1. Graph Theory and Optimization
Math 3480/5900
Graph Theory and Optimization
Topics covered:
Graph Theory
. Basic graph/network terminology and definitions
. Shortest Path Problems (SPP)
. Minimum Spanning Trees (MST)
. Maximum Flow Problems (max flow)
. Minimum Cost Flow (min cost)
Integer Programming Problems
. Linear Programming(LP) Review
. Branch and bound solution approach
. Preprocessing and Cutting Plane improvements
Dynamic Programming Problems
. Deterministic
. Probabilistic

2. Why networks? Advantages: problem representation
. Real networks from communication, transportation, etc.
. Logical processes, stages
. Easier task if modeling with pictures
problem solution
. Networks more efficient than standard LP methods
solution presentation
. Clear, concise, easy to grasp
Disadvantages:
terminology is not standardized

3. Example Entire: graph, directed graph, digraph, network3 2 6 1 4 5
Entire: graph, directed graph, digraph, network
Circles: nodes, vertices, points
Arrows: arcs, edges, links
G = (V, E) represents graph G with vertex set V and edge set E. Edges may be directed or undirected.

4. Examples 2 1
G1= (N1, E1)
N1 = {1, 2, 3, 4}
E1 = {(1,2), (1,4), (2,4), (3,1), (3, 4)} 3 4 a
G2= (N2, E2)
N2 = {a, b, c}
E2 = {{a,b}, {a,c}, {b,c}} b c

5. Applications
Transportation - city traffic, plane routes, truck shipping
Telecommunications - wiring, tower locations
Sciences - DNA sequencing, chemical reactions, social interactions
Business - project management, inventory planning, scheduling
Mathematics - approx. func., dynamic programs, difference equations
Industry - resource assignments
Agriculture, energy, medicine, ...