Graph Theory Problem & Linear Formulation

1. Graph Problems and
Their Linear Problem
Formulations
Guided by Presented by
Dr. Hemal V Shah Dharmesh R Tank
Associate Professor MTech-CE(III)
UVPCE UVPCE
08/10/2014

2. Outline
What 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

3. 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.

4. Linear Problem Formulation
xj = decision variables
bi = constraint levels
cj = objective function coefficient
aij = constraint coefficients

5. 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.

6. Outline
What 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

7. 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.

8. LP Formulation for MAD
If 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

9. 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

10. Outline
What 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

11. 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.

12. LP Formulation for TSP
One Way is add the constraint that, for an arbitrary vertex v, the set S of
edges in the solution must contain no cycle in G.
Therefore the amounts to checking the set of edges in S with no
adjacent to v is of maximal average degree strictly less than 2.

14. Applications
Synchronous Optical Networking (SONET)
Airplane Path Decider

15. Outline
What is Graph Problem??
What is Linear Problem Formulation ??
Problems:-
Maximum Average Degree
Traveling Salesman Problem
Linear arboricity
Acyclic edge coloring
Edge-disjoint spanning trees
Steiner tree
H-minor
Assignment

16. 3. Linear Arboricity
The arboricity of an undirected graph is the
minimum number of forests into which its edges can
be partitioned. Equivalently it is the minimum number
of spanning forests needed to cover all the edges of
the graph.
The linear arboricity of a graph G is the least number
k such that the edges of G can be partitioned into k
classes, each of them being a forest of paths (the
disjoints union of paths { trees of maximal degree 2).

17. LP Formulation for Linear Arboricity

18. Outline
What is Graph Problem??
What is Linear Problem Formulation ??
Problems:-
Maximum Average Degree
Traveling Salesman Problem
Linear arboricity
Acyclic edge coloring
Edge-disjoint spanning trees
Steiner tree
H-minor
Assignment

19. 4.Acyclic edge coloring
An 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

20. Assignment
1. 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?

21. Thank you