J. Edmonds, "Submodular functions, matroids and certain polyhedra," in Proc. Calgary Int. Conf. Combinatorial Structures and Applications, 1970, pp. 69--87.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....exercise to show that DV V Recall that a sampling set need not have uniform sample spacing in order to have uniform density. 25 imply that the outer bound on the density region specified by the system of inequalities in (69) forms a contra polymatroid [23, 24]. Consequently, every constraint in (69) is active, i.e. the equality in each constraint in (69) holds for some point in the region. We now present a simple example to illustrate the necessary conditions for stable MIMO sampling. Example 1. Consider a MIMO channel with inputs and ....

....sampling, the above region is an outer bound on the density region for consistency. Next, observe that . Consequently, we can use Proposition 1 to show that Z DV 35 imply that the system of inequalities in (94) forms a polymatroid [23, 24], implying that every constraint in (94) is active for some point in the region. We now present at an example to illustrate the results for consistent reconstruction. Example 2. Let the MIMO channel and the input spectral supports be as defined in Example 1. We seek necessary conditions on the ....

J. Edmonds, "Submodular functions, matroids and certain polyhedra," in Proc. Calgary Int. Conf. Combinatorial Structures and Applications, Calgary, Alberta, June 1969, pp. 69--87.

....1) normalized) 2) if (nondecreasing) 3) submodular) The polyhedron is a contra polymatroid if satisfies 1) normalized) 2) if (nondecreasing) 3) supermodular) If satisfies the three properties, is called a rank function in both cases. Polymatroids were introduced by Edmonds [4] where he proved the following key properties. If is a permutation on the set , define the vector by and for . Lemma 3.2: Let be a polymatroid. Then is a vertex of for every permutation . Also, any vertex of strictly inside the positive orthant must be for some . Moreover, if is a given vector ....

J. Edmonds, "Submodular functions, matroids and certain polyhedra," in Proc. Calgary Int. Conf. Combinatorial Structures and Applications. (Calgary, Alta, June 1969), pp. 69--87.

.... maximum cardinality of a vector in a 2 lattice polyhedron is the minimum capacity of a cover, generalizes such classic special cases as Konig s Theorem [23] Menger s Theorem [26] Dilworth s Theorem [7] and Edmonds theorems for cardinality matroid intersection [8] and polymatroid intersection [11]. In fact, the methods used to prove this result are by now rather standard. There are, however, more intimate duality relationships for these problems. For example, Shapley and Shubik [29] showed that the collection of optimal dual solutions to a bipartite matching problem forms a lattice and ....

....1, 2 . The set P(#, #) x#IIR L # ( S ) x # # ( S ) for each S # # , 4 is called a 2 lattice polyhedron and each vector x # P(#, #) is called a 2 lattice vector. Examples of 2 lattice polyhedra include bipartite matching polyhedra [19, 24] the intersection of two integral polymatroids [11], and the perfectly matchable subgraph polytope of a bipartite graph [1] In this paper we consider the relationships between the problem of finding a 2 lattice vector with maximum sum of components: max # ##L x(#) s.t #(S)x # #(S) for each S # # (2.2) x # 0 and the dual problem: min # ....

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J. Edmonds, "Submodular Functions, Matroids and Certain Polyhedra," in: Combinatorial Structures and their Applications, R. Guy et al., eds., Proceedings of the Calgary International Conference (Gordon and Breach, New York, 1970) 67--87.

....in the form of SSC and then deriving their approximability results as applications of these heuristics. 2. Preliminaries he concept of a submodular function frequently plays a vital role in combinatorial theorems and algorithms, and its importance in discrete optimization has been well studied [6], 8] 20] 29] Definition 1: Let f be a real valued function defined on all the subsets of a finite set N . f is nondecreasing if f( S)# f( T ) for S # T # N . f is submodular if f( S) f( T ) # f( S#T) f( S#T ) for S, T # N . he pair( N,f) is called a submodular system on N ....

J. Edmonds, "Submodular functions, matroids, and certain polyhedra," in Combinatorial Structures and Their Applications, eds. R. Guy, H. Hanani, N. Sauer, and J. Schonheim, pp.69--87, 1970.

