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Combinatorial Optimization Top Ten List Discrete Mathematics 2000 as selected by William R. Pulleyblank T. J. Watson Research Center IBM Corporation Yorktown Heights, NY Euler's Theorem 1736 Theorem: A graph has a Euler tour if and only if there are zero or two nodes of odd degree L. Euler, Solutio Problematis ad Geometriam Situs Pertinentis, Commentarii Academiae Scientiarum Imperalis Petropolitanae 8 (1736) 128-140. " As far as the problem of the seven bridges of Koenigsberg is concerned, it can be solved by making an exhaustive list of all possible routes, and then finding whether or not any route satisfies the conditions of the problem. Because of the number of possibilities, this method of solution would be too difficult and laborious, and in other problems with more bridges, it would be impossible..." " So whatever arrangement may be proposed, one can easily determine whether or not a journey can be made, crossing each bridge once, by the following rules: If there are more than two areas to which an odd number of bridges lead, then such a journey is impossible. If, however, the number of bridges is odd for exactly two areas, then the journey is possible if it starts in either of these areas. If, finally, there are no areas to which an odd number of bridges leads, then the required journey can be accomplished starting from any area. " Algorithm first published 137 years later C. Hierholzer, Ueber die Moeglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechnung zu umfahren, Mathematische Annalen 6 (1873), 30-32. - start at an odd degree node; - walk until you are stuck * what's left has no odd degree nodes - solve the remaining problem and splice in the answer. see N.L. Biggs, E.K.Lloyd, R.J.Wilson, Graph Theory 1736-1936, Oxford University Press, 1976. Chinese Postman problem Guan Meigu, Graphic Programming using odd and even points, Chinese Math. 1 (1962) 273-277 The streets were given lengths, and any (even) number of odd degree intersections was permitted. The problem now became: Find the shortest tour traversing each street at least once. Solved polynomially in J. Edmonds and E.L. Johnson, Matching, Euler Tours and the Chinese Postman, Mathematical Programming 5 (1973) 88-124. Many extensions - one in particular - P.D. Seymour, The matroids with the max-flow min-cut property, J. Combin. Theory Ser. B 23 (1977) 189-222. Herbert J. Ryser, Combinatorial Mathematics, The Mathematical Association of America, 1963. " Combinatorial mathematics cuts across the many subdivisions of mathematics, and this makes a formal definition difficult. ... Two general types of problems appear throughout the literature. In the first the existence of the prescribed configuration is in doubt, and the study attempts to settle this issue. These we call existence problems. In the second the existence of the configuration is known, and the study attempts to determine the number of configurations or the classification of these configurations according to types. These we call enumeration problems. " March 1971 -- Second Louisiana Conference on Combinatorics, Graph Theory and Computing, LSU, Baton Rouge, Ryser gave an invited address and included a third type of problem: How do you efficiently find or construct the configurations? Max-flow Min-cut Theorem 1956 Given a directed graph G = (V, E),with finite arc capacities ( uj : j? E), and designated source and sink nodes s and z, the maximum value of an s - z flow is the minimum capacity of an s - z cut. L.R. Ford Jr. and D.R. Fulkerson, Maximal flow through a network, Canadian J. of Math. 8 (1956) 399-404. A. Kotzig, "Suvislost' a Pravidelina Suvislost Konecnych Grafor", Bratislava: Vysoka Skola Economicka (1956). P. Elias, A. Feinstein and C.E. Shannon, A note on the maximum flow through a network, IRE Transactions on Information Theory IT2 (1956) 117-119. Equivalent Theorems: Menger's Theorem (1927) (edge or node disjoint paths) Dilworth's Theorem (1950) (antichains in partially ordered sets) P. Hall's Theorem (1935) (Systems of Discrete Representatives) Koenig's Theorem (1931) (maximum matching and minimum cover in a bipartite graph) Hoffman's Circulation Theorem (1960) Gale's Supply Demand Flow (1957) ... These were all linked by total unimodularity - Hoffman and Kruskal (1956) Write down a "natural" linear programming formulation, then there will be an integral optimal solution ... and some extensions ... Flow augmenting paths Dinit (1970) and Edmonds-Karp (1972) - shortest flow augmenting paths give polynomial performance Preflow-push algorithms Generalized flows Multicommodity flows ... Seymour's Theorem (1980): [A good characterization of total unimodularity] - a matrix is totally unimodular if and only if it can be composed from network matrices, their transposes, and two exceptions. P.D. Seymour, Decomposition of regular matroids, J. of Combin. Theory Ser. B 28 (1980) 305-359. Nonbipartite matching algorithm and polyhedral characterization 1965. Jack Edmonds, Paths, trees and flowers, Canadian J. of Maths. 