J. Edmonds, Submodular functions, matroids and certain polyhedra, in: R. Guy, ed., Combinatorial Structures and Their Applications (Gordon and Breach, New York, 1970) 69-87.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....compute the robustness function for a transversal matroid, but design a more efficient algorithm for solving problem (1) in the computation of a set with smallest rate. Let M k = D k ; I k ) M jDw k ) D w k . For any set S D k and vector x 2 IR we let x(S) e2S x(e) From results in [5, 8], it follows that for any vector x 2 IR , min f x(S) rank (D k Gamma S; M k ) j S D k g = max fy(D k ) j y 2 P (M k ) and y x g, where P (M k ) is the matroid polyhedron for M k . Problem (1) then can be formulated as max fy(D k ) j y 2 P (M k ) and y c= g: 2) We derive an expression ....

....j. Then, f(T ) f(T ) jN (T [ S)j Gamma jSj jN (T = jN (T [ S)j jN (T ) Gamma N (T [ S)j jN (T ) N (T [ S)j Gamma jS [ S jN (T [ S [ T )j jN (T T ) N (S S )j Gamma jS [ S f(T [ T ) f(T T The following result by Edmonds [8, 6] allows us to give explicit expressions for the independent sets of M k and for its matroid polyhedron P (M k ) Theorem 4.1. Edmonds) Let g : 2 7 IR be a submodular function and H = f S j S F and jS F j g(F F g. Then H is the family of independent sets of a matroid on F, and ....

J. Edmonds, Submodular functions, matroids and certain polyhedra, in Combinatorial Structures, R.K. Guy et al. (Editors), Gordon and Beach, New York 1970, pp . 69--87.

....is the matroid intersection problem. We let (M 1 ; M 2 ) denote the maximum size of a common independent set of M 1 and M 2 . The matroid intersection problem is a generalization of the maximum matching problem for bipartite graphs, and can be solved by 20 augmenting path methods; see Edmonds [3]. In this section we consider the case that M 1 and M 2 are linear matroids. A matroid is linear if it is given to us as a column matroid of a matrix. We formulate the linear matroid intersection problem as a matrix rank problem. Consider a bipartite graph G = V; E) with bipartition (V r ; V ....

....is a matching if and only if it is a common independent set of M r and M c . Therefore, the matroid intersection problem is indeed a generalization of the maximum matching problem for bipartite graphs. As well as nding an ecient algorithm for solving the matroid intersection theorem, Edmonds [3] also proved a remarkable min max theorem. Theorem 10.1 (Edmonds) If M 1 = S; I 1 ) and M 2 = S; I 2 ) are matroids with rank functions r 1 ( and r 2 ( respectively, then max(jAj : A 2 I 1 I 2 ) min(r 1 (X) r 2 (S X) X S) Note that, if A is a common independent set of M 1 and M ....

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.

....of determining the rank of a mixed matrix contains the linear matroid intersection problem. Murota [18] studies mixed matrices extensively, and shows that computing their rank is in fact equivalent to linear matroid intersection. In particular, we can compute the rank of Q X using Edmonds [5] matroid intersection algorithm. 2 Randomized algorithms The matrix rank formulations do not immediately provide ecient algorithms, as we cannot e ciently perform basic operations on a matrix with indeterminate entries. For example, the determinant of a mixed matrix is a polynomial that may ....

....1.6 For any graph G = V; E) 2 (G) jV j (odd(G A(G) jA(G)j) Moreover, D(G) D(G A(G) Theorems 1.4 and 1.6 shall be proved in Section 3. 4 Deterministic algorithms Edmonds has ecient augmenting path algorithms for both the matching problem [6] and the matroid intersection problem [5]. Therefore, the ranks of Tutte matrices and mixed matrices can be computed eciently. We describe a di erent approach, based on evaluations. Let Q X be an evaluation of a mixed matrix Q X . For an indeterminate z in X , we denote by X(z a) the evaluation of X obtained from X be ....

