H. N. Gabow and M. Stallmann, "Efficient Algorithms for Graphic Matroid Intersection and Parity," Automata, Language and Programming: 12 Colloq., Lecture Notes in Computer Science, Vol. 194, 210--220, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....edges to connect the triangles. Second, it is not known whether a heaviest triangular structure in G can be found in polynomial time. Graphical matroid parity is used to find a largest triangular structure in an unweighted graph [CFFK98] graphical matroid parity being solvable in polynomial time [GS85]; weighted graphical matroid parity is not known to be NP hard or in P. Third, although in the unweighted case, proving that a heaviest (i.e. largest) triangular structure in G has weight at least (1=3 ffl)Opt(G) is not hard (indeed, a greedily constructed triangular structure suffices) ....

H.N. Gabow and M. Stallmann, "Efficient Algorithms for Graphic Matroid Intersection and Parity," Automata, Language and Programming: 12 Colloq., Lecture Notes in Computer Science 194 (1985), 210--220.

....edges and have exactly one source and one sink, see Figure 4. Phi Phi Phi Phi Theta Fig. 4: Two planar source sink graphs 45 46 GRAPHS [4. 1 Reachability in planar source sink graphs [25] O(logn) y upward planar source sink graphs grid graphs y O(log n) [27, 19] Upward planarity testing Monotone point location y O(logn) 7] Tab. 1: Overview of graph results with references. Results from this thesis are marked with y. Preliminaries A graph is embeddable on a surface if it can be drawn on the surface such that the edges do not intersect except ....

Harold N. Gabow and Matthias Stallman. Efficient algorithms for graphic matroid intersection and parity. In Proc. 12th ICALP, volume 194 of Lecture Notes in Computer Science, pages 210--220. Springer Verlag, Berlin, 1985.

....edges to connect the triangles. Second, it is not known whether a heaviest triangular structure in G can be found in polynomial time. Graphical matroid parity is used to find a largest triangular structure in an unweighted graph [CFFK98] graphical matroid parity being solvable in polynomial time [GS85]; weighted graphical matroid parity is known neither to be NP Hard nor in P. Third, although in the unweighted case, proving that a heaviest (i.e. largest) triangular structure in G has weight at least (1=3 ffl)Opt(G) is not hard (indeed, a greedily constructed triangular structure suffices) ....

H. N. Gabow and M. Stallmann, "Efficient Algorithms for Graphic Matroid Intersection and Parity," Automata, Language and Programming: 12 th Colloq., Lecture Notes in Computer Science 194 (1985), 210-220.

....S is a cactus. For each two edges in F we get a triangle in S, for a total of p triangles. This completes the proof. As described by Chiba and Nishizeki [CN85] we can explicitly list all the triangles in a graph G with m edges in time O(m 3=2 ) So jE 0 j is O(m 3=2 ) Gabow and Stallmann [GS85] describe an algorithm for GMP, which runs in time O(m 0 n 0 log 6 n 0 ) where m 0 and n 0 are the number of edges and vertices, respectively, in the input graph. In our case, n 0 = n and m 0 = jE 0 j, which is O(m 3=2 ) This gives a time bound of O(m 3=2 n log 6 n) for ....

H. N. Gabow and M. Stallmann, "Efficient Algorithms for Graphic Matroid Intersection and Parity," Automata, Language and Programming: 12 th Colloq., Lecture Notes in Computer Science, Vol. 194, 210--220, 1985.

....b d c e f g (a) Figure 2: a) A planar graph (bold) and its dual (shaded) b) Primal and dual spanning trees for the graph of (a) Lemma 4 [20, pp. 289] Given a spanning tree T in G, let T be the set of dual edges fe je is not in Tg. The set T is a spanning tree for G . Corollary 1 [6, 9] T is a minimum spanning tree for G if and only if T is a maximum spanning tree for G . Given a rooted dual maximum spanning tree T , the replacement edge of f 2 T is the relevant edge of maximum cost. We use the following relationship between primal and dual edges. Lemma 5 Let f ....

H. N. Gabow and M. Stallmann. Efficient algorithms for graphic matroid intersection and parity (extended abstract). In Automata, Languages, and Programming, 12 th Colloquium, Lecture Notes in Computer Science, vol. 194, pages 210--220. Springer-Verlag, Berlin, 1985.

.... specifically, the best algorithm presented in [CFFK96] from now on denoted by A, simply outputs a maximum triangular structure in the input graph G (that is, which is a subgraph of G) Finding a maximum triangular structure in G is solvable in polynomial time, by using a matroid parity algorithm [LP86, GS85]. Given a graph H, let mts(H) denote the number of edges in a maximum triangular structure in H. Define ae(H) mts(H) jE(H)j if E(H) 6= and ae(H) 1 if E(H) If H is a maximum planar subgraph of G, then mts(G) mts(H) ae(H)jE(H)j, implying that the approximation ratio of algorithm A ....

H. N. Gabow and M. Stallmann, "Efficient Algorithms for Graphic Matroid Intersection and Parity", Automata, Language and Programming: 12 th Colloq. , Lecture Notes in Computer Science, Vol. 194, 210--220, 1985.

