A. Frank, Conservative Weightings and Ear-Decompositions of Graphs. Combinatorica 13 (1993) 65-81.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....A subset of edges J E(G) is called a join of G, if at most half the edges of each cycle in G belong to J . We consider the problem of nding a join of maximum weight that can be stated as follows: given a graph G and an edge weighting c : E(G) R, nd a join of G with maximum weight. Frank [4] proves that the problem is polynomial time solvable in the case of unit weights. In Section 2 of this paper we show that the problem is NP hard in the case of 0; 1 weights, which answers in the negative a question Translated from Discrete Analysis and Operations Research, Novosibirsk, 4, no. 3 ....

....and Operations Research, Novosibirsk, 4, no. 3 (1997) 3 8. Supported in part by DIMANET PECO, contract ERBCIPDCT 94 0623, and by the Russian Foundation for Basic Research, grant 96 01 01614. E mail address: ageev math.nsc.ru Preprint submitted to Elsevier Preprint 22 February 2000 in [4]. In Section 3 we show that in the case of series parallel graphs and arbitrary weights the problem can be solved in time O(n ) where n is the number of vertices in the graph. 2 NP hardness Theorem 1 The problem of nding a join of maximum weight in a graph G is NP hard even when restricted ....

A. Frank, Conservative weightings and ear decompositions of graphs, Combinatorica 13 (1993) 65-82.

....graphs (see Lucchesi [14] Lucchesi and Younger [15] Frank [8] or Grotschel et al. 11, p. 252] By contrast, the complexity status of CUT PACKING on general graphs still remains an open problem. In this paper we use a characterization in [1] and an algorithmic result on joins due to Frank [9] to prove that CUT PACKING is polynomially solvable when restricted to the family of Seymour graphs. To present a rigorous definition of Seymour graph we need to introduce a few more notions. At this point we only notice that the family includes both all bipartite and all series parallel graphs ....

....that the polynomial time solvability in the case of bipartite graphs was shown earlier and in a different way by D. H. Younger [20] To prove that CUT PACKING is polynomially solvable on Seymour graphs we shall use the above characterization and the following algorithmic result. Theorem 2 (Frank [9]) Given an undirected graph G, a join of maximum cardinality in G can be found in polynomial time. The running time of Frank s algorithm is bounded by O(mn) where n = jV (G)j, m = jE(G)j [9] Let G be a Seymour graph and J be a join of G with maximum cardinality returned by Frank s algorithm. ....

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Frank, A: Conservative weightings and ear-decompositions of graphs. Combinatorica 13 (1993) 65--81.

....Let G = V; E) be a given 2 EC graph, with jV j = n. W.l.o.g. we can assume that G is 2 VC. Otherwise we can solve the 2 EC problem separately on each 2 VC component. The 1st step of the algorithm nds an ear decomposition H of G with minimum number of even ears, using the algorithm of Frank [15, 7] (delete all 1 ears, since they are redundant) In the 2nd step, the algorithm performs all possible 1 opt exchanges on H w.r.t. 2 EC. The resulting ear decomposition, say H , is the output. Lemma 4.1. 7] n 1 is a lower bound on the optimum 2 EC solution in G. An ear decomposition with ....

A. Frank. Conservative weightings and eardecompositions of graphs. Combinatorica, 13, 65-81,

....in Computer Science, Centre of the Danish National Research Foundation. 1 Graph Class Complexity Approx. Achievable General NP hard [8] O( n log 2 n ) Planar NP hard [7] 2 Degree d NP hard [7] 2d 6 5 Perfect 2 Triangulated O(m) Co triangulated O(mn) Bipartite O(mn) [10] k partite k 1 Seymour O(minfmn 2 ; n 3:5 log 1:5 n p (n 2 ; n)g) Table 1: Summary of the results (and open questions) on Cut Packing. thresholds of the two problems coincide. More precisely, any lower bound on the approximability of Independent Set is also a lower bound for ....

....of the two problems coincide. More precisely, any lower bound on the approximability of Independent Set is also a lower bound for Cut Packing. Note that we are considering the case of undirected graphs, for the problem of packing directed cuts in digraphs can be solved in polynomial time (see [17, 18, 10, 12]) In [2] Ageev has shown that Cut Packing can be solved in polynomial time on Seymour graphs, a class containing bipartite and series parallel graphs. In this paper we further study the complexity of Cut Packing. First, we illustrate a stronger connection between Cut Packing and Independent ....

