CS270: Combinatorial Algorithms and Data Structures

Fall 2008

Announcements

Dec. 10		Final Exam clarifications: Problem 8: the expression should be n(n-1)/2, Problem 11: MAXCLIQUE(H)=MAXCLIQUE(G)^2
Oct. 29		Solutions to the midterm exam are now available.
Oct. 16		In Problem Set 2, Problem 1, b' should be changed to (9,7).
Oct. 10		Notes for Lecture 12 have been posted with some revisions.
Sept. 30		Luqman's office hours for Wednesday, October 1st, are moved to 3-4 for this week only.
Sept. 25		The midterm exam is scheduled for Thursday, October 16th, in class.
Sept. 25		Each member of the class can submit one late problem set without penalty provided it is submitted by the class period following the due date.
Sept. 16		Problem Set 1 is due Thursday, September 25, 2008.
Sept. 5		Lecture notes are now being posted in PDF format.
Sept. 4		Discussion sections will be held in 145 McCone M. 2-3.
Aug. 30		Discussion sections will take place on Mondays 2:00-3:00. Location to be announced.

Administrative Information

Instructor: Richard Karp (karp@cs)
Office Hours: 621 Soda, Tu. 2-3, W. 11-12
TA: Luqman Hodgkinson (luqman@berkeley.edu)
Office Hours: 751 Soda, M. 1-2, W. 4-5
Lecture: 306 Soda, Tu., Th. 12:30-2:00
Section: 145 McCone, M. 2-3
Class Website: http://www.cs.berkeley.edu/~luqman/cs270.html
Grading: Final 40%, midterm 20%, problem sets 15%, project 25%
Units: 3

Problem Sets

Problem Set 1: Due Sept. 25, 2008 (pdf)(solutions)
Problem Set 2: Due Oct 23, 2008 (pdf)(solutions)
Problem Set 3: Due Nov 20, 2008 (pdf)(solutions)

Term Project

Students may work alone or in two-person teams. There are three options:

Formulating a computational problem related to your research interests and providing and evaluating an algorithm for its solution.
Devising, implementing and testing an algorithm for the challenge problem to be presented in class.
Writing a report demonstrating your understanding of a paper that has won the Goedel Theory Prize or Kanellakis Theory and Practice Prize.

Lecture Notes

Professor Karp will provide informal notes for each lecture. There will be no textbook. Pointers to published literature will occasionally be placed on this course website.

Lecture No. 1: Aug. 28, 2008 (syllabus, notes)
Lecture No. 2: Sept. 2, 2008 (notes)
Lecture No. 3: Sept. 4, 2008 (notes)
Lecture No. 4: Sept. 9, 2008 (notes)
Lecture No. 5: Sept. 11, 2008 (notes)
Lecture No. 6: Sept. 16, 2008 (notes)
Lecture No. 7: Sept. 18, 2008 (notes)
Lecture No. 8: Sept. 23, 2008 (notes)
Lecture No. 9: Sept. 25, 2008 (notes)
Lecture No. 10: Sept. 30, 2008 (notes)
Lecture No. 11: Oct. 2, 2008 (notes)
Lecture No. 12: Oct. 7, 2008 (notes)
Lecture No. 13: Oct. 9, 2008 (notes)
Lecture No. 14: Oct. 14, 2008 (notes)
Lecture No. 15: Oct. 21, 2008 (notes)
Lecture No. 16: Oct. 23, 2008 (notes) (Marques-Silva, Sakallah, 1999)
Lecture No. 17: Oct. 28, 2008 (notes)
Lecture No. 18: Oct. 30, 2008 (notes)
Lecture No. 19: Nov. 4, 2008 (notes)
Lecture No. 20: Nov. 6, 2008 (notes)
Lecture No. 21: Nov. 13, 2008 (notes)
Lecture No. 24: Nov. 25, 2008 (notes)
Lecture No. 25: Dec. 2, 2008 (notes)
Lecture No. 26: Dec. 4, 2008 (notes)
Lecture No. 27: Dec. 9, 2008 (notes)

Tentative Course Outline

Introductory Examples: Stable marriage, Euclidean algorithm, Gaussian elimination, Fast Fourier Transform

Dynamic Programming: Longest path, sequence alignment, knapsack problem

Graph-theoretic algorithms: Bipartite matching, maximum flow, non-bipartite matching

Linear programming: Simplex algorithm, duality theorem, ellipsoid algorithm

Convex optimization: application to supervised classification (Support Vector Machine)

Randomized algorithms: min-cut, checking polynomial identities, universal hashing, pattern matching, Miller-Rabin primality test, distributed consensus (including impossibility of a deterministic algorithm in adversarial setting)

Computational complexity: P, NP, Cook's Theorem, reducibility, NP-completeness, PCP theorem, hardness of approximation

Approximation algorithms: vertex cover, set cover, primal-dual technique, max-cut via semi-definite programming

Parallel algorithms: list ranking, parallel prefix computation, connected components

On-line algorithms: competitive ratio, paging, list accessing, splay tree algorithm

Streaming algorithms: approximating frequency moments

Spectral algorithms: singular value decomposition and its applications to clustering, collaborative filtering, data mining and website ranking

Raptor codes for bulk data transmission

Discussion Section

The discussion section is optional and will cover auxiliary topics. Each week a volunteer will scribe notes to be made available to other members upon request.

