CS 598 TMC, Fall 2018
Geometric Approximation Algorithms
Instructor
Timothy Chan
(Siebel 3230, tmc "at" illinois.edu,
office hours: Tue 11:0012:00 or by appointment)
Meeting Time and Place
Wed & Fri 11:0012:15, Siebel 1103
Course Overview
This course is about approximation algorithms in
computational geometry, which deals with problems involving geometric
data (mostly in low dimensions).
We will see how approximation allows us to solve such
problems faster  obtaining linear (or even sublinear) algorithms
for many geometric problems in P,
and polynomial algorithms for many NPhard geometric problems.
We will study a variety of fundamental problems, chosen to illustrate important
techniques in the area. (Occasionally, we will touch on some recent research results and
open questions.)
Possible topics include:
 Extentrelated problems: diameter, width, minimum bounding box (techniques: grids,
epsilonkernels/coresets, polytope approximation, a recent polynomial method)
 Rangecountingrelated problems (techniques: random sampling, Chernoff bounds, discrepancy, shallow cuttings)
 Proximityrelated problems: closest pair, nearest neighbors, Euclidean spanners,
Euclidean minimum spanning trees
(techniques: grids, quadtrees, shifting, wellseparated pair decomposition)
 NPhard problems: geometric independent set, set cover, hitting set
(techniques: shifted grids/quadtrees,
planargraph separators and geometric separators, local search, LP rounding, union complexity, (<=k)levels, epsilonnets, quasisampling, multiplicative weights update, APXhardness)
 Other assorted problems: Euclidean traveling salesman,
clustering (kcenter/kmedian), embedding, shortest paths in unit disk graphs,
highdimensional minimum enclosing ball, ...
Prerequisite: Prior knowledge in computational geometry or approximation algorithms is not
required, but a solid background in algorithm
design and analysis (e.g., at the level of CS 374) is assumed.
Course Work
 3 or 4 assignments (problem sets), worth 40%
 Homework 1 (due Sep 28, Fri) [note (9/14): have changed Q3 to make it less trivial]
 Homework 2 (due Oct 17, Wed) [note: in Q1, all squares are axisaligned]
 presentation, worth 20%
 a project (reading some papers and writing a report,
or implementing some algorithms,
or doing some original research), worth 40%
(Assignments may be done in groups of at most 2; presentations and
projects are to be done individually.)
See guidelines for presentation and project.
References
There is no required textbook, but you may find the following general references useful (especially Sariel's book):

[HP] S. HarPeled,
Geometric Approximation Algorithms,
AMS, 2011

[BCKO] M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars,
Computational
Geometry: Algorithms and Applications (3rd ed.),
SpringerVerlag, 2008

[GRT] J. Goodman, J. O'Rourke, and C. D. Toth (eds.),
Handbook of Discrete and Computational Geometry (3rd ed.),
CRC Press, 2017
Lectures
(I'll provide copies of my handwritten notes, and links to relevant papers, below...)
Aug 29, 31: The diameter problem
 Constantapprox. algorithms: factor 2 and factor sqrt{3}
 (1+eps)approx. algorithms in fixed dimensions:
1. rounding points to a grid (O(n + 1/eps^{2(d1)}) time), 2. rounding directions
(O(n/eps^{(d1)/2})), 3. combination (O(n + 1/eps^{3(d1)/2})),
4. recursion in dimension (O(n+1/eps^{d(1/2)}))
 Refs: my notes,
my paper (2002)
[Sec 2, Algorithms 14]
Sep 5: The diameter problem (cont'd): polynomial method
Sep 7: The width problem
 First (1+eps)approx algorithm: rounding directions +
linear programming (O(n/eps^{(d1)/2}) time)
 Second algorithm: epskernels/coresets...
 Refs: my notes,
my paper (2002)
[Sec 3, Algorithm 1]
Sep 12, 14, 19: epskernels/coresets
 Algorithms: 1. fattening + rounding points (O(1/eps^{d1}) size),
2. fattening + rounding directions (O(1/eps^{d1}) size), 3. Dudley's technique (O(1/eps^{(d1)/2}) size)
 Applications to min enclosing ball/cylinder/box/..., minwidth enclosing annulus (by
linearization), sketching/streaming, outliers
 Refs: my notes (part 1 and part 2), Agarwal, HarPeled, and Varadarajan's
paper (2004)
and survey,
my paper (2006) [Sec 2]
Sep 21, 26: Approximate range counting problems
 "epsapproximations" (with additive error eps*n): random sampling + Chernoff (O((1/eps^2) log (1/eps)) size),
improved bounds for balls/halfspaces via discrepancy + matching with
low crossing numbers ((O~(1/eps^{22/(d+1)}) size)
 Deterministic construction of epsapproximations: Matousek's O(n)time algorithm for constant eps via mergeandreduce
 Refs: my notes, books on discrepancy theory by Matousek and
Chazelle
Sep 28, Oct 3: Approximate range counting problems (cont'd): 1+eps factor
Oct 5, 10: Proximity problems: bichromatic closest pair, nearest neighbors

fixed radius: grid methods (O((1/eps^d)n) time naively, or O((1/eps^{d/4})n) by epskernelrelated techniques)

arbitrary radius: compressed quadtrees, balanced quadtrees, shifting (O(n) space and
O(log U + 1/eps^d) or O((1/eps^d) log n) query time)

Refs: my notes, Sariel's book [Ch2],
my paper (1998) [Sec 3]
Oct 12: EMST, spanners

Method 1: Yao graph (O(n) size, O(n polylog n) time)

Method 2: wellseparated pair decomposition...