Thu Apr 16, 2015
Cutting Plane (continued)
Definition 22.1 (Chv
atal rank) The smallest t for which P (t) = PI is the Chv
atal rank of P .
Theorem 22.2 (Chv
atal, 1973) Every polytope P (possibly irrational) has finite Chv
Proof. P i
The Feb 10, 2015
A face F is a minimal face of P i it is an ane subspace.
Theorem 6.1 F = , F P = cfw_x : Ax b is a minimal face of P i F = cfw_x : A x = b for some
subsystem of Ax b.
Proof. If : Let F = cfw_x : A x = b P , F = , F is a face s
Thu Mar 5, 2015
Definition 13.1 (Polynomial Time Reduction) A polynomial time reduction of a problem to a problem
is a polynomial time (polynomial in the size(X ) computable function f such that X is a YES instance of
i f (X ) is YES instance
Thu Feb 26, 2015
Definition 11.1 (Independent system) Let N be a finite set, I 2N , the pair (N, I) is an independent
system if A I and B A B I.
The sets in I are called independent sets.
An independent system is a matroid if
A, B I, w
Tue Feb 24, 2015
Definition 10.1 (Total Dual Integrality, Edmonds-Giles) A rational linear system Ax b is totally
dual integral if integral c with z LP = max(cT x : Ax b) finite the dual min(y T b : y T A = cT , y 0) has an
integral optimal sol
Thu Feb 19, 2015
maxcfw_cT x : Ax b, x 0, x Zn . Assume that A, b are integral.
LP: maxcfw_cT x : Ax b, x 0, P = cfw_x : Ax b, x 0.
Question: When is the LP solution integral? When is P integral?
Extreme points of LP are of the form x = (xB , xN
Tue Feb 17, 2015
Theorem 8.1 (Birkho ) Let G be a bipartite graph, let P = cfw_x R|E| ,
= 1, v V, x(e)
0, e E (v) denote all the edges incident to v, P is a matching), then
(1) Perfect matching polyhedron of G is P ;
(2) P is integral
Thu Feb 12, 2015
Definition 7.1 Integer hull if P is PI = conv(P Z).
Lemma 7.2 If P is a bounded polyhedron, then PI is polyhedron.
Proof. P is bounded P Zn is finite conv(P Z) is a polytope PI is a polyhedron.
If P is an unbounded polyhedron, t
Thu Feb 5, 2015
Proof of Theorem 4.16. Only if : Let F = P cfw_x : cT x = for some c = 0 and = maxcfw_cT x : x P .
is finite implies that = mincfw_y T b : y T A = cT , y 0. Let y be the optimal solution to the dual LP
mincfw_y T b : y T A = cT
Tue Feb 3, 2015
Consider a polyhedron P = cfw_x Rn : Ax b.
Definition 4.1 An inequality aTi x bi for Ax b is called an implicit equality if aTi x
= bi ,
Notation: A= x b= is the system of implicit equality. A+ x b+ is the system of other
Tue Jan 27, 2015
Duality, weak-dual pair, strong dual pair
Example 2.1: Definition 2.2 Given a graph G = (V, E),
a matching is a subset of disjoint edge.
a vertex cover is a subset U of vertices s.t. each edge is adjacent to at least one verte
Tue Mar 3, 2015
Optimization Problem: maxcfw_cT x : x S
Decision Problem: Does a x S with value cT x K.
Related to problem instance size
Size of an instance of an optimization problem.
(1) Number of variables;
(2) Number of
Thu Mar 12, 2015
Theorem 15.1 Let a1 , . . . , am be rational vectors. The group L generated by a1 , . . . , am is a lattice.
Proof. Convert the matrix [a1 , . . . , am ] to HNF. Say HNF(A) = [b1 , . . . , br , 0, . . . , 0
Tue Apr 7, 2015
Branch and Bound
z = maxcfw_cT x : x S. Let S = S1 S2 Sk be a decomposition of S. zi = max cT x : x Si . Then
z = maxcfw_zi : i [k].
Suppose S = cfw_0, 13 . S0 = cfw_x S : x1 = 0, S1 = cfw_x S : x1 = 1.
Example 19.1: Optimal tou
Tue Mar 17, 2015
Always maintain a primal feasible solution but improve towards optimality.
Definition 16.1 A matching M E is a set of disjoint edges.
Definition 16.2 A vertex cover R V is a subset of nodes such that every edge
Thu Apr 30, 2015
Lagrangian Duality (continued)
Set Cover Problem
Given: Finite ground set N . Collection F 2N of subsets of N , each corresponding a cost cf , f F .
Goal: Find a collection of the sets in F with min cost such that the union of
Tue Apr 28, 2015
Lagrangian Duality (continued)
Lagrangian Dual Problem
wLD := mincfw_z(u) : u 0 least possible upper bound.
Lemma 25.1 (Optimality of Relaxation) If u 0 and
(i) x(u) is optimal to IP(u).
(ii) Dx(u) d.
(iii) [Dx(u)]i = di whenev
Thu Apr 23, 2015
Definition 24.1 A set S = S1 S2 with S1 , S2 Rn is a disjunction of two sets S1 and S2 .
Lemma 24.2 If
i is valid for Si , i = 1, 2, then
j=1 wj xj
is valid for S if wj mincfw_wij , w2j ,
Tue Apr 21, 2015
Cutting Plane (continued)
Corollary 23.1 Let P = cfw_x : Ax b be a rational polytope and PI is empty. Then a cutting plane proof
of 0T 1 from Ax b.
Cutting Plane Algorithms
Problem: z = maxcfw_cT x : x P Zn
Cutting Plane Proce
The Apr 9, 2015
P = cfw_x Rn : Ax b, x 0, maxcfw_cT x : x P Zn . PI = conv(P Zn ).
Definition 20.1 (Recall, Valid Inequality) An inequality wT x is a valid inequality for S Rn if
Lemma 20.2 (Farkas) Let P =
Thu Apr 2, 2015
Dynamic Programming (Bellman, 1957)
Shortest Paths Problem
Given: Directed graph G = (V, A), |V | = n, |A| = m. Arc lengths c : A R+ , source node s.
Goal: Find shortest path from s to every other nodes in G.
Thu Jan 29, 2015
3x + y
3x + y 0
x, y Z
IPs with irrational inputs, bounded, can have no optimal solution. Points can be arbitrarily close to line
3x + y = 0 but never touch it.
Definition 3.1 (1) A polyhedral co
Tue Apr 14, 2015
Cutting Planes (continued)
P = cfw_x : Ax b, PI = conv(P Zn ), P = P cfw_CG-cuts for P .
Definition 21.1 Let P (0) = P, P (1) = P , P (2) = (P (1) ) , P (3) = (P (2) ) , . . . , P (i+1) = (P (i) ) be a sequence of
Tue Mar 31, 2015
Maintain a dual feasible solution and try to find a primal feasible solution that satisfies complementary
Given: Complete Bipartite graph G = (V1 V2 , E), |V1 | = |V2 | = n we
Tue Mar 10, 2015
Talk by George Nemhauser:
Title: IP the Global Impact
4PM, April 7th (Tue)
Linear Equation Integer Feasibility Problem
Given A Zmn , b Zm .
Does Ax = b have an integer solution.
Example 14.1: Does
6 1 2