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. 11

.... are known to be NP complete like the (asymmetric) traveling salesman problem or the stable set problem (cf. 2, 7, 9] Any independence system can be represented as the intersection of nitely many matroids (see Theorem 5) One of the most important results in this context was given by Edmonds [4], who proved that the optimization problem over the intersection of two matroids is solvable in polynomial time. Algorithms for this problem were given by Edmonds [5] Frank [6] and Lawler [11, 12] Unfortunately, this approach fails in the case of three or more matroids. Since the (asymmetric) ....

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

....Sports and Culture of Japan. 1 K. MUROTA and A. TAMURA: Characterizations of M convexity 2 1 Introduction Discrete convex analysis, which was recently proposed by Murota [15, 16] is a unified framework of discrete optimization with reference to existing studies on submodular functions [5, 14, 8], generalized polymatroids [6, 7, 8] valuated matroids [3, 4, 18] and convex analysis [25] The concepts of M M convex functions play a central role in discrete convex analysis and have been investigated by Murota [15, 16, 17] Fujishige and Murota [9] Murota and Shioura [21, 22, 23] and ....

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

....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, 10]. For Mconvex 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) g( ....

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

....While we have efficient algorithms for computing the rank of a rational matrix, complications arise when A contains indeterminates, like for matching delta matroids (we will see more on this in Chapter 8) In order to prove Proposition 4. 2 we require the following fundamental theorem of Edmonds [32]. Theorem 4.3 If (V; B 1 ) and (V; B 2 ) are matroids, then conv(B 1 B 2 ) conv(B 1 ) conv(B 2 ) 2 Proof of Proposition 4.2. We first observe that, for a subset F of V , F 2 F if and only if X [ V n F ) 2 B 1 . Now suppose that x 2 conv(F ) that is, there exists 2 R F such that ....

J. Edmonds. Submodular functions, matroids and certain polyhedra. In R. Guy et al, editor, Combinatorial structures and their applications, pages 69--87. Gordon and Breach, 1970. 93

.... linear programs that strictly improve on known exponential size linear programming relaxations [51] Minimizing Submodular Set Functions The minimization of submodular set functions is a generic optimization problem which contains a large class of important optimization problems as special cases [25]. Here we will see why the ellipsoid algorithm provides a polynomial time solution method for submodular minimization. Definition 5.2 Let N be a finite set. A real valued set function f defined on the subsets of N is ffl submodular if f(X [ Y ) f(X Y ) f(X) f(Y ) for X; Y N . Example ....

J. Edmonds, Submodular functions, matroids and certain polyhedra, in Combinatorial Structures and their Applications, edited by R. Guy et al., Gordon Breach, (1970) 69-87.

....the elements of N so that w i w i 1 ; i = 1; 2; Delta Delta Delta ; n Gamma 1. Let T = OE; i = 1. 1. If w i 0 or i n, stop T is optimal, i.e. x j = 1 for j 2 T and x j = 0 for j = 2 T. If w i 0 and T [ fig 2 F, add element i to T. 2. Increment i by 1 and return to step 1. Edmonds [40, 41] derived a complete description of the matroid polytope, the convex hull of the characteristic vectors of independent sets of a matroid. While this description has a large (exponential) number of constraints, it permits the treatment of linear optimization problems on independent sets of matroids ....

....treatment of linear optimization problems on independent sets of matroids as linear programs. Cunningham [35] describes a polynomial algorithm to solve the separation problem 5 for the matroid polytope. The matroid polytope and the associated greedy algorithm have been extended to polymatroids [40,93]. The separation problem for a polymatroid is equivalent to the problem of minimizing a submodular function defined over the subsets of N , see Nemhauser and Wolsey [95] A class of submodular functions that have some additional properties can be minimized in polynomial time by solving a maximum ....

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J. Edmonds, Submodular functions, matroids and certain polyhedra, in Combinatorial Structures and their Applications, edited by R. Guy et al., Gordon Breach, (1970) 69-87.

....that augments over this family in polynomial time. The vectors that we investigate are combinatorial in the sense that they can be interpreted as paths and cycles in a network. Similar ideas have been used earlier for designing augmentation algorithms for the matroid intersection problem (see [E70], E79] L76] CCPS98] or [FSW99] To be more precise, let us consider the combinatorial optimization problem of intersecting two 0=1 integer programs, max c T x : Ax b; Cx d; x 2 f0; 1g n ; IIP) where A 2 Z m 1 Thetan , C 2 Z m 2 Thetan , b 2 Z m 1 , d 2 Z m 2 , and c ....