....Dress Wenzel [3] f#] valuated matroids and a domain reduction type polynomial time algorithmof Shioura [39]f#9 M convex f#LF2RA 45 The minimumof an L convexf#LflLfl5A can Table 1 Results in matroid theory related to convexity. Year Author Result (ca. 1935 Whitney [40] Matroid axioms 1965 Edmonds [5] Polymatroid intersection theorem 1975 Weighted intersection Edmonds [6] Lawler [22] Iri Tomizawa [17] potential char. Frank [8] weight splitting 1982 Relation to convexity Frank [9] discrete separation Fujishige [11] Fenchel duality Lovasz [23] Lovasz extension 1990 Dress Wenzel [3] 4] ....

J. Edmonds, "Submodular functions, matroids and certain polyhedra," in Combinatorial Structures and Their Applications, eds. R. Guy, H. Hanani, N. Sauer, and J. Schonheim, pp.69--87, Gordon and Breach, New York, 1970.

....theoretic approach, the rank function of a matroid is interpreted as a classical cooperative game and next, the game theoretic solution concept called core is de. ned as the set of optimal solutions of a certain linear programming problem in which the rank function of the matroid is involved. Edmonds (1970) showed that the core coincides with the convex hull of the incidence vectors corresponding to basic coalitions of the matroid. Section 3 introduces the concept of a cooperative game on a matroid as a real valued function on the matroid itself. In other words, the characteristic function of this ....

....x (S) P i2S x i and x ( 0: For every set S N,wede. nethe incidence vector e S 2 R N such that e S i : 1 for all i 2 S and e S i : 0 4 J.M.BILBAO 1 , T. S. H. DRIESSEN 2 , A. JIMNEZ LOSADA 3 , AND E. LEBRN 4 otherwise. The following theorem has been showed by Edmonds (1970) and provides one interpretation of the core of a cooperative game induced by the rank function of a matroid. Theorem 2.3. Let r :2 N Z be the rank function of a matroid M2 N and B(M) the set of basic coalitions of M. Then Core(N;r) conv e B : B 2B(M) Proof. First, we establish ....

Edmonds, J. (1970). "Submodular functions, matroids and certain polyhedra," in Combinatorial Structures and Their Applications (R. K. Guy, H. Hanani, N. Sauer, and J. Schnheim, Eds.), pp. 69--87. New York: Gordon and Beach.

....the vertices of the capacity region. Fig. 4(a) shows a two user capacity region. 4. 2 Polymatroid structure and conservation laws It is observed in [66] that the Shannon capacity region of the Gaussian multiaccess channel is a polymatroid, a class of combinatorial objects first studied by Edmonds [13]. Since this structure is central to our power control problems, we will review the general definition here. Definition 1 Let E = f1; Mg and f : 2 E be a set function. The polyhedron B(f) j f(x 1 ; xM ) X i2U x i f(U) 8U ae E; x i 0 8ig (39) is a polymatroid if the ....

....) i = 2; M (The interpretation of is such that user (M) is decoded first, user (1) is decoded last. The optimal solution to the problem (40) must be at one of these M vertices, corresponding to the M possible successive decoding orders. A well known result in polymatroid theory [13] says that the decoding ordering should be in increasing value of the coefficients i = Gamma i , i.e. the user with smallest i = Gamma i decoded first, the user with largest i = Gamma i decoded last. Note that the optimal ordering does not depend on the target rates R , although the optimal ....

Edmonds, J. (1969) "Submodular functions, matroids and certain polyhedra" Proc. Calgary Intl. Conf. on Combinatorial structures and appl. Calgary, Alberta, pp. 69-87, June.

....through the works of A. Frank, S. Fujishige, and L. Lov asz. In particular, Lov asz [22] pointed out that a set function is submodular if and only if the so called Table 1: Results in matroid theory related to convexity Year Author Result (ca. 1935 Whitney [39] Matroid axioms 1965 Edmonds [5] Polymatroid intersection theorem 1975 Weighted intersection Edmonds [6] Lawler [21] Iri Tomizawa [17] potential char. Frank [8] weight splitting 1982 Relation to convexity Frank [9] discrete separation Fujishige [11] Fenchel duality Lov asz [22] Lov asz extension 1990 Dress Wenzel [3, 4] ....

