1. Linear Programming (LP) Vector Form Maximize: cx Subject to : Ax  b c = (c 1, c 2, …, c n ) x = b = A = Summation Form Maximize:  c i x i Subject to:  a 1i x i  b 1  a 2i x i  b 2.  a mi x i  b m x1..xnx1..xn b1..bnb1..bn a 11 … a 1n …… a n1 … a mn

2. Example LP n = 2; m = 4 x 1 + x 2  max x 1  0 (-1)x 1 + 0x 2  0 x 2  0 0x 1 + (-1)x 2  0 x 1  5 (-1)x 1 + 0x 2  5 x 2  6 0x 1 + 1x 2  6 c = (1, 1) b = A = Optimal solution is the unique point of intersection of the objective with the hyperplane feasible polytope. x 2 optimal solution x 1 = 5 ; x 2 = 6 objective: x 1 + x 2 = 11 x 1 00560056 -1 0 0 -1 1 0 0 1 Feasible solution region

3. Integer Linear Programming (ILP) Vector Form Maximize: cx Subject to : Ax  b and x  {0,1} c = (c 1, c 2, …, c n ) x = b = A = Summation Form Maximize:  c i x i Subject to:  a 1i x i  b 1  a 2i x i  b 2.  a mi x i  b m and x  {0,1} x1..xnx1..xn b1..bnb1..bn a 11 … a 1n …… a n1 … a mn

4. ILP for MIS Maximum Independent Set (MIS) - Find the maximum subset of nodes in graph G = (V, E) which are pairwise non-adjacent ILP - For any v  V make a variable x v  {0, 1} x v = 0  v  MIS which means 0 is not chosen x v = 1  v  MIS which means 1 is chosen - Maximize  v  V x v Subject to:  e = (u, v)  V, x u + x v  1

5. Example ILP of MIS Max: x 1 + x 2 + x 3 + x 4 + x 5 + x 6 subject to: x 1 + x 6  1 x 1 + x 2  1 x 2 + x 3  1 x 3 + x 6  1 x 3 + x 5  1 x 6 + x 5  1 x 3 + x 4  1 x 4 + x 5  1 and x 1, x 2,…, x 6  {0,1} Graph 12 3 4 5 6

6. ILP for MaxClique ILP -  x i  max - Subject to: x i + x j  1  (i, j)  E MaxClique - Given G = (V, E) - Find X  V  x, x’  X (x, x’)  E |X|  max

7. ILP for Matching Matching - Given G = (V, E) - Find X  E  e, e’  X e and e’ don’t share endpoint. |X|  max ILP - for any e  E x e  {0, 1} + 0 is not in matching + 1 is in matching -  e  E x e  max - Subject to:  e incident to V x e  1  v  V e 2 x e 1 + x e 2 + x e 3  1 only one edge from e 1 e 3 matching can be incident to v v

8. LP relaxation (LPR) vs. ILP LP relaxation (LPR) for MAX independent set problem (MISP) gives larger value than the maximum size of independent set. MISP  x i  max, i  V x i + x j  1, for each edge (x i,x j )  E x i  {0, 1} LPR  x i  max, i  V x i + x j  1, for each edge (x i,x j )  E 0  x i  1

9. Example 1 of ILP vs. LPR  x i  max x 1 +x 6  1 x 1 +x 2  1 x 5 +x 6  1 x 5 +x 2  1 x 5 +x 4  1 x 4 +x 3  1 x 4 +x 2  1 x 2 +x 3  1 ILP x 1 = x 3 = x 5 = 1  x i = 3 LPR x i = ½  x i = 3 1 2 3 4 5 6

10. MISP Integrality Gap x 1 +x 2 + x 3  max x 1 +x 2  1 x 1 +x 3  1 x 2 +x 3  1 0  x 1  1 0  x 2  1 0  x 3  1 LPR (  )  3/2 Implies LPR (  ) = 3/2 So x 1 = x 2 = x 3 = ½  LPR (  )  3/2 ILP (  ) = 1 Integrality Gap (IG) = LPR / ILP = 3/2 What is the integrality gap for (MISP) For a complete graph ILP (Kn) = 1 LPR (Kn) = n/2 Integrality Gap (IG) = LPR / ILP Integrality gap may be as large as n/2 1 2 3