Minimum Number Of Arcs Research Articles

Realizing degree imbalances in directed graphs

In a directed graph, the imbalance of a vertex v is b(v)=d +(v)−d −(v) . We characterize the integer lists that can occur as the sequence of imbalances of a simple directed graph. For the realizable sequences, we determine the maximum number of arcs in a realization and provide a greedy algorithm that constructs realizations with the minimum number of arcs.

Minimal Oriented Graphs of Diameter 2

Let f(n) be the minimum number of arcs among oriented graphs of order n and diameter 2. Here it is shown for n>8 that (1−o(1))n log n≤f(n)≤n log n−(3/2)n.

Adding and Reversing Arcs in Semicomplete Digraphs

Let T be a semicomplete digraph on n vertices. Let ak(T) denote the minimum number of arcs whose addition to T results in a k-connected semicomplete digraph and let rk(T) denote the minimum number of arcs whose reversal in T results in a k-connected semicomplete digraph. We prove that if n[ges ]3k−1, then ak(T)=rk(T). We also show that this bound on n is best possible.

An auction algorithm for the max-flow problem

We propose a new algorithm for the max-flow problem. It consists of a sequence of augmentations along paths constructed by an auction-like algorithm. These paths are not necessarily shortest, that is, they need not contain a minimum number of arcs. However, they can be found typically with much less computation than the shortest augmenting paths used by competing methods. Our algorithm outperforms these latter methods as well as state-of-the-art preflow-push algorithms by a very large margin in tests with standard randomly generated problems.

On the minimum dummy-arc problem

A precedence relation can be represented non-uniquely by an activity on arc (AoA) directed acyclic graph (dag). This paper deals with the NP-hard problem of constructing an AoA dag having the minimum number of arcs among those that have the minimum number of nodes. We show how this problem can be reduced in polynomial time to the set-cover problem so that the known methods of solving the set-cover problem can be applied. Several special cases that lead to easy set-cover problems are discussed. Return+ I'm' ft~luiif,n Jt /Aˆ,,' t'lit'ni'r ps'iit it,, rt~pr~~~~~~t~~~~ .Jut,,' in^tiiii'is- nun-un~qtit 1 par u,t ~idplii'dirfi tm \iln/uf\rJuium l~rfutcwnrur, i iA\.-Ii f'fturiii If tr~ilt'dn prt.hI>ittr NP.JtIfti.ile la ~~,ns~ru~rt~,,~ d'm XJU .ld mwtl le numhrt m,nvn,d d.~r, purnt, wua uw mt It, ,~,mI,r~~

Score sequences of oriented graphs

AbstractWe extend Landau's concept of the score structure of a tournament to that of the score sequence of an oriented graph, and give a condition for an arbitrary integer sequence to be a score sequence. The proof is by construction of a specific oriented graph Δ(S) with given score sequence S. It is shown that Δ(S) is transitive and has the minimum number of arcs among the oriented graphs with score sequence S.

Jump number of dags having Dilworth number 2

The jump number, denoted by σ, of a directed acyclic graph (dag) G , is the minimum number of arcs that have to be added to G such that the resulting graph is still acyclic and has a hamiltonian path. We study here the particular class of dags having an induced partial order of width 2, and give a characterization of such graphs with σ( G )= i . This yields immediately a polynomial algorithm to compute the jump number in this particular class.

Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems

This paper presents new algorithms for the maximum flow problem, the Hitchcock transportation problem, and the general minimum-cost flow problem. Upper bounds on the numbers of steps in these algorithms are derived, and are shown to compale favorably with upper bounds on the numbers of steps required by earlier algorithms. First, the paper states the maximum flow problem, gives the Ford-Fulkerson labeling method for its solution, and points out that an improper choice of flow augmenting paths can lead to severe computational difficulties. Then rules of choice that avoid these difficulties are given. We show that, if each flow augmentation is made along an augmenting path having a minimum number of arcs, then a maximum flow in an n-node network will be obtained after no more than ~(n a - n) augmentations; and then we show that if each flow change is chosen to produce a maximum increase in the flow value then, provided the capacities are integral, a maximum flow will be determined within at most 1 + logM/(M--1) if(t, S) augmentations, wheref*(t, s) is the value of the maximum flow and M is the maximum number of arcs across a cut. Next a new algorithm is given for the minimum-cost flow problem, in which all shortest-path computations are performed on networks with all weights nonnegative. In particular, this algorithm solves the n X n assigmnent problem in O(n 3) steps. Following that we explore a scaling technique for solving a minimum-cost flow problem by treating a sequence of derived problems with scaled down capacities. It is shown that, using this technique, the solution of a Iiitchcock transportation problem with m sources and n sinks, m ~ n, and maximum flow B, requires at most (n + 2) log2 (B/n) flow augmentations. Similar results are also given for the general minimum-cost flow problem. An abstract stating the main results of the present paper was presented at the Calgary International Conference on Combinatorial Structures and Their Applications, June 1969. In a paper by l)inic (1970) a result closely related to the main result of Section 1.2 is obtained. Dinic shows that, in a network with n nodes and p arcs, a maximum flow can be computed in 0 (n2p) primitive operations by an algorithm which augments along shortest augmenting paths. KEY WOl¢l)S AND PHP~ASES: network flows, transportation problem, analysis of algorithms CR CATEGOI{.IES: 5.3, 5.4, 8.3