GRAPH-THEORY AND APPLICATIONS Nitesh Tripathi

MOTIVATION What is the common link between the following problems: Traffic network design and cancer research? Arranging marriages and scheduling flights? Finding cure for mental illness, computer chip design, architectural floor planning, fighting terror online? Fighting epidemics, e-commerce, designing voting schemes, job assignment, designing electrical networks, deciding on facility location, etc. etc. etc.

WHAT IS A GRAPH? V2 and v3 are adjacent. e2 is incident with v2. A set of points and lines joining these points. Formally: G=(V,E), V-vertices, E-edges. e6 v1 v4 v3 V2 and v3 are adjacent. e2 is incident with v2. e1 e2 e3 e5 v2 v5 e4

PRESENTATIONS OF GRAPHS v3 Drawing Incidence matrix e2 e3 e5 e4 v5 v2 v4 e1 e6 v1 Deg of a vertex. Sum of the degrees. Addition of all the 1’s in the matrix. Hand-shaking Lemma. e6 e5 e4 e3 e2 e1 2 1 v1 v2 v3 v4 v5 Degree of a vertex, Regular graph Hand shaking lemma

PRESENTATIONS OF GRAPHS v3 Adjacency matrix e2 e3 e5 v5 v4 v3 v2 v1 1 2 e4 v5 v2 v4 e1 e6 v1

DIRECTED GRAPHS Adjacency matrix v3 e2 e3 e5 v5 v4 v3 v2 v1 1 e4 v5 v2 1 e4 v5 v2 v4 e1 e6 v1

WEIGHTED GRAPHS Adjacency matrix v3 2 2 5 v5 v4 v3 v2 v1 3 2 1 3 v5 v2 3 2 1 3 v5 v2 v4 1 3 v1

CONNECTOR PROBLEM Design a railway network connecting a number of cities, with a minimum possible construction cost. Or electrical network, cable TV, gas, etc.

CONNECTOR PROBLEM Design a railway network connecting a number of cities, with a minimum possible construction cost. Or electrical network, cable TV, gas, etc. Problem is Tractable A variation of it is intractable!

CONNECTOR PROBLEM GRAPH-THEORY Definition: A tree is a connected acyclic graph. Connected = between any 2 vertices there is a path. Acyclic = contains no cycles. Spanning = contains all the vertices in the graph. Connector problem: Given a weighted graph G: Find a minimum weight spanning tree of G.

CONNECTOR PROBLEM GRAPH-THEORY Computing Minimum Spanning Tree is polynomial time solvable. Greedy Algorithms: Prim’s and Kruskal’s Algorithm

TRAVELLING SALESMAN PROBLEM Wonderla Intractable! Mantri Mall Nandi Hills CSA, IISc Iskcon Temple Orion Mall Summer school students begin at CSA IISc, have to visit all the places exactly once, and return to CSA in a shortest possible distance. (or time, or cost)

Travelling salesman problem-Graph Theory Given a graph (or a directed graph), does there exist a cycle in the graph that contains each vertex once? (i.e. a Hamilton cycle)? Given a complete weighted graph, find a Hamilton cycle of minimum weight.

MORE APPLICATIONS OF TSP JOB SEQUENCING ON A SINGLE MACHINE V- jobs J1,J2,…,Jn E- edge (i,j) has cost Ci,j - the time required for job Jj to be performed immediately after job Ji J2 C1,2 J1 C1,3 C2,3 J3

CHINESE POSTMAN PROBLEM A postman begins in the post office, has to traverse all the streets, and returns to the post office in a shortest possible distance. Tractable!

CHINESE POSTMAN PROBLEM- GRAPH THEORY Definition: an Euler tour is a closed walk that traverses all the edges in the graph. Problem: Does a graph have an Euler tour? Easy problem… Problem: Find a minimum Chinese Postman Tour. Tractable…

CHINESE POSTMAN PROBLEM- ANOTHER APPLICATION Testing a computer program V – states of a program E – transitions between the states Problem: Generate input data that forces the program to test all possible transitions. Another Problem: Test all pairs of consecutive transitions… (how should we do that?)

Matching Problem B1 B2 B3 B4 B5 A1 A2 A3 A4 A5 Part A Part B Definition: A matching is a set of disjoint edges in a graph. A matching is perfect if it meets every vertex in the graph..

Matching Problem Is this a maximum matching? Part A Part B A1 A2 A3 A4

Stable Marriage Problem women men w1 w2 w3 w4 w5 1 m1 1 1 Does there exist a stable marriage? 2 m2 1.5 2 2 m3 4 1.7 m4 3 2 m5 W2 prefers m5 to her spouse and m5 prefers w2 to his spouse.

Find a minimum (maximum) cost assignment of workers to machines Assignment Problem workers machines w1 w2 w3 w4 w5 1 m1 3 1 2 m2 1.5 2 2 m3 4 1.7 m4 3 2 m5 Find a minimum (maximum) cost assignment of workers to machines

Applications Assigning doctors to hospitals. Assigning people to jobs. Assigning students to dormitories. Assigning pairs of drivers to trucks. Etc. Stable Matching Problem can be solved in polynomial time using Gale-Shapley Algorithm.

TIMETABLING PROBLEMS m teachers, n classes. Teacher i is required to teach class j for Pij periods. In a given period a teacher can be in at most 1 class, and a class can have at most 1 teacher. Design a timetable with minimum no. of periods. Tractable! T1 C1 T2 C2 C3 Cn Tm

TIMETABLING PROBLEMS m teachers, n classes. Teacher i is required to teach class j for Pij periods. In a given period a teacher can be in at most 1 class, and a class can have at most 1 teacher. Design a timetable with minimum no. of periods. Tractable! Properly color the edges of G with as few colors as possible. T1 C1 T2 C2 C3 Cn Tm

EDGE COLORING OF GRAPHS Definition: A proper k- edge coloring of a graph G=(V,E) is a mapping c: E {1,2,…,k} such that adjacent edges receive distinct colors. If the maximum degree is Δ then clearly k ≥ Δ Theorem (Vizing): Any graph has either a Δ-coloring or a (Δ+1)-coloring. Designing a time-table is a proper edge coloring of a graph. Theorem A bipartite graph has a Δ-coloring

Questions?