Section No. 1: Sept. 8, 2008 History and current best complexity of matrix multiplication, Strassen's algorithm, computing matrix inverses, gcd of univariate polynomials with rational coefficients
Section No. 2: Sept. 15, 2008 History of polynomial-time algorithms for Max Flow, an O(|V|^3) algorithm for Max Flow based on Karzanov's algorithm; Scribe: Steve Hanna (sch at eecs)
Section No. 3: Sept. 22, 2008 History of Minimum Weight Triangulation (MWT) including recent NP-hardness proof, an O(n^3) dynamic programming algorithm for MWT on the point set of a convex polygon; Scribe: Sayak Ray (sayak at eecs)
Section No. 4: Sept. 29, 2008 The assignment problem with proof that extreme points of LP-relaxation are integer, total unimodularity with application to node-arc incidence constraint matrices; Scribe: Sayak Ray (sayak at eecs)
Section No. 5: Oct 6, 2008 Proof of Koenig-Egervary theorem by duality and total unimodularity, proof that every regular bipartite graph has a perfect matching, review of odd-set covers and proof of the odd-set cover theorem, proof that the extreme points of the LP-relaxation for nonbipartite matching may be fractional by duality and the odd-set cover theorem; Scribe: Adam Pauls (adpauls at eecs)
Section No. 6: Oct 13, 2008 Problem Set 2 Question and Answer Session
Section No. 7: Oct 20, 2008 History of the theory of NP-completeness, Cook reductions as a general method for proving NP-hardness with demonstration that Karp reductions are a special case of Cook reductions, pseudopolynomial-time algorithm for Subset-Sum, discussion of strong NP-completeness, reduction from Subset-Sum to Knapsack; Scribe: Grey Ballard (ballard at cs)
Section No. 8: Oct 27, 2008 Review of midterm exam solutions, review of branch-and-bound algorithm for Symmetric TSP with the LP-relaxation lower bound, definition of Gomory-Hu trees and algorithm for computing them
Section No. 9: Nov 3, 2008 The IP for TSP with a polynomial number of constraints given by Miller, Tucker, Zemlin in 1960, review of the definitions of NP-completeness, NP-hardness, NP, and decision problems, reduction from Maximum Independent Set to Maximum Clique, reduction from 3DM to X3C, reduction from X3C to Connected Colorful Subgraph, reduction from 3SAT to 0-1 Integer Programming, with mention that proof of the general case of integer linear programming being in NP (i.e. the existence of a polynomially-sized solution/certificate) is not trivial.
Section No. 10: Nov 10, 2008 Review of strong NP-hardness with proof that a pseudopolynomial-time algorithm for a strongly NP-hard problem implies P=NP, proof that under some widely-applicable conditions (that are met by Minimum Makespan) an FPTAS implies a pseudopolynomial-time algorithm, reduction from 3-Partition to Minimum Makespan where the number of machines can vary proving this version strongly NP-hard, dynamic programming algorithm for Minimum Makespan that runs in pseudopolynomial-time when the number of machines is a fixed constant
Section No. 11: Nov 17, 2008 Pseudopolynomial time dynamic programming algorithm for Minimum Makespan when m is a fixed constant, including details on how to extract the optimal schedule; also a comparison to a naive enumeration of schedules, review of the definition of FPTAS and how it may be designed using a pseudopolynomial-time algorithm and a scaling technique, review of FPTAS for Knapsack using the scaling technique with approximation analysis and time complexity analysis; Scribe: Grey Ballard (ballard at cs)
Section No. 12: Nov 24, 2008 Review of Miller-Rabin primality testing with some exercises, Karp-Rabin algorithm for pattern matching with analysis of probability of correctness.
Section No. 13: Dec 1, 2008 Freivald's technique for verifying matrix identities, verifying identities of univariate polynomials, Schwartz-Zippel Theorem for verifying identities of multivariate polynomials.
Section No. 14: Dec 8, 2008 Review of proof that NP is a subset of PCP(poly(n),1), review of test for linearity, review of self-correction mechanism for correcting any entry in a somewhat faulty truth table to that of the nearest linear function with high probability.