J. Edmonds and R. Giles. A min-max relation for submodular functions on graphs. Annals of Discrete Mathematics, 1:185--204, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....again trivial: s j 0 for j = 1; m and no TT cut exists. We are really concerned with the nontrivial case, when x is not a vertex of P , but lies on a face of P . The key to resolving TT SEP in this case is the following integer version of Farkas lemma, taken from Edmonds Giles [6] (see also Schrijver [15] chapters 4 and 5) Theorem 1 (Edmonds Giles, 1977) For a given system of linear equations of the form Ax = b, precisely one of the following statements is true: There is an integer solution to Ax = b There is some rational vector y such that y A is integral, ....

J. Edmonds & R. Giles (1977) A min-max relation for submodular functions on graphs. Ann. Discr. Math., 1, 185-204.

....discrete convex function; submodular flow; algorithm 1 Introduction The M convex submodular flow problem, introduced by Murota [16] is one of the most general frameworks of e#ciently solvable combinatorial optimization problems. It includes the minimum cost flow and the submodular flow problems [3, 7] as its special cases, and has an application to mathematical economics [19] The submodular flow problem with an M convex function admits nice optimality criteria in terms of potentials and negative cycles like the minimum cost flow problem. The concept of M convex functions was proposed by ....

....with upper and lower capacity bounds c, c the cost function # . For each X V , let # X (resp. # X) denote the set of arcs leaving (resp. entering) X. Suppose that B is a bounded M convex set. Then the integer flow version of the submodular flow problem is formulated as follows [3]. Submodular flow problem MSFP 1 (linear arc cost, integer flow) #(a)#(a) 2.1) B, 2.3) The submodular flow problem is a well behaved combinatorial problem that has nice properties such as optimality criterion in terms of potentials (dual variables) optimality criterion in terms ....

J. Edmonds and R. Giles (1977). A Min-max Relation for Submodular Functions on Graphs, Ann. Discrete Math., 1, 185--204.

....Farkas Lemma available when we require integrality and simultaneously bounds for the variables. However for an integer programming problem with unbounded variables, i.e. the question of deciding whether a given point belongs to a lattice, a Farkas type lemma exists, due to Edmonds and Giles [2]. However, to the best of our knowledge, it has not # Address: Otto von Guericke Universitat Magdeburg, Department of Mathematics IMO, Universit atsplatz 2, 39106 Magdeburg, Germany. E mail addresses: mkoeppe, weismant imo.math.unimagdeburg. de. Supported by grants FKZ 0037KD0099 and FKZ ....

J. Edmonds and R. Giles. A min-max relation for submodular functions on graphs. In P. L. Hammer, editor, Studies in Integer Programming, pages 185--204. Annals of Discrete Mathematics 1, North-Holland, Amsterdam, 1977.

....and various other fields. Examples include cut capacity functions, matroid rank functions, and entropy functions. Submodular function minimization (SFM) is the problem of finding a subset X C V with f(X) f(Y) for all Y C V. Connecting submodular functions with network flows, Edmonds and Giles [4] introduced the submodular flow prob lem, which includes network flow, matroid intersection, and directed cut covering. Since then, several combinatorial optimization problems have been shown to be special cases of submodular flow. In particular, Frank and Tardos [7] solved the minimum cost ....

J. Edmonds and R. Giles. A min-max relation for submodular functions on graphs. Ann. Discrete Math., 1:185-204, 1977.

....Emaih stmvadk.coranerce.ubc.ca. Research supported by an NSERC Operating Grant; part of this research was done during visits to SORIE at Cornell University and to LIMOS at Universit Blaise Pascal, Clermont Ferrand. i Introduction The submodular flow problem, introduced by Edmonds and Giles [4], is one of the most important frameworks of efficiently solvable combinatorial optimization problems. It includes minimum cost flow, graph orientation, polymatroid intersection, and directed cut covering problems as special cases. To talk about running time we use n for the number of nodes, h for ....

....exists an integral submodular flow. Given a price function (or node potentials) p R v, we define the reduced cost w.r.t. p as Cp(a) c(a) p(3 a) p(3 a) for each a A. The following optimality conditions for the submodular flow problem are due to Fujishige [14, Theorem 5. 3] Edmonds and Giles [4] show that the submodular flow problem is totally dual integral, which implies the integrality claim. Theorem 3.2 A submodular flow 9 is optimal if and only if there exists p R v such that: a) For any a A, Cp(a) 0 implies qo(a) l(a) and Cp(a) 0 implies qo(a) u(a) and b) The boundary ....

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J. Edmonds and R. Giles. A min-max relation for submodular functions on graphs. Ann. Discrete Math., 1 (1977), 185-204,.

