W. McCuaig, N. Robertson, P. D. Seymour, and R. Thomas. Permanents, pfaffian orientations, and even directed circuits. 1996. Preprint.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of the modi ed matrix (P olya s permanent problem, P ol 13] When does a bipartite graph have a Pfaan orientation (see [Kas 63] and [Kas 67] When is a square matrix sign nonsingular (see [BS 95] and [KLM 84] For the version above that is phrased in terms of Pfaan orientations, RST 99] presents an O(n ) algorithm, based on a structural characterization of bipartite graphs that possess a Pfaan orientation. The exact de nition of a Pfaan orientation is not important here; what is relevant is that the structural characterization is in terms of a trisum operation that pastes ....

....of a trisum operation that pastes three smaller graphs along a cycle on 4 vertices. With careful implementation, the running time of that algorithm can be reduced to O(n ) but attempts at further improvements run into serious diculty. In order to take advantage of the structure theorem of [RST 99] one needs to be able, at the very least, to eciently decide whether a 4connected bipartite graph has a 4 shredder. It is not clear to us whether the bipartite ness would help. Given that the corresponding problem for 3 shredders in 3 connected graphs was not known, we started with that as the ....

Robertson, N., Seymour, P.D. and Thomas, R., Permanents, Pfaan Orientations, and Even Directed Circuits, Ann. Math. 150 (1999), 929-975.

....are linearly independent. In [4] it was shown that Theorem 1 A family F = S 1 ; Sm ) of nonempty subsets of V = fx 1 ; x n g has the EUP if and only if its incidence matrix M is not an L matrix. When m = n, it is known that the EUP can be recognized in polynomial time [8, 9], but FEUP recognition is NP complete [3] Our paper is concerned with the complexity of recognizing the EUP and FEUP in the general case of n points and m sets. Recently, in [2] it was shown that Theorem 2 For families in which there is a bound on the number of sets whose cardinality exceeds ....

N. Robertson, P.D. Seymour, and R. Thomas, Permanents, Pfaan orientations, and even directed circuits, Annals of Math., to appear.

....Szego [21] pointed out in the same year that not all matrices are convertible. The computational problem of recognising the convertible matrices had been studied extensively and it was proved recently to admit a polynomial algorithm in a seminal work by McCuaig, Robertson, Seymour and Thomas [17]. The problem of recognizing convertible matrices is known to be equivalent to many computational problems. Let us mention just one of them here, the one which gave it a name. It is the Even Cycle Problem: given a directed graph, decide whether it contains a directed cycle of even length. Let A ....

W. McCuaig, N. Robertson, P. D. Seymour, and R. Thomas. Permanents, pfaffian orientations, and even directed circuits. 1996. Preprint.

....0012 365X 98 19.00 Copyright c # 1998 Elsevier Science B.V. All rights reserved PII S0012 365X(98)00097 1 102 G. Gutin et al. Discrete Mathematics 191 (1998) 101 107 existence of an even length cycle in a digraph (the complexity until recently was not known [9,10] W. McCuaig et al. [11] proved that it is polynomial time solvable) To see that the ADC problem generalizes the even cycle problem, replace every arc (x; y) of a digraph D by two vertex disjoint alternating paths of length three, one starting from colour 1 and the other from colour 2. Clearly, the obtained ....

W. McCuaig, N. Robertson, P.D. Seymour, R. Thomas, Permanents, pfa#an orientations, and even directed circuits, submitted.

.... to compute the determinant, while Valiant proved that the problem of computing the permanent of a (0, 1) matrix is #P complete [12] The computational problem of recognition of convertible matrices has been proved recently to admit a polynomial algorithm by McCuaig, Robertson, Seymour and Thomas [8]. An earlier paper of Galluccio and Loebl [3] contains a related algorithmic result, as well as a history of the problem. The problem of recognizing convertible matrices is equivalent to the problem of recognizing bipartite Pfa#an graphs, and to the Even Cycle Problem: given a directed graph, ....

W. McCuaig, N. Robertson, P. D. Seymour, and R. Thomas. Permanents, pfaffian orientations, and even directed circuits. 1996. Preprint.

.... Kasteleyn matrix are the probabilities of local configurations of edges [19] For this reason and others, the method could well have some bearing on arctic circle phenomena in domino and lozenge tilings [8, 16] Another recent development is a satisfactory classification of bipartite Pfa#an graphs [34]. The Gessel Viennot method was discovered by Gessel and Viennot [11, 12] in the context of enumerative combinatorics, and completely independently of the permanentdeterminant method. The method was independently anticipated by Lindstrom [23] in the context of matroid theory. It has been widely ....

N. Robertson, P. D. Seymour, and R. Thomas. Permanents, Pfa#an orientations, and even directed circuits. Preprint, 1997.

.... matrix are the probabilities of local configurations of edges [19] For this reason and others, the method could well have some bearing on arctic circle phenomena in domino and lozenge tilings [8, 16] Another recent development is a satisfactory classification of bipartite Pfaffian graphs [34]. The Gessel Viennot method was discovered by Gessel and Viennot [11, 12] in the context of enumerative combinatorics, and completely independently of the permanentdeterminant method. The method was independently anticipated by Lindstrom [23] in the context of matroid theory. It has been widely ....