J. Edmonds, Submodular Functions, Matroids and Certain Polyhedra, in: Comb. Structures and their Applications (R. K. Guy, H. Hanai, N. Sauer and J. Schonheim, eds.), Gordon and Brach, New York, 69--87, 1970 14

....systems on V . We refer to the problem of deciding whether J 1 and J 2 as the intersection problem. The intersection problem can be posed as a membership problem, since J 1 J 2 6= if and only if 0 2 J 1 Gamma J 2 . The intersection problem is well solved for matroids; see Edmonds [7] and [8]. The main result in this section implies the matroid intersection polyhedron theorem of Edmonds [8] Matroid parity problem The last two examples describe membership problems for which there exist efficient algorithms; now we shall see that the general situation is not so nice. Let G = V; E) ....

....The intersection problem can be posed as a membership problem, since J 1 J 2 6= if and only if 0 2 J 1 Gamma J 2 . The intersection problem is well solved for matroids; see Edmonds [7] and [8] The main result in this section implies the matroid intersection polyhedron theorem of Edmonds [8]. Matroid parity problem The last two examples describe membership problems for which there exist efficient algorithms; now we shall see that the general situation is not so nice. Let G = V; E) be the graph where E : f(1; 2) 3; 4) n Gamma 1; n)g, and let J be a matroid on the set ....

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

.... g : Z n Z [ f 1g with dom g 6= is said to be submodular if g(p) g(q) g(p q) g(p q) p; q 2 Z n ; where p q and p q are, respectively, the vectors of componentwise maxima and minima, i.e. p q) i = max(p i ; q i ) p q) i = min(p i ; q i ) i = 1; n: See Edmonds [3], Frank Tardos [7] Fujishige [10] Lov asz [14] and Topkis [28] for general background on submodular functions. It is shown in Favati Tardella [4] that a global minimum of an integrally convex function g over a discrete rectangle can be characterized as a local minimum, i.e. for p 2 dom g, we ....

....p 2 Z V g = supfh ffi (x) 0 g ffl (x) j x 2 Z V g: If this common value is finite, the infimum is attained by some p 2 dom g domh and the supremum is attained by some x 2 dom g ffl domh ffi . It is mentioned that Fujishige [8] formulated the matroid intersection theorem of Edmonds [3] as a Fenchel type duality for submodular and supermodular functions, whereas Murota [19] rewrote the valuated matroid intersection theorem of Murota [16] as another Fenchel type duality. 4. Proofs of Theorems on L convexity Proof of Theorem 1.2: Let g : Z n Z[f 1g be an L convex function. The ....

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

....as the solution set of a certain system of linear inequalities. Finally, Cunningham and Marsh [5] proved the total dual integrality of the system of inequalities. Given matroids M 1 ; M 2 on the same set T , a common basis of M 1 ; M 2 is a subset of T that is a basis in both matroids. Edmonds [9] gave a necessary and sufficient condition for the existence of a common basis, and polynomial time algorithms to determine whether there exists a common basis and to find a common basis of maximum weight. In analyzing such matroid algorithms , we regard each independence test as a single step ....

....is also an odd component of G[D 1 [ D 2 ] Therefore, odd(G[D 1 D 2 ] odd(G[D 1 [D 2 ] If we take S = V n(D 1 [D 2 ) it follows that odd(G Gamma S) jSj, as required. Matroid Intersection. Suppose that M 1 ; M 2 are matroids on T , each of rank r. Edmonds Matroid Intersection Theorem [9] states that there exists a common basis if and only if there does not exist a subset A of T such that r 1 (A) r 2 (TnA) r. It is easy to see that this condition is necessary. Now suppose that there does not exist a common basis. If we take G to be a perfect matching joining copies T 1 ; T 2 ....

