A Dynamic Programming Approach for the Max-Min Cycle Packing Problem in Even Graphs

Peter Recht

doi:10.4236/ojdm.2016.64027

Abstract

Let be an undirected graph. The maximum cycle packing problem in G then is to find a collection of edge-disjoint cycles Ciin G such that s is maximum. In general, the maximum cycle packing problem is NP-hard. In this paper, it is shown for even graphs that if such a collection satisfies the condition that it minimizes the quantityon the set of all edge-disjoint cycle collections, then it is a maximum cycle packing. The paper shows that the determination of such a packing can be solved by a dynamic programming approach. For its solution, an-shortest path procedure on an appropriate acyclic networkis presented. It uses a particular monotonous node potential.

Highlights

We consider a finite and undirected graph G with vertex set V = V (G) and edge-setE = E (G) that contain no loops.( ) For a finite sequence vi1 e1 vi2 e2, er −1, vir of vertices vij and pairwise distinct ( ) { } edges ej = vij, vij+1 the subgraph W of G with vertices V (W ) = and edges
It is shown for even graphs that if such a collection satisfies
{ } such that all Gi are mutually edge-disjoint and G is induced by G1, Gq

Summary

Introduction

{ } such that all Gi are mutually edge-disjoint and G is induced by G1, , Gq. If exactly s of the Gi is cycles, (G) is called a cycle packing of cardinality s. Packing edge-disjoint cycles in graphs is a classical graph-theoretical problem. The algorithmic problems concerning the construction of maximum edge-disjoint cycle packings are typically hard The paper [10] investigates a relation between a maximum cycle packing and maximum local traces for the case that G is Eulerian. We will consider even graphs and tackle the cycle packing problem by a dynamic programming approach. This theorem gives reason to consider maximum cycle packing problems of G within the framework of dynamic programming.

A Max-Min Theorem

The MMCP-Network N