N. Robertson, P. D. Seymour, and R. Thomas. Permanents, Pfaffian orientations, and even directed circuits. Preprint, 1997.

.... methods on the Ising Model [6] started the line of work on the enumeration of perfect matchings by computing pfaffians, permanents, and determinants [12] 4] 2] 24] 9] 14] 22] 18] This field continues to be a very active one, and recently McCuaig, Robertson, Seymour and Thomas [16] proved an important structure theorem for noneven digraphs. 3 Preliminary Results In carrying out our plan, we find it most convenient to use the digraph formulation. Our first result will be that there is exactly one equivalence class of noneven digraphs on N vertices with M (N ) arcs, which ....

W. McCuaig, N. Robertson, P.D. Seymour, and R. Thomas, Permanents, Pfaffian Orientations, and even directed circuits, prepint 1996.

.... [7] Kastelyn s methods on the Ising Model [6] started the line of work on the enumeration of perfect matchings by computing pfaffians, permanents, and determinants [12] 22] 14] 20] 17] This field continues to be a very active one, and recently McCuaig, Robertson, Seymour and Thomas [15] proved an important structure theorem for noneven digraphs. Next we give the basic definitions and results that will be used later. A 0; Sigma1 square matrix H is a SNS(n) pattern if each term in its determinant expansion det H = X oe2Sn ( Gamma1) sgn(oe) H 1oe(1) H 2oe(2) H noe(n) has ....

W. McCuaig, N. Robertson, P.D. Seymour, and R. Thomas, Permanents, Pfaffian Orientations, and even directed circuits, prepint 1996.

.... to compute the determinant, while Valiant proved that the problem of computing the permanent of a (0; 1) matrix is #P complete [12] The computational problem of recognition of convertible matrices has been proved recently to admit a polynomial algorithm by McCuaig, Robertson, Seymour and Thomas [8]. An earlier paper of Galluccio and Loebl [3] contains a related algorithmic result, as well as a history of the problem. The problem of recognizing convertible matrices is equivalent to the problem of recognizing bipartite Pfaffian graphs, and to the Even Cycle Problem: given a directed graph, ....

W. McCuaig, N. Robertson, P. D. Seymour, and R. Thomas. Permanents, pfaffian orientations, and even directed circuits. 1996. Preprint.

....of computing the permanent is polynomial, but exponential in the rank. A.P. Il ichev, G.P. Kogan and V.N. Shevchenko [IKS97] showed polynomial algorithms for computing the permanent of certain matrices with symmetry conditions for rows. W. McCuaig and N. Robertson and P. Seymour and R. Thomas [MRST97] discuss certain 0 Gamma 1 matrices related to graph theoretic problems and U. Feige and C. Lund [FL97] show that the complexity of computing the permanent for random matrices is hard. All these papers concern matrices over fields of characteristic 0. This paper deals with the computation of ....

W. McCuaig, N. Robertson, P. Seymour, and R. Thomas. Permanents, pfaffian orientations, and even directed circuits. In STOCS'97, pages 402--405. ACM, 1997.

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W. McCuaig, N. Robertson, P. D. Seymour, and R. Thomas. Permanents, pfaffian orientations, and even directed circuits. 1996. Preprint.

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N. Robertson, P. D. Seymour and R. Thomas, Permanents, Pfa#an orientations, and even directed circuits, Math. Ann. 150 (1999), 929--975.

....Cornuejols and Vuskovic. They conjectured that every Berge graph either belongs to one of the four classes introduced in Section 2, or has a separation of one of two kinds. This is a reasonable plan, because similar approach was successful for many other graph theory problems, for instance [26, 27, 33, 34, 40], and many others. We were able to prove this conjecture, but to deduce (1.1) we needed to prove a minor modification of it. Let us first introduce the two kinds of separation. A 2 join in G is a partition (X 1 , X 2 ) of V (G) so that there exist disjoint nonempty sets A i , B i X i (i = 1, 2) ....

N. Robertson, P. D. Seymour and R. Thomas, Permanents, Pfa#an orientations, and even directed circuits, Math. Ann. 150 (1999), 929--975.

....if and only if it can be obtained from strongly planar digraphs by means of certain composition operations. Our main tool in the proof is a characterization of bipartite graphs that have a Pfaffian orientation, found independently by McCuaig [1] and by Robertson, Seymour and the second author [6]. We present the characterization in Section 5. The rest of the paper is organized as follows. In Section 2 we mention three related results. Section 3 reduces the problem to strongly 2 connected digraphs. It is shown in Section 4 that strongly planar digraphs pack. Sections 6 8 show that the ....

