Cs 598csc: Combinatorial Optimization 1 Polyhedra and Linear Programming
نویسنده
چکیده
In this lecture, we will cover some basic material on the structure of polyhedra and linear programming. There is too abundant material on this topic to be covered in a few classes, so pointers will be given for further reading. For algorithmic and computational purposes one needs to work with rational polyhedra. Many basic results, however, are valid for both real and rational polyhedra. Therefore, to expedite our exploration, we will not make a distinction unless necessary.
منابع مشابه
Modelling Decision Problems Via Birkhoff Polyhedra
A compact formulation of the set of tours neither in a graph nor its complement is presented and illustrates a general methodology proposed for constructing polyhedral models of decision problems based upon permutations, projection and lifting techniques. Directed Hamilton tours on n vertex graphs are interpreted as (n-1)- permutations. Sets of extrema of Birkhoff polyhedra are mapped to tours ...
متن کاملCs 598csc: Combinatorial Optimization
Now we turn our attention to the proof of Theorem 1. Proof of Theorem 1. Let x be an extreme solution for the polytope (∗). Suppose that M has a loop e. Since rM ({e}) = 0, it follows that x(e) = 0 and we are done. Therefore we may assume that M does not have any loops and thus the polytope (∗) is full dimensional1. Now suppose that x(e) ∈ (0, 1) for all elements e ∈ S. Let n denote the number ...
متن کاملCs 598csc: Approximation Algorithms
Let G = (V,E) be an undirected graph with arc weights w : V → R+. Define xv for each vertex v as follows: xv = 1, if v is in the vertex cover; xv = 0, if v is not chosen. Our goal is to find min ( ∑ v∈V wvxv), such that xu + xv ≥ 1, ∀e = (u, v) ∈ E, xv ∈ {0, 1}. However, we can’t solve Integer Linear Programming (ILP) problems in polynomial time. So we have to use Linear Programming (LP) to app...
متن کاملCs 598csc: Combinatorial Optimization
One of several major contributions of Edmonds to combinatorial optimization is algorithms and polyhedral theorems for matroid intersection, and more generally polymatroid intersection. From an optimization point of view, the matroid intersection problem is the following: Let M1 = (S, I1) and M2 = (S, I2) be two matroids on the same ground set S. Then I1 ∩ I2 is the collection of all sets that a...
متن کاملCs 598csc: Combinatorial Optimization
Proof: We show one direction, the one useful for applications. Suppose A is TUM, consider P = {x | x ≥ 0, Ax ≤ b}. Let y = kx∗ be an integral vector where x∗ ∈ P . We prove by induction on k that y = x1 + x2 + . . . .xk for integral vectors x1, x2, . . . , xk in P . Base case for k = 1 is trivial. For k ≥ 2, consider the polyhedron P ′ = {x | 0 ≤ x ≤ y;Ay − kb + b ≤ Ax ≤ b}. P ′ is an integral ...
متن کامل