Maximal Concurent Limited Cost Flow Problems on Extended Multi-commodity Multi-cost Network

Graphs are excellent mathematical tools applied in many fields such as transportation, communication, informatics, economy,.... A network and a flow network is a useful device to solve many problems in many fields in reality. However, most of the network applications in traditional graphs have only considered the weights of edges and vertexes independently, in which the length of a path is the sum of weights of the edges and the vertexes on the path. However, in many practical problems, weights at a vertex are not the same for all paths passing the vertex, but depend on the edges coming to and leaving the vertex. For example, the transit time on the transport network depends on the direction of transportation: turn right, turn left or go straight, even some directions are forbidden. Furthermore, on a network, there are many types of commodities, each of which are at different costs. Types of commodities share the capacity of edges and vertexes. Therefore, it is necessary to study a network with multiple commodities at multiple costs. The article builds a model of extended multi-commodity multi-cost network in order to modelise practical problems more exactly and effectively. The maximal concurent multi-commodity multi-cost flow limited cost problems, that are modelized by implicit linear programming problems. On the basis of duality theory in linear programming, an effective polynomial approximation algorithm is developed.


Introduction
Flows on networks are excelent mathematical means used in many applications as communication, transportation, economy, informatics, ….. So far, most of the network applications in traditional graphs have only considered the weights of edges and vertexes independently, in which the length of a path is the sum of weights of the edges and the vertexes on the path. However, in many practical problems, the weight at one node is not the same for all paths passing through that node, but also depends on coming and leaving edges. For example, the transit time on the transport network depends on the direction of transportation: turn right, turn left or go straight, even some directions are forbidden. The idea of using duality theory of linear programming to solve these problems is motivated by the work [1]. Paper [2] proposes switching cost only for directed graphs. Therefore, it is necessary to build an extended mixed network model in order to apply more accurate and effective modeling of practical problems. Multi-commodity flow in traditional network problems have been studied in the papers [3][4][5][6][7][8][9][10][11]. Multi-commodity singlecost flow problems in extended transport networks are studied in the papers [12][13][14][15][16][17][18][19][20][21][22]. The papers [23,24] study maximal multi-commodity multi-cost flow problems. The papers [25,26] study maximal multi-commodity multi-cost flow limited cost problems. The papers [27] and [28] study maximal concurent flow problems on extended multi-commodity multi-cost networks.
Given a multi-cost multi-commodity network G=(V, E, ce, ze, cv, zv, {be i , bv i , q i |i=1, 2,.., r}). Assume, for each commodity type i, i=1, 2, …, r, there are k i source-target pairs (s i,j , t i,j ), j=1, 2, …, k i , each pair assigned a quantity of comodity of type i, that is necessary to move from source node s i,j to target node t i,j .
Denote P i,j the set of paths from node s i,j to node t i,j in G, which commodity of type i can be passed through, i=1, 2, …, r, j=1, 2, …, k i . Set For each path p ∈P i,j , i=1, 2, …, r, j=1, 2, …, k i , denote x i,j (p) the flow of converted commodity of type i from the source node s i,j to the target node t i,j along the path p.
Let P i,e denote the set of paths in P i passing through the edge e, ∀e∈E.
Let P i,v denote the set of paths in P i passing through the node v, ∀v∈V.
A set is called a multi-commodity flow on the extended multi-cost multi-commodity network, if it satisfies the edge capacity constraints: (4) and the vertex capacity constraints: The expressions , i=1, 2, …, r, j=1, 2, …, k i (6) are called the flow value of commodity kind i of the source-target pair (s i,j ,t i,j ) of the multi-commodity flow F.
The expresstions are called the flow value of commodity kind i of the multi-commodity flow F. The expresstion source-target pairs (s i,j , t i,j ), j=1, 2, …, k i , each pair assigned a quantity D i,j of commodity of kind i, that is required to transferred from source vertex s i,j to target vertex t i,j . Given a limited cost B.
The mission of the problem is to find a maximum number λ such that there exists a flow converting λ.D i,j unit of commodity kind i, i=1, 2, …, r, from source vertex s i,j to target vertex t i,j , ∀j = 1, 2, …, k i , and the total cost doex not exceed the limited cost B. Set The problem is expressed by an implicit linear programming model (P) as follows: The dual linear programming problem of (P), called (D), is constructed as follows: each edge e∈E is assigned a dual variable le(e), each vetex v∈V is assigned a dual variable lv(v), each requirement d ij is assigned a dual variable z ij , ∀i=1, 2, …, r, ∀j=1, 2,..., k i , and the cost constraint is assigned a dual variable ϕ. The problem (D) states as following: , , Now, given p∈P a path from vertex u to vertex v through edges e i , i=1, 2, …, (h+1), and vertex u i , i=1, 2, …, h, as follows p = [u, e 1 , u 1 , e 2 , u 2 , …, e h , u h , e h+1 , v] We define the path length of p, denoted by length i (p), depending on the variables le(e), lv(v) by the following formula: Denote dist i,j (le,lv,ϕ) the shortest path length from s i,j to t i,j calculated by function length i (p), ∀i=1, 2, …, r, ∀j=1, 2,..., k i .
The problem (D) is equivalent to the problem (D α ) such that their optimal value are equal and the optimal solution of one problem derives the optimal solution of the other problem and vice versa.
Prove Denote min(D) and min(D α ), respectively, the optimal values of the problem (D) and the problem (D α ). Given functions le: E→R * , lv:V→R * . Set is an accepted solution of (D) and On the contrary, let (le,lv,ϕ, z i,j ) be an admitted solution of (D). Then, we have: From (11) and (12) it follows min(D) = min(D α ). Next, if (le,lv,ϕ, z i,j ) is an optimal solution of the problem is an optimal solution of problem (D). Conversely, if (le,lv,ϕ, z i,j ) is an optimal solution of the problem (D), then (le,lv,ϕ, z i,j ) is an optimal solution of the problem (D α ).