....let G 1 ; G 2 be subgraphs of G 0 such that G 1 [ G 2 = G 0 ; G 1 G 2 = C, and V (G 1 ) V (G 2 ) 6= 6= V (G 2 ) V (G 1 ) Let G be obtained from G 0 by deleting all the edges of C . In this case we say that G is the 4 sum of G 1 or G 2 along C. This is a slight departure from the definition in [6], but the class of simple graphs obtainable according to our definition is the same, because we allow parallel edges. Let G 0 be a bipartite graph, let C be a central circuit of G 0 of length 4, and let G 1 ; G 2 ; G 3 be three subgraphs of G 0 such that: G 1 [ G 2 [ G 3 = G 0 and for distinct ....

[Article contains additional citation context not shown here]

N. Robertson, P. D. Seymour and R. Thomas, Permanents, Pfaffian orientations, and even directed circuits, Ann. Math. 150 (1999), 929--975.

.... the determinant, while Valiant proved that the problem of computing the permanent of a 0; 1 matrix is #P complete (see [12] The computational problem of recognition of convertible matrices has been proved recently to admit a polynomial algorithm by McCuaig, Robertson, Seymour and Thomas (see [8]) An earlier paper of Galluccio and Loebl (see [3] contains a related algorithmic result, as well as the history of the problem. The recognition of convertible matrices is equivalent to the problem of recognition of bipartite Pfaffian graphs, and to the Even Cycle Problem : given a directed ....

W. McCuaig, N. Robertson, P. D. Seymour, and R. Thomas. Permanents, pfaffian orientations, and even directed circuits. 1996. Preprint. 18

....graph has a Pfaffian orientation is equivalent to: ffl P olya s permanent problem [24] ffl the even directed cycle problem [34] 41] 43] 44] ffl a hypergraph problem [32] ffl the sign nonsingular matrix problem [9] 18] 42] THEOREM. McCuaig [23] Robertson, Seymour and Thomas [29]) A bipartite graph has a Pfaffian orientation if and only if it can be obtained from planar bipartite graphs and the Heawood graph by means of 0 , 2 and 4 sums. Figure 2. 4 sum. COROLLARY. There exists a cubic algorithm to solve the above mentioned problems. 8 LINKLESS EMBEDDINGS A ....

N. Robertson, P. D. Seymour and R. Thomas, Permanents, Pfaffian orientations, and even directed circuits, manuscript.

....0 is an integer, if every matching of size at most k can be extended to a perfect matching. A 2 extendable connected bipartite graph is called a brace. It is easy to see that the problem of finding Pfaffian orientations of bipartite graphs can be reduced to braces. The following is our main result [7]. 1.2) A brace has a Pfaffian orientation if and only if it is either isomorphic to the Heawood graph, or can be obtained by repeated application of the sum operation, starting from planar braces. By [15] this also solves the even circuit problem for directed graphs (see [9, 10, 12, 13] by [8] ....

N. Robertson, P. D. Seymour and R. Thomas, Permanents, Pfaffian orientations, and even directed circuits, manuscript.

....minor isomorphic to K 3;3 . Theorem 11.1 is a beautiful result, but, unfortunately, it does not seem to imply a polynomial time algorithm to test if a given bipartite graph has a Pfaffian orientation. The next theorem, proven independently McCuaig [34, 35] and by Robertson, Seymour and Thomas [53], can be used to design such an algorithm. We say that a bipartite graph is a brace if every matching of size at most two can be extended to a perfect matching. An argument similar to the one in the proof of Theorem 1.2 shows that it suffices to characterize braces that have a Pfaffian ....

....characterization. Theorem 11.2 A brace has a Pfaffian orientation if and only if either it is isomorphic to the Heawood graph, or it can be obtained from planar braces by repeated applications of the C 4 sum operation. Using Theorem 11.2 we were able to design a polynomial time algorithm [53] to decide if an input graph has a Pfaffian orientation, formally as follows. Theorem 11.3 There exists an O(n 3 ) algorithm that, given an input graph G on n vertices, either outputs a Pfaffian orientation of G, or a valid statement that G has no Pfaffian orientation. Let me now describe ....

N. Robertson, P. D. Seymour and R. Thomas, Permanents, Pfaffian orientations, and even directed circuits, manuscript.

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N. Robertson, P.D. Seymour and R. Thomas, Permanents, Pfaan orientations, and even directed circuits, Ann. Math. (2) 150 (1999), 929-975.

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N. Robertson, P.D. Seymour and R. Thomas, Permanents, Pfa#an orientations, and even directed circuits, Ann. Math. (2) 150 (1999), 929--975.

No context found.

Robertson, N., P.D. Seymour and R. Thomas, Permanents, Pfaan orientations, and even directed circuits. Ann. of Math. (2) 150, 1999, 929-975.

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McCuaig, W., Robertson, N., Seymour, P.D. and Thomas, R., Permanents, Pfaan Orientations, and Even Directed Circuits (Extended abstract), Proc. STOC (1997).

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