in consultation with a retired army general, Frank Ross, who had been chief of the Army’s Transportation Corps in Europe. We thought of modeling a rail system as a network. At first it didn’t make sense, because there’s no reason why the crossing point of two lines should be a special sort of node. But Ross realized that, in the region we were studying, the “divisions” (little administrative districts) should be the nodes. The link between two adjacent nodes represents the total transportation capacity between them. This made a reasonable and manageable model for our rail system.

The Harris-Ross report stresses that specialists remain needed to make up the model (which is always a good tactics to get a new method accepted):

The ability to estimate with relative accuracy the capacity of single railway lines is largely an art. Specialists in this field have no authoritative text (insofar as the authors are informed) to guide their efforts, and very few individuals have either the experience or talent for this type of work. The authors assume that this job will continue to be done by the specialist.

The authors next dispute the naive belief that a railway network is just a set of disjoint through lines, and that cutting these lines would imply cutting the network:

It is even more difficult and time-consuming to evaluate the capacity of a railway network comprising a multitude of rail lines which have widely varying characteristics. Practices among individuals engaged in this field vary considerably, but all consume a great deal of time. Most, if not all, specialists attack the problem by viewing the railway network as an aggregate of through lines.

The authors contend that the foregoing practice does not portray the full flexibility of a large network. In particular it tends to gloss over the fact that even if every one of a set of independent through lines is made inoperative, there may exist alternative routings which can still move the traffic.

This paper proposes a method that departs from present practices in that it views the network as an aggregate of railway operating divisions. All trackage capacities within the divisions are appraised, and these appraisals form the basis for estimating the capability of railway operating divisions to receive trains from and concurrently pass trains to each neighboring division in 24-hour periods.

Whereas experts are needed to set up the model, to solve it is routine (when having the ‘work sheets’):

The foregoing appraisal (accomplished by the expert) is then used in the preparation of comparatively simple work sheets that will enable relatively inexperienced assistants to compute the results and thus help the expert to provide specific answers to the problems, based on many assumptions, which may be propounded to him.

For solving the problem, the authors suggested applying the ‘flooding technique’, a heuristic described in a RAND Report of August 5, 1955 by A. W. Boldyreff [1955a]. It amounts to pushing as much flow as possible greedily through the network. If at some vertex a ‘bottleneck’ arises (that is, more trains arrive than can be pushed further through the network), the excess trains are returned to the origin. The technique does not guarantee optimality, but Boldyreff speculates:

In dealing with the usual railway networks a single flooding, followed by removal of bottlenecks, should lead to a maximal flow.

Presenting his method at an ORSA meeting in June 1955, Boldyreff [1955b] claimed simplicity:

The mechanics of the solutions is formulated as a simple game which can be taught to a ten-year-old boy in a few minutes.

The well-known flow-augmenting path algorithm of Ford and Fulkerson [1955], that does guarantee optimality, was published in a RAND Report dated only later that year (December 29, 1955). As for the simplex method (suggested for the maximum flow problem by Ford and Fulkerson [1954]) Harris and Ross remarked:

The calculation would be cumbersome; and, even if it could be performed, sufficiently accurate data could not be obtained to justify such detail.

Figure 2: From Harris and Ross [1955]: Schematic diagram of the railway network of the Western Soviet Union and Eastern European countries, with a maximum flow of value 163,000 tons from Russia to Eastern Europe, and a cut of capacity 163,000 tons indicated as “The bottleneck”.

The Harris-Ross report applied the flooding technique to a network model of the Soviet and Eastern European railways. For the data it refers to several secret reports of the Central Intelligence Agency (C.I.A.) on sections of the Soviet and Eastern European railway networks. After the aggregation of railway divisions to vertices, the network has 44 vertices and 105 (undirected) edges.

The application of the flooding technique to the problem is displayed step by step in an appendix of the report, supported by several diagrams of the railway network. (Also work sheets are provided, to allow for future changes in capacities.) It yields a flow of value 163,000 tons from sources in the Soviet Union to destinations in Eastern European ‘satellite’ countries (Poland, Czechoslovakia, Austria, Eastern Germany), together with a cut with a capacity of, again, 163,000 tons. So the flow value and the cut capacity are equal, hence optimum.

In the report, the minimum cut is indicated as ‘the bottleneck’ (Figure 2). While Tolstoĭ and Harris-Ross had the same railway network as object, their objectives were dual.

Acknowledgements. I thank Sasha Karzanov for his efficient help in finding Tolstoĭ’s paper in the (former) Lenin Library in Moscow, Irina V. Karzanova for accurately providing me with an English translation of it, Kim H. Campbell and Joanne McLean at Air Force Pentagon for declassifying the Harris-Ross report, and Richard Bancroft and Gustave Shubert at RAND Corporation for their mediation in this.