....integral (TDI) if for every c 2 Z such that j minfb y = c; y 0gj 1, there exists an integral vector y , A = c, y 0 with b = minfb y = c; y 0g. The TDIness of the system Ax b has an important consequence for polyhedra and a geometric meaning. Theorem 3.12. [29] If Ax b is TDI and b is integral, then fx 2 R : Ax bg is integral. Let c be an integral vector that lies in the cone generated by all the rows of A. Among all possible ways of writing c as a conic combination of the row vectors of A, let S be the set of shortest conic combinations with ....

J. Edmonds and R. Giles (1977), A min-max relation for submodular functions on graphs, in Studies in Integer Programming, P.L. Hammer et al. eds, Annals of Discrete Mathematics 1, 185 - 204.

....E F # ) and H # = V s, K F # ) satisfy (1) with respect to k and l, respectively, can be solved by finding an integer valued z C(pG ) C(pH ) for which z(V ) is minimum. By Frank s [6] g polymatroid intersection theorem (see also Frank and Tardos [9, Theorem 1.4, Proposition 4. 1] and [5]) the system S(p H ) G (A) p H (A) is a submodular flow system and hence we can find an integer valued x # S(p H ) minimizing x(V ) in polynomial time (see e.g. Cunningham and Frank [4] Summarizing our observations we obtain the following algorithm for the simultaneous ....

J. Edmonds, R. Giles, A min-max relation for submodular functions on graphs, Annals of Discrete Mathematics 1 (1977) 185-204.

....Q 2 6 Q 6 C 3 2 C 3 Figure 1: Classes of clutters. A linear system Ax b is Total Dual Integral (TDI) if the linear program min wx subject to Ax b has an integral optimal dual solution y for every integral w for 5 which the linear program has a nite optimum. Edmonds and Giles [20] proved that, if Ax b is TDI and b is integral, then P = fx : Ax bg is an integral polyhedron. The proof of the Edmonds Giles theorem can be found in Schrijver [52] pages 310 311, or Nemhauser and Wolsey [40] pages 536 537. It follows that C(M) has the MFMC property if and only if (6) has an ....

J. Edmonds and R. Giles, A Min-Max Relation for Submodular Functions on Graphs, Annals of Discrete Mathematics 1 (1977) 185-204.

....cannot be used as a recognition of balancedness. 4 Total Dual Integrality A system of linear constraints is totally dual integral (TDI) if, for each integral objective function vector c, the dual linear program has an integral optimal solution (if an optimal solution exists) Edmonds and Giles [31] proved that, if a linear system Ax b is TDI and b is integral, then fx : Ax bg is an integral polyhedron. 7 Theorem 4.1 (Fulkerson, Ho man, Oppenheim [34] Let A = 0 B A 1 A 2 A 3 1 C A be a balanced 0; 1 matrix. Then the linear system 8 : A 1 x 1 A 2 x ....

J. Edmonds, R. Giles, A min-max relation for submodular functions on graphs, Annals of Discrete Mathematics 1 (1977) 185-204.

....) Proof. Consider (i) Proposition 4.2 states (H; p) p(F ) Because H is ideal, H; p) H; p) i.e. p(F ) H; p) This implies by Proposition 4.2(ii) that M is F flowing with costs p. Consider (ii) If H is nonideal then for some p : E(M ) Z , H; p) H; p) [5]. Let F 0 be an optimal solution to (a) Then p(F 0 ) H; p) and F 0 2 b(H) Proposition 4.2(ii) states M is not F 0 flowing with costs p. We leave the next result as an easy exercise. Corollary 4.5. A binary clutter H is ideal if and only if u(H) is F flowing for every F 2 b(H) ....

J. Edmonds and R. Giles. A min-max relation for submodular functions on graphs. Annals of Discrete Math., 1:185--204, 1977.

....In Appendix A, we discuss in more detail the algorithms for the case where all orientation constraints are set at equality and the requirement function is crossing supermodular. This 3 version of the problem can be solved optimally in polynomial time via work of Frank [9] and Edmonds and Giles [3]. We also describe a clever proof of Younger [21] which had not previously appeared. 2 Preliminaries We denote a directed graph by D = V; A) For S V , denote by ffi (S) resp. ffi Gamma (S) the set of arcs with tail in S (resp. V Gamma S) and head in V Gamma S (resp. S) A pair of ....

J. Edmonds, R. Giles, A min-max relation for submodular functions on graphs, Annals Discrete Math., 1 (1977), pp. 185--204.

....a dicut is equal to the maximum number of pairwise disjoint dijoins. Let # be a digraph and let # # # # ## be a weight function. The weight of a dicut is the sum of the weights of its arcs. The following conjecture is a weighted version of Woodall s conjecture. Conjecture 1. 2 (Edmonds and Giles [2]) For every nonnegative integer arc weight function #, the minimum weight of a dicut is equal to the maximum number of dijoins such that no arc # is contained in more than #### of these dijoins. Schrijver [6] exhibited an example showing that this conjecture is not true (see next section) ....

