Combinatorial Methods Introduction

For example, the subsets of {1, 2, 3, 4} are {}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4} (for a total of 16 subsets).

Example Applications:
• Vertex Cover of a Graph - minimum number of vertices needed to touch all edges of a graph.
• Knapsack Problem - determine maximum number of items that can be taken on a trip given some constraints.
• Set Packing - smallest number of subsets that together cover each item exactly once, useful for finding independent sets and scheduling airline flight crews to airplanes.
• Set Partitions - divide the elements 1..n into nonempty subsets, useful for vertex coloring and connected components.
• Boolean Logic Minimization - given a boolean function mapping the 2^k input vectors to the values 0 or 1, find a simplest circuit that exactly implements this function.

arrangements or orderings of items.
• The number of permutations on a set with n objects is n!.
  For example, the permutations of {1,2,3,4} are 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321 (for a total of 24 permutations).

Example Applications:
• Traveling Sales Problem - least-cost order to visit n cities.
• Bandwith Problems - order the vertices of a graph on a line so as to minimize the length of the longest edge.
• Graph Isomorphism - order the vertices of one graph so that it is identical to another.

• The number of combinations of r objects from a set with n objects is nCr.
  For example, the combinations of {1,2,3,4} taken 2 at a time are {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4} (for a total of 6 = 4!/(2!2!) subsets).
  The general formula is nCr = n! / [(n-r)! r!]. This is basically a subset problem where you specify the number of elements in the subset.

• The number of simple paths from a vertex to another vertex in a given graph is more complicated.

Example:

			a	  b
			o ------- o
			|	  |
	G:		|	  |
			o ------- o -------- o
			c	  d	     e

		
			a
		       / \
		      b   c
		     /     \
		    d       d
	           / \     / \
	          e   c	  b   e