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

Let ( ) ( ) ( ) , G V G E G = be an undirected graph. The maximum cycle packing problem in G then is to find a collection { } 1 2 , , , s C C C  of edge-disjoint cycles i C in 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 quantity ( ) ( ) ( ) ( )2 2 =1 s i i E C E G E + − ∑  on 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 * A -shortest path procedure on an appropriate acyclic network N  is presented. It uses a particular monotonous node potential.

in 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 quantity 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 * A -shortest path procedure on an

Introduction
We consider a finite and undirected graph G with vertex set ( ) V V G = and edge-set ( ) For a finite sequence ( ) , , , , , , , , ,  and edges ( ) { } If W is closed, i.e.
( ) ( ) ( ) ( ) , where ( ) is called a cycle packing of cardinality s .The family of cycle-packings of cardinality s is denoted by . If no confusion is possible, we will write  instead of ( ) Packing edge-disjoint cycles in graphs is a classical graph-theoretical problem.There is a large amount of literature concerning conditions that are sufficient for the existence of certain numbers of disjoint cycles which may satisfy some further restrictions.An overview of related references is given in [1].Practical applications of cycle packings are mentioned in the papers [2] [3] [4] [5].The algorithmic problems concerning the construction of maximum edge-disjoint cycle packings are typically hard (e.g.see [6] [7] [8]).A simple greedy-type heuristic for the problem is presented in [7], which iteratively looks for cycles of small length and removes the corresponding edges from the current graph until there is no cycle left.A different approach to tackle the problem is to relate maximum cycle packings of G to maximum cycle packings of subgraphs of G.In [1] it is described how ( ) is known, then [9] shows how to construct G from one of a finite number of graphs by a series of simple graph operations.The paper [10] investigates a relation between a maximum cycle packing and maximum local traces for the case that G is Eulerian.For v V ∈ , an Eulerian subgraph ( ) with start vertex v can be extended to an Eulerian tour in ( ) T v .Traces were first considered in [11] and [12].
In [13] bounds on ( ) G ν are presented if G is a polyhedral graph.These bounds depend on the size, the order or the number of faces of G, respectively.Polyhedral graphs are constructed that attain these bounds.
In the present paper, we will consider even graphs and tackle the cycle packing problem by a dynamic programming approach.The main idea is, instead of regarding the length ( ) .
In Section 2, we prove a max-min theorem that relates a minimizer *  of L to a maximum cycle packing of G.This theorem gives reason to consider maximum cycle packing problems of G within the framework of dynamic programming.In section 3, therefore, the problem is transformed into a shortest path problem on some appropriate acyclic networks N  .In order to avoid unnecessary excessive calculations in N  , suitable bounds on the length of an optimal paths are used.These bounds can be incorporated into an * A -algorithm.The algorithmic scheme of the procedure is presented in Section 3.2.

A Max-Min Theorem
then can be represented by , , , , still contain cycles of G.For an even graph G, it may occur that ( ) In these cases, we will write For the purpose of proving the crucial Lemma 1, consider particular subsets s  of s  , defined by 1) be the graph induced by G and a single edge as an additional component.For We will use induction on and let us assume that for all even graphs G such that ( ) Let G be an even graph such that ( ) , , , , Hence, . Applying the induction assumption to G  , we then get: . From this we finally conclude , we denote the family of all cycle packings of G.We get Theorem 1.Let G be even, ( ) be a minimizer of L on ( ) . We can assume that ( ) For this, consider the non-even graph K and some even graph H that contains at least one cycle C G′ , , , , Consider the packing , , , , , \ .
We conclude, that there must be a minimizer , , , , , , , , Applying Lemma 1 to the even graph where the last inequality is strict if ˆˆˆ, , , , , ) ( ) where the inequality is strict, if Therefore, ( ) ( ) is the max-min cycle value of G.
The determination of a max-min cycle packing *  will be called the max-min cycles packing problem (mmcp-problem) of G. Clearly, max-min cycle packings, in general, are not unique.
The following theorem relates the determination of ( ) * L G to the determination of the max-min cycle values for even subgraphs H G ⊂ .
Theorem 2. Let G be even.Then The proof of Theorem 2 immediately induces Corollary 1.

A Shortest Path Approach for the MMCP-Problem
Theorem 2 gives reason to treat the mmcp-problem as a shortest path problem within a suitable weighted acyclic network ( )

The MMCP-Network
 N Let the edges in E be labelled, i.e. ( ) { }  ( ) , , , , We will identify the set X of nodes 1 in N  with the set of even subgraphs of G.Each node x X ∈ corresponds to some specific even subgraph H of G (we will write H X ∈ ).Nodes in N  are also assigned to stages 0,1, 2, , i.e.
For the construction of N  , the nodes and edges are defined inductively: • The unique node in 0 X corresponds to the subgraph 0 G of G with empty edge set.Assume that the set of nodes We call to be a successor of • As edge weights we set ( ) ( ) 2 1 , : 1 For N  we will use "nodes", in G we use "vertices".
Clearly, ( ) is acyclic and the number of stages in N  cannot exceed with starting node 0 G and end node H.In a canonical way, any path ( ) , , , , \ the cycles used in the successive expansions of the corresponding nodes.
Obviously, G is reachable in N


, but not all even subgraphs of G have this property.
Hence, not every cycle packing ( ) is induced by some paths ( ) be a cycle-packing of H of cardinality s.Then there is a path ( ) Without loss of generality, we can assume that the cycles in ( ) The last inequality is true, since In [14], it is described how information of a monotonous node potential could be incorporated into a searching strategy for the shortest path procedure.Such an  , i e u v = must be generated.This makes it necessary to identify all simple paths between u and v in the graph \ G H .Typically, this subproblem is attacked by using DFS procedures.In general, it is NP-hard ( [15]).
Step 3 incorporates the stopping rule ( A terminates in step 3 if G is expanded from some H for the first time.Since it is possible that the graph G may appear in N  at different stages, it must be guaranteed that * A doesn't stop at a "wrong" node that corresponds to G.  E G =be an undirected graph.The maximum cycle packing problem in G then is to find a collection { }

3 K
and an additional edge to G. Obviously, must also be max-min.We get and a specific successor j G is generated by expanding 1 j G − .• An edge in U corresponds to some specific cycle in G. Edges exist only between nodes at consecutive stages.An edge ( ) if for the corresponding subgraphs j G is a successor of 1 j G − .

2 .
An A * -Shortest Path Algorithm For an even subgraph H G ⊂ , let ( ) * l H denote the length of the shortest cycle in H, then, l G H E G H = ⋅ is a lower bound for the max-min cycle value monotonous node potential.The scheme of such an * A -search is outlined as follows.The determination of H in step 1 requires the determination of the girth of \ G H .This can efficiently be done by the shortest paths procedures.For the expansion of H in step 2, the value ( ) * 0 i H and the set C of all cycles in \ G H that contain ( ) * 0 ∪ = ) and the elimination of super- fluous nodes (and sub-paths) according to Prop. 4. Coming from step 2, * and .Such an identification, preferably, should be done as early as possible in the calculations.The following proposition gives conditions for such a situation.They can be checked during the shortest path procedure.If such a condition * is a max-min cycle packing of G if and only if * * P G H be the last common node of the paths ( ) is expanded at some iteration of * A , there must be a subgraph H on the subpath that belongs to X when * A terminates.Since this node is never selected in step 1 until * A stops (otherwise it would have been deleted from X in * Let * * P G ν .Since * H * P G ν * A terminates at stage ( ) G ν , i.e. * is maximum.