Results 1 
8 of
8
Graph minor algorithm with the parity condition
"... We generalize the seminal Graph Minor algorithm of Robertson and Seymour to the parity version. We give polynomial time algorithms for the following problems: 1) the parity Hminor (Odd Kkminor) containment problem, and 2) the disjoint paths problem with k terminals and the parity condition for ea ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
We generalize the seminal Graph Minor algorithm of Robertson and Seymour to the parity version. We give polynomial time algorithms for the following problems: 1) the parity Hminor (Odd Kkminor) containment problem, and 2) the disjoint paths problem with k terminals and the parity condition for each path, as well as several other related problems. We present an O(mα(m,n)n) time algorithm for these problems for any fixed k, where n,m are the number of vertices and the number of edges, respectively, and the function α(m,n) is the inverse of the Ackermann function (see Tarjan [69]). Note that the first problem includes the problem of testing whether or not a given graph contains k disjoint odd cycles
ODD CYCLE TRANSVERSALS AND INDEPENDENT SETS IN FULLERENE GRAPHS
, 2013
"... Abstract. A fullerene graph is a cubic bridgeless plane graph with all faces of size 5 and 6. We show that every fullerene graph on n vertices can be made bipartite by deleting at most √ 12n/5 edges, and has an independent set with at least n/2 − √ 3n/5 vertices. Both bounds are sharp, and we chara ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. A fullerene graph is a cubic bridgeless plane graph with all faces of size 5 and 6. We show that every fullerene graph on n vertices can be made bipartite by deleting at most √ 12n/5 edges, and has an independent set with at least n/2 − √ 3n/5 vertices. Both bounds are sharp, and we characterise the extremal graphs. This proves conjectures of Doˇslić and Vukičević, and of Daugherty. We deduce two further conjectures on the independence number of fullerene graphs, as well as a new upper bound on the smallest eigenvalue of a fullerene graph. 1.
Odd cycles in planar graphs
, 2005
"... Given a graph G = (V,E), an odd cycle cover is a subset of the vertices whose removal makes the graph bipartite, that is, it meets all odd cycles in G. A packing in G is a collection of vertex disjoint odd cycles. This thesis addresses algorithmic and structural problems concerning odd cycle covers ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Given a graph G = (V,E), an odd cycle cover is a subset of the vertices whose removal makes the graph bipartite, that is, it meets all odd cycles in G. A packing in G is a collection of vertex disjoint odd cycles. This thesis addresses algorithmic and structural problems concerning odd cycle covers and packings. In particular, we consider the two NPhard problems of finding a maximum packing and a minimum covering. In 1994 Brass [53] conjectured that τ, the minimum size of an odd cycle cover, is at most twice ν, the maximum size of a packing. The conjecture is known to be false in general [11, 41]. We prove here that τ ≤ 10ν for planar graphs. Our structural results leads to the first constant approximation algorithm for the packing problem. The covering problem was shown to be tractable for graphs of constant sized solutions [42]. We give a linear time algorithm for the covering problem restricted to the case where the graphs have constant sized solutions and are planar.
Edgedisjoint Odd Cycles in 4edgeconnected Graphs
"... Finding edgedisjoint odd cycles is one of the most important problems in graph theory, graph algorithm and combinatorial optimization. In fact, it is closely related to the wellknown maxcut problem. One of the difficulties of this problem is that the ErdősPósa property does not hold for odd cycl ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Finding edgedisjoint odd cycles is one of the most important problems in graph theory, graph algorithm and combinatorial optimization. In fact, it is closely related to the wellknown maxcut problem. One of the difficulties of this problem is that the ErdősPósa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an integer f(k) satisfying the following: For any 4edgeconnected graph G = (V, E), either G has edgedisjoint k odd cycles or there exists an edge set F ⊆ E with F  ≤ f(k) such that G − F is bipartite. We note that the 4edgeconnectivity is best possible in this statement. Similar approach can be applied to an algorithmic question. Suppose that the input graph G is a 4edgeconnected graph with n vertices. We show that, for any ε> 0, if k = O((log log log n) 1/2−ε), then the edgedisjoint k odd cycle packing in G can be solved in polynomial time of n.
On largest volume simplices and subdeterminants
, 2014
"... We show that the problem of finding the simplex of largest volume in the convex hull of n points in Qd can be approximated with a factor of O(log d)d/2 in polynomial time. This improves upon the previously best known approximation guarantee of d(d−1)/2 by Khachiyan. On the other hand, we show that t ..."
Abstract
 Add to MetaCart
We show that the problem of finding the simplex of largest volume in the convex hull of n points in Qd can be approximated with a factor of O(log d)d/2 in polynomial time. This improves upon the previously best known approximation guarantee of d(d−1)/2 by Khachiyan. On the other hand, we show that there exists a constant c> 1 such that this problem cannot be approximated with a factor of cd, unless P = NP. Our hardness result holds even if n = O(d), in which case there exists a c ̄ dapproximation algorithm that relies on recent sampling techniques, where c ̄ is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a d × n matrix.
ProjectTeam Mascotte Méthodes Algorithmiques, Simulation et Combinatoire pour l’OpTimisation des
"... c t i v it y e p o r t 2009 Table of contents ..."
(Show Context)