Algorithm
Ideas Algorithm is implemented through several phases. Each phase consists of k loops, k=k 1 +k 2 + … +k r . At the loop [i,j], ∀i=1, 2, …, r, ∀j=1, 2,..., k i , of a phase t we move d i,j converted units of commodity of kind i from source vertex s i,j to target vertex t i,j . This move is implemented in several steps.
Algorithm ◊ Input: Extended multi-cost multi-commodity network G=(V,E, ce, ze, cv, zv, {be i , bv i , q i |i=1, 2, …, r}). Assume, for each commodity of kind i, i=1, 2, …, r, there are k i source-target pairs (s i,j , t i,j ), j=1, 2, …, k i , each pair assigned a quantity of commodity D ij of kind i, that is necessary to move from source vertex s i,j to target vertex t i,j . Given a limited cost B and an approximation ratio ω. Let denote n=|V|, m=|E|.
◊ Output: Maximal concurent multi-commodity flow F represents a set of converted commodity flows at edges Maximal concurent multi-commodity flow rF represents a set of real commodity flows at edges ;// f i,j (e) denote optimal flow } while (D(t) < 1) //Modifying the resulting flows F ; //The End. Proof of algorithm ◊ Remarks: In (t−1) phases of implementation of the above algorithm, ∀i=1, 2, …, r, j=1, 2, …, k i , we have transferred (t−1).d i,j units of converting commodity type i from s i,j to t i,j. However, the transferred flow may exceed the throughout capacity of the edges.
On the other hand, the number of times to send ce(e).ze(e) converted unit of commodity over each edge e∈E is at least fe(e)/(ce(e).ze(e)) and the number of times to send cv(v)zv(v) converted unit of commodity through each node v∈V is at At this point, the edge and node functions will satisfy the following inequality: ) ( Thus, divide fe(e) by δ ε 1 log 1+ , ∀e ∈ E, that follows fv(v) is divided by δ ε 1 log 1+ , we receive accepted flows.
The above analysis shows that after (t−1) of phases, ∀i=1, 2, …, r, j=1, 2, …, k i , we have moved (t−1).d i,j converted units from s i,j to t i,j . However, in order for the flow to be accepted, we must divide the flow by Next, we have ) ( From (15) and (16) we receive: For β ≥ 1, we have With regard to stop condition of algorithm D(t) ≥ 1, we have On the other hand, by duality, we have γ ≥ 1.

Proof
We have ϕ 0 = δ /B. After (t−1) phases we get D(t−1) < 1, t.e. , the total cost after does not exceed B.  3 .log 2 (m+n+1)), where m is the number of edges, n is the number of vertices of the network, k = k 1 +…+k r , cemax = max{ce(e).ze(e) | e∈E }, First, we find the number of phases the algorithm has taken. According to the proof of lemma 4.2 above and for β = λ, we have

Algorithm Complexity
where ε and δ depend on ω. Besides, t depends on β.
Further, denote imax, jmax indexes satisfying On the other hand, each phase implements k loop, so the loop number is k.t. Consider the loop transfering d i,j converted units of commodities from s i,j to t i,j , i=1, 2, …, r, j=1, 2, …, k i . Since cmin is the the minimal capacity of edges and nodes, the 1 number of steps required to execute the loop is not exceeded (d i,j /cmin+1). The main procedure in each step, finding the shortest path from s i,j to t i,j , has a complexity of n 3 [8]. So the complexity of the loop is (d i,j /cmin+1).n 3

Conclusions
The contribution defines the maximal concurent limited cost flow problems on extended multi-commodity multi-cost networks, that can be more exactly and effectively applied to model many practical problems. The maximal concurent limited cost flow problems are modeled as implicit linear optimization problems. On the base of dual theory in linear optimization, an effective polynomial approximate algorithm is developed. Correctness and algorithm complexity are proved.