17 (1965) 449-467. There exists a polynomially bounded algorithm which finds a maximum matching in a nonbipartite graph. Generalized the Hungarian method of Kuhn 1955 and Munkres 1957 for bipartite graphs Alternating paths not enough You shrink the odd sets! Paths, trees and flowers (Edmonds 1965) - 2. Digression ... I am claiming, as a mathematical result, the existence of a good algorithm for finding a maximum cardinality matching in a graph. ... an algorithm whose difficulty grows only algebraically with the size of the graph... ...For practical purposes the difference between algebraic and exponential order is often more crucial than the difference between finite and non-finite. also, Cobham (1965) Foundation of Polyhedral Combinatorics The convex hull of the incidence vectors x of the matchings of a nonbipartite graph is defined by xj m 0 for all j ? E, ?@ 5 (xj : j (v)) [ 1 for all v ?V, ? 5 (xj : j E(S)) [ (|S| -1)/2 for all odd cardinality S <V. J. Edmonds, Maximum matching and a polyhedron with 0-1 vertices, J. Res. Nat. Bureau of Standards B 69 (1965) 125-130. Extensions: bidirected capacitated weighted matchings - includes network flows, edge covers and natchings parity constraints - includes T-joins, cuts and postman problems stable sets in claw free graphs path systems Matroid Intersection Theorem 1970 Matroids had been introduced by Hassler Whitney in 1935 as an abstraction of linear independence. A family F of (so called independent) subsets of a set E forms a matroid if 1. all subsets of an independent set are independent; 2. for any A < E, all maximal independent subsets of A have the same cardinality - denoted by r(A). (If 1. is satisfied, then F is called an independence system.) Examples - edge sets of forests in a graph, linearly independent sets of columns of a matrix, systems of distinct representatives of a set family, ... Matroid Optimization Problem: Let each element e c E have a weight we. Find an independent set I for which 5 (we: ec I) is maximized. Theorem (Rado 1957, Edmonds 1970): An independence system is a matroid if and only if the greedy algorithm solves the Matroid Optimization Problem, for any vector w of weights. Theorem (Edmonds 1970): The incidence vectors x of the independent sets of a matroid are the vertices of the polyhedron defined by xe m 0 for all ec E, 5(xe: e? A) [ r(A) for all A ` E. Matroid Intersection Theorem (Edmonds(1970): Let M1 = (E, F1) and M2 = (E, F2) be matroids on a set E. Then maximum |I| = minimum {r1(S)+r2(E\S)}. c 3 ` I F1 F2 S E Weighted Intersection Theorem (Edmonds 1970): Let M1 = (E, F1) and M2 = (E, F2) be matroids on a set E. Let each element e c E have weight we. Then x is the incidence vector of a maximum weight set independent in both matroids if and only if x is a basic optimal solution to the linear program maximize wx subject to xe m 0 for all ec E, 5(xe: e? A) [ r1(A) for all A ` E, 5(xe: e? A) [ r2(A) for all A ` E. Consequences: Koenig's Theorem for maximum matchings in bipartite graphs; Edmonds' Matroid Partition Theorem Perfect's Theorem on Systems of Common Representatives Extensions: Polymatroids and polymatroid intersection Submodular and supermodular functions Unification with nonbipartite matching: Matroid parity problem, or matroid matching - cardinality case for linear matroids (Lovasz 1980) Cook's Theorem 1971 If there exists a polynomially bounded algorithm for satisfiability, then if a class of decision problems has a polynomial length certificate for a "YES" answer, then there is a polynomial time algorithm to solve any decision problem in the class. S.A. Cook, The complexity of theorem-proving procedures, Proc. 3rd Ann. ACM Symp. on Theory of Computing, Association for Computing Machinery, New York (1971). NP - problems having short "YES" certificates - polynomially solvable on nondeterministic Turing machines P - problems solvable in polynomial time Cook's Theorem: SATISFIABILITY is NP-complete - a polynomial algorithm for SATISFIABILITY implies P = NP is P = NP ? R. M. Karp 1972 Reducibility among combinatorial problems in R.E. Miller and J.W. Thatcher (eds.), Complexity of Computer Computations, Plenum Press, New York, 85-103. Showed that many well known problems were NP-complete - Hamiltonian circuit; large independent set of nodes; scheduling problems; 3-d matching ... M.R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979 * More than 300 NP-complete problems Conventional Wisdom: P CNP. Garey and Johnson - 12 open problems: 1. Graph Isomorphism open 2. Subgraph homeomorphism for a fixed graph P 3. Graph genus NP-complete 4. Chordal graph completion NP-complete 5. Chromatic Index NP-complete 6. Spanning tree parity P 7.
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