Graph problem & lp formulation

Graph Problems and

Their Linear ProblemFormulationsGuided by Presented by

Dr. Hemal V Shah Dharmesh R Tank Associate Professor MTech-CE(III)

UVPCE UVPCE

08/10/2014

OutlineWhat is Graph Problem??

What is Linear Problem Formulation ??

Problems:-

Maximum Average Degree

Traveling Salesman Problem

Acyclic edge coloring

Edge-disjoint spanning trees

Steiner tree

Linear arboricity

H-minor

Assignment

Graph Problem

A problem that appears intractable may prove to be a few lines with the proper linear formulation or data structure.

To solving a graph related problem, it’s necessary to recognizing that it is a graph problem.

More difficult than it sounds.

If we are required to find a path of any sort, it is a graph problem.

Keywords : vertices, nodes, edges, connections, connectivity, paths, cycles and direction.

Nearly all graph problems will use a grid or network in the problem.

Linear Problem Formulation

xj = decision variablesbi = constraint levelscj = objective function coefficientaij = constraint coefficients

Steps for Linear Problem Formulation

Step 1: Identify variables.

Step 2: Write down the objective function( max or min).

Step 3: Write down the constraints with a system of inequalities.

Step 4: Find the feasible solution with graph representation.

Step 5: Calculate the coordinates of the vertices of feasible solutions.

Step 6: Calculate the optimal value of the objective function at each of the vertices for maximum or minimum values.



Problems:-





Steiner tree

Linear arboricity

H-minor

Assignment

1. Maximum Average Degree

The average degree of a graph G is defined as ad(G) = 2 E(G) / V(G)

The maximum average degree of G is meant to represent its densest part, and is formally defined as :

mad(G) = max ad(H)

Let D be a directed graph which is the disjoint union of E(G) and V (G).

Each edge will then have a flow of 2 (a source and the necessary edges) to distribute among its two endpoints.

LP Formulation for MADIf H Є G is the densest subgraph in G, its E(H) edges will send a flow of 2E(H) to their V (H) vertices, such feasible only if Z ≥ 2E(H)/ V(H).

An elementary application of the max-flow/min-cut theorem, or bipartite matching theorem

Example: set of authors who wrote at least one paper in the

period between 1974 and 2004.

http://www.nature.com/srep/2012/120625/srep00469/fig_tab/srep00469_F1.html



Problems:-





Steiner tree

Linear arboricity

H-minor

Assignment

2.Traveling Salesman Problem

TSP is a Hamiltonian cycle whose weight

(the sum of the weight of its edges) is minimal.

Both the objective and the constraint that

each vertex must have exactly two neighbors.

But this produce solutions set of edges

with several cycles.

LP Formulation for TSPOne Way is add the constraint that, for an arbitrary vertex v, the set S ofedges in the solution must contain no cycle in G.

Therefore the amounts to checking the set of edges in S with noadjacent to v is of maximal average degree strictly less than 2.

Applications

Synchronous Optical Networking (SONET)

Airplane Path Decider



Problems:-



Linear arboricity



Steiner tree

H-minor

Assignment

3. Linear ArboricityThe arboricity of an undirected graph is theminimum number of forests into which its edges canbe partitioned. Equivalently it is the minimum numberof spanning forests needed to cover all the edges ofthe graph.

The linear arboricity of a graph G is the least numberk such that the edges of G can be partitioned into kclasses, each of them being a forest of paths (thedisjoints union of paths { trees of maximal degree 2).

LP Formulation for Linear Arboricity



Problems:-



Linear arboricity



Steiner tree

H-minor

Assignment

4.Acyclic edge coloringAn edge coloring with k colors is said to be acyclic if it is proper (each color class is a matching { maximal degree 1), and if the union of the edges of any two color classes is acyclic.

The corresponding LP is almost a copy of the previous one

Except that we need to ensure that

different classes are acyclic

Assignment1. The Sureset Concrete Company produces concrete. Two ingredients in concrete are sand (costs $6 per ton) and gravel (costs $8 per ton). Sand and gravel together must make up exactly 75% of the weight of the concrete. Also, no more than 40% of the concrete can be sand and at least 30% of the concrete be gravel. Each day 2000 tons of concrete are produced. To minimize costs, how many tons of gravel and sand should be purchased each day?

Thank you

Graph problem & lp formulation

Engineering

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