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

....no double claw, the 2 level planarization problem is equivalent to the maximum forest problem. It is well known that this problem can be solved in polynomial time by a simple greedy algorithm. Moreover, the structure of the associated weighted forest polytope has been well studied (see, e.g. [Edm70]) The inequalities of the weighted forest polytope are still valid for our polytope 2LPS(G) even if the graph G contains double claws. And, as we will see in our computational experiments, they are quite useful in practice. Lemma 3.7 Let G = L; U; E) be a 2 level graph. The forest ....

Edmonds, J.: Submodular functions, matroids and certain polyhedra. in: Combinatorial Structures and Their Applications, Gordon and Breach, London (1970) 69 -- 87

....the robustness function for a transversal matroid, but design a more efficient algorithm for solving problem (1) in the computation of a set with smallest rate. Let M k = D k ; I k ) M jDw k ) D w k . For any set S D k and vector x 2 IR D k we let x(S) P e2S x(e) From results in [5, 8], it follows that for any vector x 2 IR D k , min f x(S) rank (D k Gamma S; M k ) j S D k g = max fy(D k ) j y 2 P (M k ) and y x g, where P (M k ) is the matroid polyhedron for M k . Problem (1) then can be formulated as max fy(D k ) j y 2 P (M k ) and y c= g: 2) We derive an ....

.... jN (T [ S)j jN (T 0 [ S 0 ) Gamma N (T [ S)j jN (T 0 [ S 0 ) N (T [ S)j Gamma jS [ S 0 j Gamma jS S 0 j jN (T [ S [ T 0 [ S 0 )j jN (T T 0 ) N (S S 0 )j Gamma jS [ S 0 j Gamma jS S 0 j f(T [ T 0 ) f(T T 0 ) 2 The following result by Edmonds [8, 6] allows us to give explicit expressions for the independent sets of M k and for its matroid polyhedron P (M k ) Theorem 4.1. Edmonds) Let g : 2 F 7 IR be a submodular function and H = f S j S F and jS F 0 j g(F 0 ) for all F 0 F g. Then H is the family of independent sets of a ....

J. Edmonds, Submodular functions, matroids and certain polyhedra, in Combinatorial Structures, R.K. Guy et al. (Editors), Gordon and Beach, New York 1970, pp . 69--87.

....by means of expansions and their connectivity. Examples of these functions are the rank functions of matroids; they are extreme in the above cone if and only if the matroid is connected after deleting all its loops. For other results on semimodular functions we refer the reader to [3] [2] and [8] Another linear transformation of H 2 gives the cone fg 2 R P ; g(I) f(N Gamma I) Gamma f(N) I ae N; f 2 H 2 g consisting of the normalized (g( 0) nonnegative and convex set functions. Its extreme functions, which are interesting from the game theory point of view, were ....

J. Edmonds, Submodular functions, matroids and certain polyhedra, in: R. Guy, ed., Combinatorial Structures and Their Applications (Gordon and Breach, New York, 1970) 69-87.

....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 closely related to convex functions (as we explain later; see Theorem 4. 5) and are often easy to optimize [6, 9, 23, 24, 25]. Huffman coding gives a significant example of the importance of submodularity in algorithms. 2. Ordered Sequences, Rooted Binary Trees, and Huffman Codes. 2.1. Ordered Sequences. By a sequence we mean an ordered collection of nonnegative real values x = h x1 x2 Delta Delta Delta xn i: ....

....gw is submodular over the lattice of trees. which helps explain why efficient algorithms for finding optimal trees are possible at all. 4.1. Submodularity. Most work on submodular functions assumes the lattice is the lattice of subsets of a given set, the case originally emphasized by Edmonds [6]. However, the definition applies to any lattice: Definition 4.1. 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) 16 D.S. PARKER AND P. RAM for all x; y 2 L. Equivalently, f is submodular if a differential inequality holds: ....

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J. Edmonds, Submodular Functions, Matroids and Certain Polyhedra, in Combinatorial Structures and their Applications, R. Guy et al., eds., Gordon & Breach, 1970, pp. 69--87.

....vertices have degree at most 2, then a depth first traversal of T with short cuts will yield a tour whose length is at most twice OPT. We now claim that the problem of finding T can be viewed as that of finding the minimum weight, maximum cardinality subset in the intersection of two matroids [29, 15, 16]. The two matroids in this case are: M 1 , the matroid of all bipartite forests, and M 2 , the matroid of all bipartite subgraphs whose blue vertices have degree at most 2. For completeness we review here the definition of a matroid, following the standard text [29] APPROXIMATING CAPACITATED ....