....as a function of the output change [17, 40] The main dynamic problems considered on directed graphs include shortest paths and transitive closure. For lack of space, we do not include in this chapter dynamic algorithms for planar graphs, which have received considerable attention in recent years [6, 7, 11, 12, 18, 21, 22, 28, 32, 34, 42, 46, 47, 48, 51], and focus our attention to general undirected graphs only. The remainder of the chapter is organized as follows. In Section 2 we give some preliminary definitions and a little terminology. Dynamic tree problems are considered in Section 3, while in Section 4 we describe partially dynamic ....

H. N. Gabow and M. Stallman. Efficient algorithms for graphic matroid intersection and parity. In Proc. 12th Int. Coll. Automata, Languages, and Programming, pages 210--220. Lecture Notes in Computer Science 194, Springer-Verlag, Berlin, 1985.

.... vice versa) in time O(n) For lack of space, we omit the proof of this lemma, which is used implicitly in [LP86] As described by Chiba and Nishizeki [CN85] we can explicitly list all the triangles in a graph G with m edges in time O(m 3=2 ) So jE 0 j is O(m 3=2 ) Gabow and Stallmann [GS85] describe an algorithm for GMP, which runs in time O(m 0 n 0 log 6 n 0 ) where m 0 and n 0 are the number of edges and vertices, respectively, in the input graph. In our case, n 0 = n and m 0 = jE 0 j, which is O(m 3=2 ) This gives a time bound of O(m 3=2 n log 6 n) for ....

H. N. Gabow and M. Stallmann, "Efficient Algorithms for Graphic Matroid Intersection and Parity," Automata, Language and Programming: 12 th Colloq., Lecture Notes in Computer Science, vol. 194, 210--220, 1985.

....edges to connect the triangles. Second, it is not known whether a heaviest triangular structure in G can be found in polynomial time. Graphical matroid parity is used to find a largest triangular structure in an unweighted graph [CFFK96a] graphical matroid parity being solvable in polynomial time [GS85]; weighted graphical matroid parity is known neither to be NP Hard nor in P. Third, although in the unweighted case, proving that a heaviest (i.e. largest) triangular structure in G has weight at least (1=3 ffl)Opt(G) is not hard (indeed, a greedily constructed triangular structure suffices) ....

H. N. Gabow and M. Stallmann, "Efficient Algorithms for Graphic Matroid Intersection and Parity," Automata, Language and Programming: 12 th Colloquium, Lecture Notes in Computer Science 194 (1985), 210-220.

....as Voronoi diagrams. Algorithms have been proposed for maintaining the embedding of a planar graph [29] and for incremental planarity testing [2, 3] The dynamic minimum spanning tree problem has been considered by Spira and Pan [28] Chin and Houck [7] Frederickson [10] and Gabow and Stallmann [11]. Frederickson gives an algorithm based on topology trees that runs in O( p m) time per operation on general graphs, and O( log n) 2 ) time on plane graphs. As Frederickson notes, the minimum spanning tree for a general graph being modified on line by edge additions alone can be main2 tained ....

....on plane graphs. As Frederickson notes, the minimum spanning tree for a general graph being modified on line by edge additions alone can be main2 tained in O(log n) amortized 1 or worst case time per operation, using the dynamic tree data structure of Sleator and Tarjan [26] Gabow and Stallmann [11] improve Frederickson s bound for planar graphs to O(log n) time per operation for the case of a fixed graph with changing edge weights. Their method also uses the dynamic tree data structure. In this paper we present a data structure and an algorithm for maintaining a minimum spanning forest of ....

[Article contains additional citation context not shown here]

H. N. Gabow and M. Stallmann. Efficient algorithms for graphic matroid intersection and parity (extended abstract). In Automata, Languages, and Programming, 12 th Colloquium, Lecture Notes in Computer Science, vol. 194, pages 210--220. Springer-Verlag, Berlin, 1985.

....S is a cactus. For each two edges in F we get a triangle in S, for a total of p triangles. This completes the proof. As described by Chiba and Nishizeki [CN85] we can explicitly list all the triangles in a graph G with m edges in time O(m 3=2 ) So jE 0 j is O(m 3=2 ) Gabow and Stallmann [GS85] describe an algorithm for GMP, which runs in time O(m 0 n 0 log 6 n 0 ) where m 0 and n 0 are the number of edges and vertices, respectively, in the input graph. In our case, n 0 = n and m 0 = jE 0 j, which is O(m 3=2 ) This gives a time bound of O(m 3=2 n log 6 n) for ....

H. N. Gabow and M. Stallmann, "Efficient Algorithms for Graphic Matroid Intersection and Parity," Automata, Language and Programming: 12 th Colloq., Lecture Notes in Computer Science, Vol. 194, 210--220, 1985.

....of 2 Theta n matrices, and our goal is to find a maximum size sub collection such that they can be stacked into a single matrix of maximum rank. This problem was solved by Lov asz [4] and much more efficient algorithms were later found by Orlin and Vande Vate [5] and Gabow and Stallmann [3] for graphic matroids. Numerous graph problems can be expressed in the language of linear matroids, therefore one should expect that a number of them would be solved using an algorithm for matroid parity. Known examples are finding a minimum feedback vertex set in a cubic graphs, equivalently, a ....

H.N. Gabow and M. Stallmann. Efficient algorithms for graphic matroid intersection and parity. Proc. ICALP'85. Lecture Notes in Computer Science, 194: 210--220, 1985.

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H. N. Gabow and M. Stallmann, "Efficient Algorithms for Graphic Matroid Intersection and Parity," Automata, Language and Programming: 12 Colloq., Lecture Notes in Computer Science, Vol. 194, 210--220, 1985.

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