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A. Frank, Conservative Weightings and Ear-Decompositions of Graphs. Combinatorica 13 (1993) 65-81.

....the independent sets whose contraction allows an odd ear decomposition form the family of feasible sets of an (up to a twist) representable Delta matroid. Some combinatorial properties of related matrices will also be exhibited. 1 Introduction The parameter (G) introduced by A. Frank in [Frank], which denotes the minimum number of even ears an ear decomposition of graph G can have, can be characterized in several other ways: G) is the minimum number of edges whose contraction results in a factor critical graph [Frank] A much deeper result in [Frank] is that it can be related to the ....

....1 Introduction The parameter (G) introduced by A. Frank in [Frank] which denotes the minimum number of even ears an ear decomposition of graph G can have, can be characterized in several other ways: G) is the minimum number of edges whose contraction results in a factor critical graph [Frank]. A much deeper result in [Frank] is that it can be related to the covering radius of the cycle code ae(G) of G by the following formula ae(G) rk (G) G) 2 , and both (G) and ae(G) can be computed in polynomial time. G) can also be expressed as the corank of a suitable evaluation of the ....

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Frank, A. (1993) Conservative Weightings and Ear-Decompositions of Graphs, Combinatorica 2 247-274

....variant of Cycle Packing. The input is a graph Basic Research in Computer Science, Centre of the Danish National Research Foundation. 1 Graph Class Complexity Hardness of Approx. Approx. Achievable General NP hard [13] APX hard O(log n) Planar NP hard 2 [6] Planar Eulerian O(mn) [11, 6] G( NP hard [13] 2(1 ) Table 1: Summary of the results (and open questions) on Cycle Packing. whose edge set is bicolored, i.e. partitioned into, say, grey and black edges. One is to nd the largest collection of edge disjoint alternating cycles. A cycle is alternating if it ....

A. Frank, Conservative Weightings and Ear-Decompositions of Graphs. Combinatorica 13 (1993) 65-81.

....connected. Our method is based on a matching theory result of Andr as Frank, namely, there is a good characterization for the minimumnumber of even length ears over all possible ear decompositions of a graph, and moreover, an ear decomposition achieving this minimum can be computed efficiently, [4]. Recall that the 2 approximation heuristic starts with an arbitrary ear decomposition of G. Instead, if we start with an ear decomposition that maximizes the number of 1 ears, and if we discard all the 1 ears, then we will obtain the optimal solution. In fact, we start with an ear decomposition ....

....L (G) denote jV j (G) Gamma 1. A join of a graph G is an edge set J E(G) such that for (the edge set of) every cycle Q E(G) we have jJ Qj jQj=2. For example, any matching is a join. Let (G) denote the maximum size of a join of the graph G. The proof of the next result appears in [4], see Theorem 4.5 and Section 2 of [4] Theorem 3.1 (A. Frank [4] Let G = V; E) be a 2 edge connected graph. An ear decomposition P 0 P 1 : P k of G having (G) even ears can be computed in time O(jV j Delta jEj) Moreover, L (G) 2 (G) Proposition 3.2 For every 2 node ....

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A. Frank, "Conservative weightings and ear-decompositions of graphs," Combinatorica 13 (1993), 65--81.

....connected. Our method is based on a matching theory result of Andr as Frank, namely, there is a good characterization for the minimum number of even length ears over all possible ear decompositions of a graph, and moreover, an ear decomposition achieving this minimum can be computed efficiently, [2]. Recall that the 2 approximation heuristic starts with an arbitrary ear decomposition of G. Instead, if we start with an ear decomposition that maximizes the number of 1 ears, and if we discard all the 1 ears, then we will obtain the optimal solution. In fact, we start with an ear decomposition ....

....ear decompositions. For example: G) 0 if G is a factor critical graph (e.g. G is an odd clique K 2 1 or an odd cycle C 2 1 ) G) 1 if G is an even clique K 2 or an even cycle C 2 , and (G) Gamma 1 if G is the complete bipartite graph K 2; 2) Theorem 3.1 (A. Frank [2]) Let G = V; E) be a 2 edge connected graph. i) There is a good characterization for , namely, G) 2 (G) Gamma jV j 1, where (G) denotes the maximum cardinality of a join of G. ii) An ear decomposition P 0 P 1 : P k of G having (G) ears of even length can be computed in time ....