....finding a minimum weight bipartite spanning tree where the blue vertices have degree at most two, is equivalent to the problem of finding a minimum weight common base of M 1 and M 2 . This is a special case of the matroid intersection problem, which was first solved in polynomial time by Edmonds [15, 16]. Other authors [9] have exploited the special structure of problems such as ours to improve the running times. We obtain the following theorem. Theorem 2.1. There is a polynomial time 2 approximation algorithm for the Bipartite TSP. 2.1. Extension to Finite Capacity Vehicles. We will now show ....

J. Edmonds. Submodular functions, matroids and certain polyhedra. In Combinatorial Structures and their Applications, Proceedings of Calgary International Conference, pages 69-- 87, 1970.

....vertices have degree at most 2, then a depth first traversal of T (with short cuts) will yield a tour whose length is at most twice OPT. We now claim that the problem of finding T can be viewed as that of finding the minimum weight, maximumcardinality subset in the intersection of two matroids [27, 12, 13]. The two matroids in this case are: M 1 , the matroid of all bipartite forests, and M 2 , the matroid of all bipartite subgraphs whose blue vertices have degree at most 2. For completeness we review here the definition of a matroid, following the standard text [27] Definition 1 A matroid M = ....

....finding a minimumweight bipartite spanning tree where the blue vertices have degree at most two, is equivalent to the problem of finding a minimum weight common base of M 1 and M 2 . This is a special case of the matroid intersection problem, which was first solved in polynomial time by Edmonds [12, 13]. Other authors [8] have exploited the special structure of problems such as ours to improve the running times. We obtain the following theorem. Theorem 1 There is a polynomial time 2 approximation algorithm for the Bipartite TSP. 2.1 Finite Capacity Robot Arms We will now show how to obtain a ....

J. Edmonds. Submodular functions, matroids and certain polyhedra. In Combinatorial Structures and their Applications, Proceedings of Calgary International Conference, pages 69--87, 1970.

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J. Edmonds, Submodular functions, matroids and certain polyhedra, in: R. Guy, ed., Combinatorial Structures and Their Applications (Gordon and Breach, New York, 1970) 69-87.

.... 81 3 5841 6920, facsimile: 81 3 5841 6879 1 Introduction Discrete convex functions have long been attracting research interest in the area of discrete optimization. Miller [15] was a forerunner in the early 1970 s. The relationship between submodularity and convexity was discussed in Edmonds [3], and deeper understanding of this relationship was gained in 1980 s by Frank [5] Fujishige [6] and Lovasz [13] see also [7] Favati and Tardella [4] introduced integrally convex functions to show a local characterization for global minimality, and Dress Wenzel [2] considered valuated matroids ....

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

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J. Edmonds. Submodular functions, matroids and certain polyhedra. In R. Guy, H. Hanani, N. Sauer, and J. Sch onheim, editors, Combinatorial Structures and Their Application, pages 69--87. Gordon and Breach, 1970.

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J.Edmonds, Submodular functions, matroids and certain polyhedra, in Combinatorial Structures and their Applications, R.Guy et al, eds., Gordon and Breach, 1970, pp69-87. 18

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J. Edmonds: Submodular functions, matroids and certain polyhedra, in: R. Guy, H. Hanani, N. Sauer, and J. Schonheim, eds., Combinatorial Structures and Their Applications, Gordon and Breach, New York,

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Edmonds, J.: Submodular functions, Matroids and certain Polyhedra. Proceedings of Calgary International Conference on Combinatorial structures and their applications, 1969.

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J. Edmonds, \Submodular functions, matroids and certain polyhedra", in: R.K. Guy, et al. (eds.) Combinatorial Structures and their Applications, Gordon and Breach, New York, pp. 69-87.

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J. Edmonds [1970]: Submodular functions, matroids and certain polyhedra. In Combinatorial Structures and Their Applications, R. Guy et al., eds., Gordon and Breach, New York, 69-87.

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