M. Conforti, G. Cornu'ejols, A. Kapoor, M.R. Rao and K. Vuskovi'c, Balanced matrices, in Mathematical Programming: State of the Art 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....been discovered. Several of these have come from studying properties of the polytopes of restricted formulations of CNF formulas as linear programs. For example, the extended Horn formulas of Chandru and Hooker [14] and the formulas corresponding to balanced matrices due to Conforti and Cornuejols [17], both of which rely on unit resolution for solutions. These have been generalized by the SLUR formulas [53] Others are a consequence of fast matrix decompositions. For example, the q Horn class, originally discovered by Boros and others [8, 9, 10] was shown by Truemper [59, 60] to be a special ....

M. Conforti, G. Cornuejols, A. Kapoor, K. Vuskovic, and M.R. Rao. Balanced Matrices. In J.R. Birge and K.G. Murty, eds. Mathematical Programming: State of the Art. Braun-Brumeld, United States. Produced in association with the 15th International Symposium on Mathematical Programming, University of Michigan, 1994.

....(introduced in [14] which themselves generalize the class of totally unimodular 0, 1 matrices. It is known that every balanced 0, 1 matrix is perfect, and every totally unimodular 0, 1 matrix is balanced. For the definitions and related results we refer the reader to the survey paper [6]. The study of perfect 0, 1 matrices was initiated by the analogy to the well studied class of perfect 0, 1 matrices. A 0, 1 matrix A is said to be perfect if the associated set packing polytope P (A) x Ax # 1, 0 # x # 1 is integral. Perfect matrices have many interesting properties, ....

M. Conforti, G. Cornuejols, A. Kapor, K. Vuskovic and M. Rao. Balanced Matrices. in The State of the Art: Mathematical Programming, Ann Arbor, August, 1994.

....F areexactlythe2 connected 2 cycled graphs. 1 Introduction A graph is said to be 2 cycled if each of its edges is contained in exactly two chordless cycles. The 2 cycled graphs arise in connection with the study of balanced signing of graphs and matrices by Truemper [3] and by Conforti et al. [2], as indicated in the next three paragraphs. A signed graph is a graph G = V,E) together with a mapping f : E # 1, 1 . Consider a mapping # : C # 0, 1, 2, 3 ,whereC is the set of chordlesscyclesofG.If# e#C f (e) # #(C) mod4)forallC #C,wesay that the signed graph is # balanced. A ....

M.Conforti,G.Cornuejols, A. Kapoor, K. Vuskovic, and M.R. Rao. Balanced matrices. In J.R. Birge and K.G. Murty, editors, Mathematical Programming State of the Art 1994, pages 1--33. The University of Michigan, 1994.

....FRANC OIS MARGOT Finally, we introduce a new class of clutters called weakly binary. They can be viewed as a generalization of binary and of balanced clutters. A 0,1 matrix is balanced if it does not have A(C 2 k ) as a submatrix, k 3 odd, where as above C 2 k denotes an odd hole. See [4] for a survey of balanced matrices) We say that a clutter C has an odd hole C 2 k if A(C 2 k ) is a submatrix of A(C) An odd hole C 2 k of C is said to have a non intersecting set if 9S 2 E(C) such that S V (C 2 k ) A clutter is weakly binary if, in C and all its minors, all ....

Conforti M., Cornuejols G., Kapoor A., Vu#skovic K. (1994) Balanced matrices, Math. Programming, State of the Art 1994 (J. R. Birge, K. G Murty, eds.), 1-33.

....are exactly the 2 connected 2 cycled graphs. 1 Introduction A graph is said to be 2 cycled if each of its edges is contained in exactly two chordless cycles. The 2 cycled graphs arise in connection with the study of balanced signing of graphs and matrices by Truemper [3] and by Conforti et al. [2], as indicated in the next three paragraphs. A signed graph is a graph G = V; E) together with a mapping f : E Gamma f 1; Gamma1g. Consider a mapping ff : C Gamma f0; 1; 2; 3g, where C is the set of chordless cycles of G. If Sigma e2C f(e) j ff(C) mod 4) for all C 2 C, we say that the ....

M. Conforti, G. Cornu'ejols, A. Kapoor, K. Vuskovi'c, and M.R. Rao. Balanced matrices. In J.R. Birge and K.G. Murty, editors, Mathematical Programming State of the Art 1994, pages 1--33. The University of Michigan, 1994.

