Feb. 22 - the adjacency matrix and the integrality condition (p.36-38)
Let G=(V,E) be a graph with n vertices. Choose a labeling v_1,.,v_n
of the vertices. A basic construction for much of graph theory:
The _adjacency matrix_ A=A(G) is the matrix whose (i
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 8 February 24th, 2014
Overview
Last week we talked about the Simplex algorithm. Today well broaden the scope of the objectives
we can optimize using linear programs, and in particular
CS 221: Computational Complexity
Prof. Salil Vadhan
Lecture Notes 1
January 25, 2010
1
Scribe: Richard Liu
Course Overview
Reading: Arora-Barak, Introduction.
Computational complexity aims to understand the limits of ecient computation. We ask:
Which prob
CS 221: Computational Complexity
Prof. Salil Vadhan
Lecture Notes 10
March 3, 2010
1
Scribe: Jonathan Ullman
Circuit Size Bounds
Proposition 1 Every function f : cfw_0, 1n cfw_0, 1 can be computed by a circuit of size O(n2n )
over the basis S = cfw_, , .
CS 221: Computational Complexity
Prof. Salil Vadhan
Lecture Notes 11
March 5, 2010
1
Circuit Complexity Diagram
2
Scribe: Michael Jemison
Randomized Computation
Now well study the complexity theory of randomized algorithms algorithms that may make
random
CS 221: Computational Complexity
Prof. Salil Vadhan
Lecture Notes 12
March 8, 2010
1
Scribe: David Wu
Recap
Recall our denition of the basic complexity classes involving randomized computation:
Denition 1
L BPP, RP, co-RP i there exists a PPT (probabilist
CS 221: Computational Complexity
Prof. Salil Vadhan
Lecture Notes 14
March 22, 2010
1
Scribe: Kyu Bok Lee
Agenda
#P-completeness (cont.)
Todas Theorem
Approximate Counting vs. Uniform Sampling (more on next lecture)
2
#P-Completeness (cont.)
Denition 1
CS 221: Computational Complexity
Prof. Salil Vadhan
Lecture Notes 13
March 10, 2010
1
Scribe: James Williamson
Randomized Reductions
We consider Unique SAT, a promise problem
USATY
= cfw_ | has exactly one satisfying assignment
USATN
= cfw_ | is unsatisa
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 18 April 7th, 2014
Overview
The methods we discussed until now were focused on optimizing objectives for which feasible solutions were in Rn . We will now start talking about combinato
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 9 February 26th, 2014
Overview
Today we will recapitulate basic definitions and properties from multivariate calculus which we will
need in the coming lectures about Convex Optimizatio
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 19 April 9th, 2014
Overview
In the previous lecture we started talking about combinatorial optimization. In general, we
would like to know whether problems that were are interested in
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 1 January 27th, 2014
Overview
In this course we will cover convex and combinatorial optimization and emphasize the deep connections between these two areas. Convex optimization is a ce
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 6 February 12th, 2014
Overview
In our previous lecture we explored the concept of duality which is the cornerstone of Optimization
Theory. The goal of todays lecture is to explore the
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 2 January 29th, 2014
Overview
In our previous lecture we discussed several motivating examples for optimization in applications
that use data. Today, we will begin the first part (out
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 4 February 5th, 2014
Overview
In our previous lecture we discussed criteria for testing whether an LP in infeasible and whether it is
unbounded. We then proved Weierstrass theorem whic
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 3 February 3rd, 2014
Overview
In our previous lecture we discussed several motivating examples for linear optimization. We
used the problem about optimizing ad budget to show that find
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 5 February 10th, 2014
Overview
In our previous lecture we discussed extreme points, and showed that if an LP has an optimal
solution, then it is an extreme point. In this lecture were
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 7 February 19th, 2014
Overview
In our previous lecture we saw the application of the strong duality theorem to game theory, and
then saw how that is applied to learning theory where we
AM 221: Advanced Optimization
Spring 2014
Prof. Yaron Singer
1
Lecture 10 March 3rd, 2014
Overview
Last week we reviewed results from multivariate calculus in preparation for our journey into convex
optimization. Today well talk about characterizations of
CS 221: Computational Complexity
Prof. Salil Vadhan
Lecture Notes 8
February 19, 2010
1
Scribe: Kevin Lee
Agenda
1. Provably Intractable Problems
2
Provably Intractable Problems
Some lower bounds that we proved so far:
1. Hierarchy theorems give us unnatu
March 5: more about the outer automorphism of S_6,
and Aut(S_n) in general (cf. Chapter 6 of the textbook)
Last time we constructed a total by starting from two disjoint
synthemes, finding the four disjoint from both of them, and
showing that one is a dea
March 3: Uniqueness and automorphism group of Pi_4 (day 2)
Recall: we began with any order-4 projective plane Pi and one of its
6!*168 ordered ovals O = cfw_p1,p2,p3,p4,p5,p6. This let us
identify 15 of the lines: the 15 secants s_cfw_i,j. That leaves
15
Feb. 24 - Moore graphs; two more conditions: "absolute" and Krein
(pages 41-42)
Recall: the adjacency matrix A of a strongly regular graph with
parameters (n,k,\lam,\mu) satisfies A\j = k\j and
(p.37, 2.13) A^2 = k I + \lam A + \mu (J-I-A).
Thus any eigen
Feb.5 and 8 Important examples of designs, II:
Hadamard matrices, particularly Sylvester and Paley
NB Problems 6 and 7 of the present problem set are postponed till
the next week: we won't get to intersection triangles or to the
relevant use of inclusion/
Feb.3 [& 5] Important examples of designs I:
projective planes and higher-dimensional projective spaces
Usually when we introduce a new kind of mathematical structure
we give a selection of important examples, and transformations that
construct new exampl
Feb. 10 - new designs from old:
complement; Hadamard 3-designs; derived designs
NB again we choose a different (but still logical) order of
presentation from the textbook's.
Complementary design (p.13, eqs. 1.37-1.38 and Prop. 1.39):
if \D = (X,\B) is a t
Feb. 12 - intro to arcs and ovals; intersection triangles
You're already had to look up "oval" on page 17 to construct a
(3,6,22) Steiner system from the square Paley 2-(11,5,2) design.
An arc in a 2-design is a set S of points no three of which are
cont
Feb. 19 - intro to _strongly regular graphs_ (ch.2)
Def. 2.1 (p.29): graph G = (V,E) where V is a finite set of "vertices"
(singular = "vertex") and E is a collection of two-element
subsets of V ("edges"). Equivalently, a finite set with a
relation that i