First meeting: Tuesday, Jan. 5, in MC 2018 B, 10:00am
Many of the problems that arise in practical applications of discrete
optimization are NP-hard, that is optimal solutions cannot be computed
in polynomial-time modulo the P not equal NP conjecture. Current
research is focusing on the design of polynomial-time approximation
algorithms for such problems.
The course will study some of the successful paradigms for designing
approximation algorithms and for proving approximation guarantees: the
greedy method, formulating and solving LP (linear programming)
relaxations, the (LP based) primal-dual method, randomized rounding,
semidefinite programming relaxations, approximation of metrics, etc.
Some lectures will focus on the hardness of approximating specific problems.
The plan is to cover a good fraction of Vazirani's book in the lectures, and
also to cover some of the more recent results.
(Similar courses have been offered in W'05, W'08, W'09 and the plan
is to offer this course regularly.)
Assignments 50% Project 50%
Main Text:
Vijay Vazirani,
Approximation Algorithms.
( homepage).
Text on Computational Complexity:
Sanjeev Arora and Boaz Barak,
Complexity Theory: A Modern Approach.
( homepage).
PDF files for each chapter and the whole book are on the web;
some of the relevant parts are
Chapter 2, NP and NP completeness,
( PDF),
Chapter 18, PCP and hardness of approximation,
( PDF).
David Williamson, Lecture Notes on Approximation Algorithms, Fall 1998. IBM Research Report RC 21273, February 1999. ( homepage).
Michel Goemans has course notes on randomized algorithms, approximation algorithms, etc. ( homepage).
Cheriyan and Ram Ravi, Lecture Notes on Approximation Algorithms for Network Problems, ( lec notes page).
Hochbaum, D.S. (1996). Approximation algorithms for NP-hard problems. (Boston: PWS publishing co.).
P.Crescenzi, and V.Kann. A compendium of NP optimization problems ( homepage).
David Johnson's NP-Completeness Columns (PDF files for all)
The NP-Completeness Column: The Many Limits on Approximation, David S. Johnson, ACM Transactions on Algorithms (TALG) 2(3):473-489 (July 2006)
Charikar (Princeton), CS 594: Limits on Approximation, S07
Chekuri (UIUC), CS 598: Approximation Algorithms, F06
Gupta & Ravi (CMU), 15-854: Approximation Algorithms, F05
Guruswami & O'Donnell (U.Wash.), CSE 533: The PCP Theorem and Hardness of Approximation, F05
Khot (NYU), G22.3033-007: Probabilistically Checkable Proofs and Hardness of Approximation, W08
Roughgarden (Stanford), CS359: Hardness of Approximation, W07
Salavatipour (UA), CMPUT 675: Topics on Approximation Algorithms and Approximability, F07
Sudan (MIT), Approximability of Optimization Problems, F99
Fujito's survey paper, Approximation Algorithms for Submodular Set Cover with Applications, survey, IEICE Trans Inf Syst (2000)
Students are allowed to collaborate on the assignments to the extent of formulating ideas as a group. Each student is expected to write up the solutions by himself or herself. All hints, collaboration, outside help etc. should be explicitly listed in your submission.
Assignment 1, due Jan. 19 (Tue), in class
PDF
Update for Q4(b) of Ass.1: Each subset has unit cost, thus c_j=1 for all j.
Assignment 2, due Jan. 29 (Fri), 5pm (last update: Jan.26) PDF
Assignment 3, due Feb. 12 (Fri), 5pm (last update: Feb.2) PDF
Assignment 4, due Mar. 26 (Fri), 5pm (last update: Mar.13) PDF
Due dates for submissions/talks:
note on topic (1 page) Feb.26 (Fri)
report on key results (5 pages) Mar.19 (Fri), 5pm
prelim project report (around 10-15 pages) April.5 (Mon), 3pm
talks April.6,7,8 during 10am-3:30pm
final project report (around 10-15 pages) April.15 (Thu), 5pm
N.Bansal, R.Khandekar, V.Nagarajan,
Additive Guarantees for Degree Bounded Direct Network Design (PDF, IBM report),
STOC 2008.
Also posted on this web page.
M.X.Goemans, papers on SDPs The original "max cut paper" (with D.P.Williamson) is posted here, as well as two survey papers.
D.E.Knuth, The sandwich theorem (postscript or PDF) Expository notes on Lovasz's theta function.
L.Lovasz, Semidefinite programs and combinatorial optimization, 2000 (postscript, 54 pages)
L.Trevisan, Inapproximability of Combinatorial Optimization Problems, Jul.2004, revised: Feb.2010
Indiana University, Bloomington, summer school on Analysis and geometry in the theory of computation, 2009 Tutorial notes on Sparsest Cuts and {L_1} embeddings, and UGC, semi-definite programs, and constraint satisfaction. Also see overview by J.Lee