18.415/6.854 Advanced Algorithms
September 17, 2008
Lecture 5
Lecturer: Michel X. Goemans
Today, we continue the discussion of the minimum cost circulation problem. We rst review the
Goldberg-Tarjan algorithm, and improve it by allowing more exibility in
18.415/6.854 Advanced Algorithms
September 10, 2008
Lecture 3
Lecturer: Michel X. Goemans
1
Introduction
Today we continue our discussion of maximum ows by introducing the fattest path augmenting
algorithm, an improvement over the Ford-Fulkerson algorithm
18.415/6.854 Advanced Algorithms
Problem Set Solution 4
Lecturer: Michel X . Goemans
1. I n class, we have seen Klein's cycle cancelling algorithm for the Min Cost Circulation Problem (MCCP). This algorithm requires O (mCU) i terations in the worst
case,
18.415/6.854 Advanced Algorithms
Problem Set Solution 1
1 . Consider P = cfw_ x : A x 5 b, x 2 0 ), where A is m x n . Show that if
x is a vertex of P then we can find sets I and J with the following
properties.
c
c
( a) I cfw_ I,. . . , m , J cfw_ I 7 .
18.415/6.854 Advanced Algorithms
Problem Set Solution 3
1 . Consider the following optimization problem:
G iven c E Rn , c 2 0, n e ven, f ind
mincfw_cTx :
xi,i 2 1
x
n
V c (1,. . . , n), I 1 = 2,
S
S
In class, it was shown that this can be solved by the
18.415/6.854 Advanced Algorithms
Problem Set Solution 6
Lecturer: Michel X . Goemans
1 . The betweenness p roblem is defined as follows: We are given n and a set T of m
triples of the elements of (1,. . . , n). We say that an ordering 7r of (1, . . . ,n)
18.415/6.854 Advanced Algorithms
Problem Set Solution 5
Lecturer: Michel X . Goemans
1. Consider the linear programming relaxation of the vertex cover problem seen in
class.
Min
C wixi
s ubject to:
( a) Argue that any basic feasible solution x of the abov
18.415/6.854 Advanced Algorithms
October 15, 2008
Lecture 11
Lecturer: Michel X. Goemans
In this lecture, we will start continuing from where we left in the last lecture on linear
programming. We then argue that LP N P co N P . In the end of this lecture,
18.415/6.854 Advanced Algorithms
October 8, 2008
Lecture 10
Lecturer: Michel X. Goemans
Last lecture we introduced the basic formulation of a linear programming problem, namely the
problem with the objective of minimizing the expression cT x (where c Rn ,
18.415/6.854 Advanced Algorithms
Problem Set Solution 2
1. T he Min s-t-Cut problem is the following:
G i v e n a n undirected graph G = (V, E), a w eight function w : E + R+ ,
a nd t w o vertices s , t E V , find
Min s - t
- Cut(G) = mincfw_w(S(S)
: S c
18.415/6.854 Advanced Algorithms
October 6, 2008
Lecture 9
Lecturer: Michel X. Goemans
9
Linear Programming
Linear programming is the class of optimization problems consisting of optimizing the value of a
linear objective function, subject to linear equa
18.415/6.854 Advanced Algorithms
September 15, 2008
Goldberg-Tarjan Min-Cost Circulation Algorithm
Lecturer: Michel X. Goemans
1
Introduction
In this lecture we shall study Kleins cycle cancelling algorithm for nding the circulation of minimum
cost in gr
Figure 1: Virtual tree (left) and corresponding rooted tree (right).
make sure that the path from v to the root is solid and that the splay tree representing the path to
which v belongs is rooted at v . We can describe this operation in three steps. In ou
18.415/6.854 Advanced Algorithms
September 29, 2008
Lecture 7 - Dynamic Trees
Lecturer: Michel X. Goemans
1
Overview
In this lecture, we discuss dynamic trees, a sophisticated data structure introduced by Sleator and
Tarjan. Dynamic trees allow to provid
18.415/6.854 Advanced Algorithms
September 24, 2008
Lecture 6 - Splay Trees
Lecturer: Michel X. Goemans
1
Introduction
In this lecture, we investigate splay trees, a type of binary search tree (BST) rst formulated by
Sleator and Tarjan in 1985. Splay tree