....be colored either red or blue in suchaway that every edge with at least two nodes contains both a red node and a blue node. Hall s theorem [26] about the existence of a perfect matching in a bipartite graph extends to balanced hypergraphs [14] Further results on balanced matrices are surveyed in [12]. Totally unimodular matrices A matrix is totally unimodular if every square submatrix has a determinant equal to 0, 1 or 1. A consequence of Theorem 1.1 is the following result, proved in Section 2 of this paper. Theorem 1.2 If a 0,1 matrix is balanced but not totally unimodular, then its ....

M. Conforti, G. Cornu'ejols, A. Kapoor, M.R. Rao and K. Vuskovi'c, Balanced matrices, in Mathematical Programming: State of the Art 1994.

....number of nonzero entries per row and per column whose sum of the entries is congruent to 2 mod 4. In this tutorial, we survey what is currently known about balanced matrices, including polyhedral results, structural theorems and recognition algorithms. A previous survey on this topic appears in [19]. 2 2 Integral Polytopes A polytope is integral if all its vertices have only integer valued components. Given an n m 0; 1 matrix A, the set packing polytope is P (A) fx 2 R n : Ax 1; 0 x 1g; where 1 denotes a column vector of appropriate dimension whose entries are all equal to ....

M. Conforti, G. Cornuejols, A. Kapoor, K. Vuskovic and M. R. Rao, Balanced matrices, in: Mathematical Programming, State of the Art

....a polynomial time recognition algorithm for these matrices. In this paper, using a similar approach, we give a polynomial time recognition algorithm for balanced 0# Sigma1 matrices, using a decomposition result derived in the companion paper [1] For a survey of results on balanced matrices, see [2]. Aconvenient setting for working with balanced 0# Sigma1 matrices is to consider their signed bipartite graph representations. A signedgraph G is a graph together with an assignmentof 1 or ;1weights to the edges. Givena0# Sigma1 matrix A,thesignedbipartite graph representation of A is a ....

M. Conforti, G. Cornu'ejols, A. Kapoor, K. Vuskovi'c, Balanced matrices, in Mathematical Programming: State of the Art 1994, J.R. Birge and K.R. Murty eds., The Universityof Michigan Press (1994) 1-33. 26

....inferencecan be solvedinpolynomial time by linear programming when the corresponding 0# Sigma1 matrix A is balanced. In fact SAT and logical inference can be solved by repeated application of unit resolution when the underlying 0# Sigma1 matrix A is balanced [5] These results are surveyed in [7]. 3 Balanceable 0# 1 Matrices In this section, we consider the following question: given a 0# 1 matrix, is it possible to turn some of the 1 s into ;1 s in order to obtain a balanced 0# Sigma1 matrix A 0# 1 matrix for which such a signing exists is called a balanceable matrix. Givena0# 1 ....

M. Conforti, G. Cornu'ejols, A. Kapoor, M. R. Rao, K. Vuskovi'c, Balanced matrices, in Mathematical Programming: State of the Art 1994, J.R. Birge and K.G. Murty eds., The University of Michigan Press (1994) 1-33.

....given set of clauses defines a balanced 0; Sigma1 matrix, then as shown by Conforti and Cornuejols [26] the satisfiability problem 21 is trivial to solve and the associated MAXSAT problem is solvable in polynomial time by linear programming. A survey of balanced matrices is in Conforti et al. [29]. A = 2 6 6 6 6 6 6 6 4 1 1 0 0 1 1 1 1 1 0 1 0 1 0 0 1 3 7 7 7 7 7 7 7 5 A = 2 6 6 6 6 6 6 6 4 1 1 0 0 1 1 1 0 1 1 1 1 3 7 7 7 7 7 7 7 5 Figure 2. A Balanced Matrix and A Perfect Matrix Definition 4.9 A 0; 1 matrix A is perfect if the set packing ....

M.Conforti, G.Cornuejols, A.Kapoor, K.Vuskovic, and M.R.Rao, Balanced Matrices, in Mathematical Programming, State of the Art 1994 (J.R.Birge and K.G.Murty eds.), University of Michigan, 1994.

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M. Conforti, G. Cornujols, A. Kapoor, K. Vusk ovi c, and M. R. Rao. Balanced matrices. In J. R. Birge and K. G. Murty, editors, Mathematical Programming: State of the Art 1994, pages 1 -- 33. The University of Michigan, 1994.

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