J. Edmonds, "Submodular functions, matroids and certain polyhedra," in Combinatorial Structures and Their Applications, eds. R. Guy, H. Hanani, N. Sauer, and J. Schonheim, pp.69--87, Gordon and Breach, New York, 1970.

....then ae is submodular and B = B(ae) Thus there is a one to one correspondence between M convex set B and submodular set function ae. In particular, B Z V is M convex if and only if B = B(ae) Z V for some submodular ae. The correspondence B ae is a restatement of a well known fact [4, 8]. For M convex sets B 1 ; B 2 Z V , it holds that B 1 B 2 = B 1 B 2 Z V and B 1 B 2 = B 1 B 2 . It is also true that a submodular set function ae corresponds one to one to a positively homogeneous L convex function g. The correspondence g 7 ae is given by the restriction ae(X) ....

Edmonds, J.: `Submodular functions, matroids and certain polyhedra', Combinatorial Structures and Their Applications, in N. Sauer R. Guy, H. Hanani and J. Sch onheim (eds.). Gordon and Breach, New York, 1970, pp. 69--87.

....top (bottom) In this way, each 2 Pi defines a rate tuple R 2 D with R i = I(X i ; Y j [ j2 Gamma1 (f1; Delta Delta Delta; i) Gamma1g) X j ) 38) It was observed by Hanly, Tse, and Whiting [16, 17] that R is in fact a bounded polymatroid. Polymatroids were introduced by Edmonds [20], who also derived their basic properties. For an introduction into polymatroid theory see, e.g. 21] The following lemma is a direct consequence of the fact that R is a bounded polymatroid. Lemma 14 (See [21] D = convfR : 2 Pig and each R , 2 Pi, is a vertex of D. C: The Degree ....

J. Edmonds, "Submodular functions, matroids, and certain polyhedra," in Combinatorial Structures and Their Applications (R. Guy, H. Hanani, N. Sauer, and J. Schonheim, eds.), pp. 69--87, New York: Gordon and Breach, 1970.

....we go further to clarify its pervasive role. By viewing the space of trees as a lattice, a variety of new theorems and algorithms become possible. For example, the objective functions commonly used in evaluating codes are submodular on this lattice. Submodular functions are often easy to optimize [1, 12, 14, 13, 3], and were shown by Lov asz to be closely related to convex functions [14] Huffman coding gives an important new example of the significance of submodularity in basic algorithms. 2 Ordered Sequences, Rooted Binary Trees, and Huffman Codes 2.1 Ordered Sequences By a sequence we mean an ordered ....

....trees (or a logarithmic variant on the majorization lattice of densities) gives a significant example of a submodular function. This is interesting also because most work on submodular functions assumes the lattice is the lattice of subsets of a given set, the case originally emphasized by Edmonds [1]. Definition 8 A real valued function f : L defined on a lattice hL; v; u; ti is submodular if f(x u y) f(x t y) f(x) f(y) for all x; y 2 L. 123456788 1329 123457777 1340 123466677 1324 123555677 1351 Path Length Weighted Sequence ....

J. Edmonds, "Submodular Functions, Matroids and Certain Polyhedra", in Combinatorial Structures and their Applications, R. Guy, H. Hanani, N. Saauer, J. Schonheim, eds., 69--87, Gordon and Breach, 1970.

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J. Edmonds, "Submodular functions, matroids and certain polyhedra," in Proc. Calgary Int. Conf. Combinatorial Structures and Applications, 1970, pp. 69--87.

No context found.

J. Edmonds, "Submodular functions, matroids and certain polyhedra," in Proc. Calgary Int. Conf. Combinatorial Structures and Applications (Calgary, Alta, June 1969), pp. 69--87.

No context found.

J. Edmonds, "Submodular functions, matroids and certain polyhedra", in Combinatorial Structures and their Applications (R.K. Guy et al. Eds.), Gordon and Breach, New York, 1970.

No context found.

J. Edmonds (1970) "Submodular functions, matroids and certain polyhedra" in: Combinatorial Structures and Their Applications (R. Guy, et al., eds) Gordon and Breach, 69--87.

No context found.

J. Edmonds (1970) "Submodular functions, matroids and certain polyhedra" in: Combinatorial Structures and Their Applications (R. Guy, et al., eds) Gordon and Breach, 69--87.

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