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A. Frank, "Conservative weightings and ear-decompositions of graphs," Combinatorica 13 (1993), 65--81.

....connected. Our method is based on a matching theory result of Andr as Frank, namely, there is a good characterization for the minimum number of even length ears over all possible ear decompositions of a graph, and moreover, an ear decomposition achieving this minimum can be computed efficiently, [4]. Recall that the 2 approximation heuristic starts with an arbitrary ear decomposition of G. Instead, if we start with an ear decomposition that maximizes the number of 1 ears, and if we discard all the 1 ears, then we will obtain the optimal solution. In fact, we start with an ear decomposition ....

.... For example: G) 0 if G is a factor critical graph (e.g. G is an odd clique K 2 1 or an odd cycle C 2 1 ) G) 1 if G is an even clique K 2 or an even cycle C 2 , and (G) Gamma 1 if G is the complete bipartite graph K 2; 2) The proof of the next result appears in [4], see Theorem 4.5 and Section 2 of [4] Theorem 5 (A. Frank [4] Let G = V; E) be a 2 edge connected graph. An ear decomposition P 0 P 1 : P k of G having (G) even ears of length 2 can be computed in time O(jV j Delta jEj) Proposition 6. Let G be a 2 node connected graph. An ....

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A. Frank, "Conservative weightings and ear-decompositions of graphs," Combinatorica 13 (1993), 65--81.

....see Welsh [165] Many problems that are difficult for general linear codes, are polynomial for cycle codes of graphs. Examples are decoding, see Ntafos and Hakimi [124] finding a basis of minimal total weight, see Chickering et al. 40] Horton [85] and computing the covering radius, see Frank [66]. Theorem 4.4 is due to McLoughlin [122] The complexity of the Weight of error problem with preprocessing (P 9 ) was proved in Bruck and Naor [33] Their reduction is from K 3 . The proof that we give, following Lobstein [24, pp. 121 123] has the advantage of being valid for any fixed code ....

A. Frank, "Conservative weightings and ear-decompositions of graphs," Combinatorica 13 (1) (1993), 65--81.

....incident with an odd number of non loop edges in Sg (in some papers, T joins are also required to be acyclic) A T cut is an edge cut in G which contains an odd number of vertices of T on each shore. T joins and T cuts are closely related to matchings and have been studied by various authors [31, 13, 34, 35, 37, 38, 39, 40, 45, 9]. One easily checks that, when jT j is even, the cycles of a graft G T are precisely the cycles of G, together with sets of the form f g [ J where J is any T join in G. The cocycles of G T are precisely the cuts of G which are not T cuts, together with the sets of the form f g[B where B is any ....

A. Frank, Conservative weightings and ear-decompositions of graphs. in "Integer Programming and Combinatorial Optimization (IPCO) Proceedings ", R. Kannan, W. R. Pulleyblank, Eds., University of Waterloo Press, Waterloo, 1990, pp.217-230.

....paths. Section 4 is devoted to the basic min max results and their relationship to conservative weightings. The fundamental theorem of A. Sebo on the structure of distances is proved in Section 5. A consequence, due to E. Korach and M. Penn, and its recent extension by A Frank and Z. Szigeti [1993] will then be derived from Sebo s theorem. Section 6 includes two theorems concerning the special role the complete graph K 4 plays in the theory. The first one is due to E. Korach while the second one is Seymour s theorem on max flow min cut matroids specialized to T joins. In Section 7 we apply ....

....in polynomial time. Can we compute in polynomial time, as well A weaker problem for a graft is to decide in polynomial time whether = This is weaker indeed as is computable in polynomial time. But even this second question is NP complete, a result due to M. Middendorf and F. Pfeiffer [1993]: THEOREM 3.1 Determining whether (G; T ) G; T ) is NP complete even for planar graphs G. In particular, computing (G; T ) is NP complete. ffl In spite of this negative result, there are important special classes of grafts when = holds true. This is the case, for example, if G is ....

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A. Frank, Conservative weightings and ear-decompositions of graphs, Combinatorica, 1993. 13 (1) pp. 65-81.

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A. Frank, Conservative Weightings and Ear-Decompositions of Graphs. Combinatorica 13 (1993) 65-81.

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