J. Edmonds and R. Giles. A min-max relation for submodular functions on graphs. Annals of Discrete Math., 1:185--204, 1977.

....that min b T y : A T y = c, y # 0 #, there exists an integral vector y # # Z m , A T y # = c, y # # 0 with b T y # = min b T y : A T y = c, y # 0 . The TDIness of the system Ax # b has an important consequence for polyhedra and a geometric meaning. Theorem 3.12. [29] If Ax # b is TDI and b is integral, then x # R n : Ax # b is integral. Let c be an integral vector that lies in the cone generated by all the rows of A. Among all possible ways of writing c as a conic combination of the row vectors of A, let S be the set of shortest conic combinations ....

J. Edmonds and R. Giles (1977), A min-max relation for submodular functions on graphs, in Studies in Integer Programming, P.L. Hammer et al. eds, Annals of Discrete Mathematics 1, 185 - 204.

....orientations of the current edge at least one will always do) The question naturally emerges: when does there exist a k edge connected orientation of a mixed graph This can be answered with the use of submodular flows. The notion of submodular flows was introduced by J. Edmonds and R. Giles [1977]. They proved (among others) that the submodular flow polyhedron is integral. It was observed in [Frank 1982] that there is a strong link between 0 Gamma 1 valued submodular flows and orientations of graphs. For example, the integrality of the submodular flow polyhedron easily implies ....

....f entered by f . Clearly, T : fT f : f 2 Fg is a cross free family on U (depending only on T ) and its pre image Gamma1 (T ) is a cross free family on V . We will say that the pair (T; is a tree representation of Gamma1 (T ) It is not difficult to prove (see, Edmonds and Giles [1977]) that any cross free family on V has such a tree representation. For example, if A = fA 1 ; A k g is a partition of a subset A of V , then T is an out directed star, that is, T has node set fu 0 ; u 1 ; u k g and edge set fu 0 u 1 ; u 0 u k g. Furthermore, s) u i if ....

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J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Annals of Discrete Mathematics 1, (1977), 185-204.

....of the following fundamental theorem helps us. THEOREM 6.1 [Lucchesi and Younger, 1978] In a directed graph D the maximum number, of disjoint directed cuts is equal to the minimum number, of edges covering all the directed cuts. This theorem has an extension by J. Edmonds and R. Giles [1977] to describe minimum weight coverings of directed cuts and in [Frank, 1981] a strongly polynomial algorithm is constructed to compute the minimum. A polynomial time algorithm is called strongly polynomial if it uses, beside ordinary data manipulation, only basic operations such as comparing, ....

....(G; H) denote the minimum value of an admissible sub partition. We have and although equality does not hold in general, in important special cases it does. For a subset A T the notation (A; T Gamma A; G) will be abbreviated to (A; G) or to (A) when no confusion can arise. B.V. Cherkasskij [1977] and L. Lov asz [1976] proved the following theorem: THEOREM 7.7 For an inner Eulerian pair (G; T ) the maximum number of edge disjoint T paths is equal to ( P (t) t 2 T ) 2. Furthermore, there is a family of disjoint sets fX(t) t 2 Tg such that t 2 X(t) V and dG (X t ) t) t 2 T ) ....

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J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Annals of Discrete Mathematics 1, (1977), 185-204. 24

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J. Edmonds and R. Giles. A min-max relation for submodular functions on graphs. Annals of Discrete Mathematics, 1:185--204, 1977.

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J. Edmonds and R. Giles. A min-max relation for submodular functions on graphs. Annals of Discrete Mathematics, 1:185--204, 1977.

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J. Edmonds and R. Giles. A min-max relation for submodular functions on graphs. Annals of Discrete Mathematics, 1:185--204, 1977.

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Edmonds, J., Giles, R. (1977): A min-max relation for submodular functions on graphs. Ann. Discrete Math. 1, 185--204

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J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Ann. Discrete Math. 1 (1977), 185-204.

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Jack Edmonds and F. Rick Giles. A min-max relation for submodular functions on graphs. In P. L. Hammer, editor, Studies in Integer Programming, pages 185-204. Annals of Discrete Mathematics 1, North-Holland, Amsterdam, 1977.

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Edmonds, J., Giles, R.: "A min-max relation for submodular functions on graphs" Annals of Discrete Mathematics 1, 1977, 185-204.

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J. Edmonds, R. Giles, A min-max relation for submodular functions on graphs, Annals Discrete Math., 1 (1977), pp. 185-204.

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Edmonds J., Giles R. (1977) A Min-Max relation for submodular functions on graphs, Annals of Discrete Math. 1, 185-204.

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J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Annals of Discrete Mathematics 1, (1977), 185-204.

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