A polynomial time solution algorithm is described to find a smallest subset R of nodes of a directed graph D = (V, A) such that, for every node v is an element of V - R, there are k edge-disjoint paths from R to v and...
详细信息
A polynomial time solution algorithm is described to find a smallest subset R of nodes of a directed graph D = (V, A) such that, for every node v is an element of V - R, there are k edge-disjoint paths from R to v and there are / edge-disjoint paths from v to R. (C) 2004 Elsevier B.V. All rights reserved.
We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring. The problem of deciding whether a given digraph has an out-colouring with only two colou...
详细信息
We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a 2-out-colouring) is NP-complete. We show that for every choice of positive integers r,k there exists a k-strong bipartite tournament, which needs at least r colours in every out-colouring. Our main results are on tournaments and semicomplete digraphs. We prove that, except for the Paley tournament P7, every strong semicomplete digraph of minimum out-degree at least three has a 2-out-colouring. Furthermore, we show that every semicomplete digraph on at least seven vertices has a 2-out-colouring if and only if it has a balanced such colouring, that is, the difference between the number of vertices that receive colour 1 and colour 2 is at most one. In the second half of the paper, we consider the generalization of 2-out-colourings to vertex partitions (V1,V2) of a digraph D so that each of the three digraphs induced by respectively, the vertices of V1, the vertices of V2 and all arcs between V1 and V2, have minimum out-degree k for a prescribed integer k >= 1. Using probabilistic arguments, we prove that there exists an absolute positive constant c so that every semicomplete digraph of minimum out-degree at least 2k+ck has such a partition. This is tight up to the value of c.
It had been a long-standing open problem to devise a combinatorial polynomial algorithm for minimizing submodular functions. Iwata-Fleischer-Fujishige (IFF) and Schrijver resolved the problem independently and simulta...
详细信息
It had been a long-standing open problem to devise a combinatorial polynomial algorithm for minimizing submodular functions. Iwata-Fleischer-Fujishige (IFF) and Schrijver resolved the problem independently and simultaneously. The present article is basically a survey, though not comprehensive, on submodular function minimization algorithms and also describes additional results and research subjects. In this article we first give an outline of the IFF algorithm and then show two modifications of the original IFF algorithm, which seem to be promising in practical implementation. We also describe a brief history of developments in submodular function minimization. Finally, we furnish some possible future research subjects in submodular function minimization.
Stiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321-324] proved that if every vertex v in a graph G has degree d (v) >= a (v) + b(v) + I (where a and b are arbitrarily given nonnega...
详细信息
Stiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321-324] proved that if every vertex v in a graph G has degree d (v) >= a (v) + b(v) + I (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A, B) such that d(A) (v) >= a (v) for every v is an element of A and d(B) (v) >= b(v) for every v is an element of B. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7-91 and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237-239] strengthened this result, proving that it suffices to assume d(v) >= a + b (a, b >= 1) or just d(v) >= a + b - 1 (a, b >= 2) if G contains no cycles shorter than 4 or 5, respectively. The original proofs contain nonconstructive steps. In this paper we give polynomial-time algorithms that find such partitions. Constructive general izati ons for k-partitions are also presented. (c) 2006 Elsevier B.V. All rights reserved.
Gutjahr, Welzl and Woeginger found polynomial-time algorithms for a number of digraph homomorphism problems. These algorithms are based on the (X) under bar -enumeration, the C-k-extended (X) under bar -enumeration an...
详细信息
Gutjahr, Welzl and Woeginger found polynomial-time algorithms for a number of digraph homomorphism problems. These algorithms are based on the (X) under bar -enumeration, the C-k-extended (X) under bar -enumeration and the (X) under bar -graft construction. In this note, we show how the last two methods can be combined to obtain new polynomial-time algorithms, which also work for list homomorphisms. In the process, we are able to extend results of Bang-Jensen and Hell, dealing with homomorphisms to bipartite tournaments, to list homomorphisms. (C) 2011 Elsevier B.V. All rights reserved.
We consider the two problems of finding the maximum number of node disjoint triangles and edge disjoint triangles in an undirected graph. We show that the first (respectively second) problem is polynomially solvable i...
详细信息
We consider the two problems of finding the maximum number of node disjoint triangles and edge disjoint triangles in an undirected graph. We show that the first (respectively second) problem is polynomially solvable if the maximum degree of the input graph is at most 3 (respectively 4), whereas it is APX-hard for general graphs and NP-hard for planar graphs if the maximum degree is 4 (respectively 5) or more. (C) 2002 Elsevier Science B.V. All rights reserved.
In this paper we propose a long-step logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along proj...
详细信息
In this paper we propose a long-step logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along projected Newton directions are taken until the iterate is in the vicinity of the center associated with the current value of the barrier parameter. We prove that the total number of iterations is O(square-root nL) or O(nL), depending on how the barrier parameter is updated.
We consider the one-machine scheduling problem to minimize the number of late jobs under the group technology assumption, where jobs are classified into groups and all jobs from the same group must be processed contig...
详细信息
We consider the one-machine scheduling problem to minimize the number of late jobs under the group technology assumption, where jobs are classified into groups and all jobs from the same group must be processed contiguously. This problem is shown to be strongly NP-hard, even for the case of unit processing time and zero set-up time. A polynomial time algorithm is developed for the restricted version in which the jobs in each group have the same due date. However, the problem is proved to be ordinarily NP-hard if the jobs in a group have the same processing time as well as the same due date.
This paper considers a scheduling problem in which a single operator completes a set of n jobs requiring operations on two machines. The operator can perform only one operation at a time, so when one machine is in use...
详细信息
This paper considers a scheduling problem in which a single operator completes a set of n jobs requiring operations on two machines. The operator can perform only one operation at a time, so when one machine is in use the other is idle. After developing general properties of optimal schedules, the paper develops efficient algorithms for minimizing maximum lateness. The algorithm has time complexity O(n) (after due date sorting) for both the open shop and flow shop cases.
The computational complexity of inverse mimimum capacity path problem with lower bound on capacity of maximum capacity path is examined, and it is proved that solution of this problem is NP-complete. A strong polynomi...
详细信息
The computational complexity of inverse mimimum capacity path problem with lower bound on capacity of maximum capacity path is examined, and it is proved that solution of this problem is NP-complete. A strong polynomial algorithm for a local optimal solution is provided.
暂无评论