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[1996] K. S Alexander, A conversation with Ted Harris, Statistical Science 11 (1996) 150–158.

[1976] L. J Billera, W. F Lucas, Delbert Ray Fulkerson August 14, 1924 – January 10, 1976, Mathematics of Operations Research 1 (1976) 299–310.

[1955a] A. W Boldyreff, Determination of the Maximal Steady State Flow of Traffic through a Railroad Network, Research Memorandum RM-1532, The RAND Corporation, Santa Monica, California, [5 August] 1955 [published in Journal of the Operations Research Society of America 3 (1955) 443– 465].

[1955b] A. W Boldyreff, The gaming approach to the problem of flow through a traffic network [abstract of lecture presented at the Third Annual Meeting of the Society, New York, June 3–4, 1955], Journal of the Operations Research Society of America 3 (1955) 360.

[1954] L. R Ford, D. R Fulkerson, Maximal Flow through a Network, Research Memorandum RM-1400, The RAND Corporation, Santa Monica, California, [19 November] 1954 [published in Canadian Journal of Mathematics 8 (1956) 399–404].

[1955] L. R Ford, Jr., D. R Fulkerson, A Simple Algorithm for Finding Maximal Network Flows and an Application to the Hitchcock Problem, Research Memorandum RM-1604, The RAND Corporation, Santa Monica, California, [29 December] 1955 [published in Canadian Journal of Mathematics 9 (1957) 210–218].

[1962] L. R Ford, Jr., D. R Fulkerson, Flows in Networks, Princeton University Press, Princeton, New Jersey, 1962.

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[1955] T. E Harris, F. S Ross, Fundamentals of a Method for Evaluating Rail Net Capacities, Research Memorandum RM-1573, The RAND Corporation, Santa Monica, California, 1955.

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[1948] Tj. C. Koopmans, Optimum utilization of the transportation system, in: The Econometric Society Meeting (Washington, D.C., September 6– 18, 1947; D. H Leavens, ed.) [Proceedings of the International Statistical Conferences – Volume V], 1948, pp. 136–146 [reprinted in: Econometrica 17 (Supplement) (1949) 136–146] [reprinted in: Scientific Papers of Tjalling C. Koopmans, Springer, Berlin, 1970, pp. 184–193].

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[1959] A. L Lur’e, Methods of establishing the shortest running distances for freights on setting up transportation systems [in Russian], in: Primenenie matematiki v èkonomicheskikh issledovaniyakh [Russian; Application of Mathematics in Economical Studies] (V. S Nemchinov, ed.), Izdatel’stvo Sotsial’no-Èkonomicheskoĭ Literatury, Moscow, 1959, pp. 349–382 [English translation in: The Use of Mathematics in Economics (V. S Nemchinov, ed.), Oliver and Boyd, Edinburgh, 1964, pp. 323–355].

[1949] J. Robinson, On the Hamiltonian Game (A Traveling Salesman Problem), Research Memorandum RM-303, The RAND Corporation, Santa Monica, California, 1949.

[1950] J. Robinson, A Note on the Hitchcock–Koopmans Problem, Research Memorandum RM-407, The RAND Corporation, Santa Monica, California, 1950.

[1930] A. N Tolstoĭ, Metody nakhozhdeniya naimen’shego summovogo kilometrazha pri planirovanii perevozok v prostranstve [Russian; Methods of finding the minimal total kilometrage in cargo-transportation planning in space], in: Planirovanie Perevozok, Sbornik pervyĭ [Russian; Transportation Planning, Volume I], Transpechat’ NKPS [TransPress of the National Commissariat of Transportation], Moscow, 1930, pp. 23–55.

[1939] A. Tolstoĭ, Metody ustraneniya neratsional’nykh perevozok pri planirovanii [Russian; Methods of removing irrational transportation in planning], Sotsialisticheskiĭ Transport 9 (1939) 28–51 [also published as ‘pamphlet’: Metody ustraneniya neratsional’nykh perevozok pri sostavlenii operativnykh planov [Russian; Methods of Removing Irrational Transportation in the Construction of Operational Plans], Transzheldorizdat, Moscow, 1941].

Alexander Schrijver CWI Science Park 123 1098 XG Amsterdam The Netherlands

                   lex.schrijver@cwi.nl

Original publication of this article: A. Schrijver, On the history of the transportation and maximum flow problems, Mathematical Programming, 91 (2002) 437-445.