Algorithmic Aspects of Flows in Networks
Springer Science & Business Media, 30 thg 4, 1991 - 203 trang
fEt moi, . . . . sifavait sucommenten rcvenir, One service mathematics has rendered the jen'yseraispointall,,: human race. It hasput rommon senseback JulesVerne whereit belongs, on the topmost shelf next tothedustycanisterlabelled'discardednon Theseriesis divergent; thereforewemaybe sense'. ahletodosomethingwithit. EricT. Bell O. Heaviside Mathematicsisatoolforthought. Ahighlynecessarytoolinaworldwherebothfeedbackandnon linearitiesabound. Similarly, allkindsofpartsofmathematicsserveastoolsforotherpartsandfor othersciences. Applyinga simplerewritingrule to thequoteon theright aboveonefinds suchstatementsas: 'One service topology hasrenderedmathematicalphysics . . . '; 'Oneservicelogichasrenderedcom puterscience . . . ';'Oneservicecategorytheoryhasrenderedmathematics . . . '. Allarguablytrue. And allstatementsobtainablethiswayformpartoftheraisond'etreofthisseries. This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumeshaveappeareditseemsopportunetoreexamineitsscope. AtthetimeIwrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by puttingforth new branches. It also happens, quiteoften in fact, that branches which were thought to becompletely disparatearesuddenly seento berelated. Further, thekindandlevelofsophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially)in regionaland theoretical economics; algebraic geometryinteractswithphysics; theMinkowskylemma,codingtheoryandthestructure of water meet one another in packing and covering theory; quantum fields, crystal defectsand mathematicalprogrammingprofit from homotopy theory; Liealgebras are relevanttofiltering; andpredictionandelectricalengineeringcanuseSteinspaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics','CFD', 'completelyintegrablesystems','chaos, synergeticsandlarge-scale order', whicharealmostimpossibletofitintotheexistingclassificationschemes. They drawuponwidelydifferentsectionsofmathematics. " By andlarge, all this stillapplies today. Itis still truethatatfirst sightmathematicsseemsrather fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is neededandsoarebooks thatcanhelp mathematiciansand scientistsdoso. Accordingly MIA will continuetotry tomakesuchbooksavailable. If anything, the description I gave in 1977 is now an understatement.
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MINIMUMCOST FLOW PROBLEMS
DETECTING NETWORK STRUCTURE
SOLUTION OF NETWORK FLOW PROBLEMS WITH ADDITIONAL CONSTRAINTS
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additional constraints Ahuja applied arc i,j arg min assume basic solution begin bicriteria blocking flow bound breakpoints calculated called cap(i,j Cap(X,X capacity Chapter cographic column complexity Computational results consider corresponding cost defined denoted determined directed graph dual e-optimal efficient solution embedded network end end EPS1 extreme points feasible solution flow augmenting path given Goldberg graph G implementation implies incidence matrix integer investigated iteration layered graph Lemma lexicographic linear programming Mathematics maximum flow problem minimal minimum cut minimum-cost flow problem multicriteria network flow network flow problem network simplex method network structure NP-complete NP-hard objective function objective space obtain optimal solution Orlin polynomial preflow procedure RELABEL respectively rows shortest path shortest path problem shown in Figure simplex algorithm solving spanning tree subgraphs subset Table Tarjan Theorem tion totally unimodular upper variables vector vertex vertices Xeff