Helly Theorems
and Generalized Linear Programming
b y
Annamaria Beatrice Amen ta
B A Y ale Univ ersit y
A dissertation submitted in partial satisfaction of the
requiremen ts for the degree of
Do ctor of Philosoph y
in
Computer Science
in the
GRADUA TE DIVISION
of the
UNIVERSITY of CALIF ORNIA at BERKELEY
Committee in c harge
Professor Raim und Seidel
Professor John Cann y
Professor William Th urston
Helly Theorems
and Generalized Linear Programming
Cop yrigh t
b y
Annamaria Beatrice Amen ta
Abstract
Helly Theorems and Generalized Linear Programming
b y
Annamaria Beatrice Amen ta
Do ctor of Philosoph y in Computer Science
Univ ersit y of California at Berk eley
Professor Raim und Seidel Chair
This thesis establishes a connection b et w een the Helly theorems a collection of
results from com binatorial geometry and the class of problems whic h w e call Generalized
Linear Programming or GLP whic h can b e solv ed b y com binatorial linear programming
algorithms lik e the simplex metho d W e use these results to explore the class GLP and
sho w new applications to geometric optimization and also to pro v e Helly theorems
In general a GLP is a set of constrain ts and a function to b e minimized whic h
ob ey certain com binatorial conditions Linear programming is an example A Helly the
orem is also dened b y its com binatorial structure W e observ e that there is a Helly
theorem ab out an y GLP whic h is that the minim um is no greater than m if and only if
the minim um of ev ery subproblem with d constrain ts is no greater than m
W e use this observ ation to pro v e Helly theorems Then w e giv e a paradigm whic h
usually allo ws us to construct a GLP corresp onding to a giv en Helly theorem Most of
our algorithmic results are based on this paradigm W e sho w that in there are GLPs in
whic h the constrain ts or ob jectiv e function are not only nonlinear but also noncon v ex or
disconnected
W e giv e n umerous applications concen trating on exp ected O n time algorithms
Some examples are that the largest axisaligned b o x in the the in tersection of a family
of con v ex sets in xed dimension and the translation and scaling whic h minimizes the
Hausdor distance b et w een t w o con v ex p olygons in the plane can b e found b y GLP An
example of a second family of results is a GLP to nd the smallest factor b y whic h a family
of b o xes can b e scaled around their cen ters so as to admit a h yp erplane transv ersal th us
tting a h yp erplane to the family of cen ters
The author w as supp orted b y an National Science F oundation Graduate F ello w
ship an A T
T Graduate Researc h Program for W omen Gran t and a UC Presiden t s
Dissertation Y ear F ello wship Some of this w ork w as done while visiting A T
T Bell Labs
at Murry Hill and the F reie Univ ersiat at Berlin
Committee Chairman Raim und Seidel
i
Ac kno wledgmen ts
I ha v e man y p eople to thank for the w onderful formativ e y ears I sp en t at Berk e
ley First among them is m y advisor Raim und Seidel Raim und alw a ys treated me with
resp ect encouraging me to follo w m y inclinations while pro viding critical feedbac k He
w as patien t as I learned ho w to talk and write ab out math Whenev er I kno c k ed on his
do or I knew that he w ould b e gen uinely glad to see me this made a tremendous dierence
I w ould nev er ha v e gone to graduate sc ho ol without the Reen try Program and I
thank ev ery one in v olv ed in it It is a mo del armativ e action program
I w as luc ky to b e at a sc ho ol that tak es graduate instruction seriously I esp ecially
proted from the teac hing of professors Karp and V azirani
I am proud to ha v e had access to the exp erience and insigh t of Nimro d Megiddo
Tw o summers at Bell Labs in Murry Hill thanks to the Graduate Researc h Program
for W omen w ere imp ortan t in m y dev elopmen t I thank m y men tor Stev e F ortune Ken
Clarkson and P eter Shor for crucial encouragemen t Another summer at Thinking Ma
c hines in Cam bridge w as a v aluable and pleasan t exp erience Finally I thank Raim und for
bringing me to the F reie Univ ersiat at Berlin where Emo W elzl Jirk a Matou sek Helm ult
Alt and their studen ts made me feel w elcome
My fello w Berk eley studen ts w ere deligh tful and fascinating It w as with the
other theory girls Ronitt Rubinfeld Sandy Irani Diane Hernek and Dana Randall
that I learned ho w to b e a w oman scien tist and enjo y it Ronitt s faith in me has made
the dierence more than once Sandy has b een an ideal buddy in b oth w ork and pla y
and Diane Dana and I giggled our w a y through the whole exp erience Mik e Lub y has
b een in v aluable as an honored friend and advisor Will Ev ans has b een a great friend
an inspiring example and the p erfect ocemate Ash u Rege w as an essen tial civilizing
presence in the oce
Renn y Lucas w as m y rst Berk eley friend and still one of the b est I will alw a ys
v alue her go o d advice and her sense of h umor Bec ky Gross s aection and en th usiasm
touc h ev ery one s heart esp ecially mine
ii
The graphicsb o ys P aul Hec kb ert Seth T eller Henry Moreton Mik e Hohmey er
and Eric Enderton taugh t me a great deal and deligh ted me with their individual w arm th
and collectiv e liv eliness I thank Carlo Sequin for fostering their en vironmen t
Nerv ous for Nigel Dan Jurafsky DeKai W u T erry Regier Erin Dare and
sometimes Eric Enderton w as the fundamen tal force that made Berk eley CS a scene
rather than a mere departmen t I hop e to alw a ys b e their n um b er one fan and I thank
them for k eeping me up so late Lik e the rest of the AI cro wd including Mik e Sc hi and
the actual Nigel W ard they alw a ys had something in teresting to talk ab out
One of the pleasures of academia is meeting p eople from all o v er The Zlatevs
from Bulgaria and Monica from Berlin w ere an education My Nap olitan friend Geppino
Pucci can alw a ys mak e me laugh I thank him from somewhere deep in the pit of m y
stomac h
Some p eople from m y past made graduate sc ho ol p ossible Both T om Ha y es and
Sally Calhoun ga v e me great jobs that pushed me forw ard And the deluxe Jim McGun
taugh t me t w o essen tial skills during our man y y ears as compa neros de traba jo ho w to
hac k and ho w to part y
So man y other p eople w ere imp ortan t Ramon Caceres F racesca Barrien tos
Amie Wilkinson Sally Flo yd Susan Mathews and Jane Edw ards all help ed in their v ery
dieren t w a ys Da vid W olfe sho w ed me the rop es The other grad studen ts included so
man y p eople who w ere b oth brillian t and truly nice Marshall Bern Leo Guibas and
Marco P ellegrini w elcomed me to computational geometry and Phil Klein and Dann y
Sleator made me feel w elcome in the larger theory comm unit y
I w as luc ky to ha v e family nearb y I b enetted from the supp ort and p ersp ectiv e
of m y sister Da vida all through m y y ears in Berk eley Her honest y alw a ys k ept me from
heading o in to the ozone I thank her m y paren ts and m y brother T om In this momen t
of ac hiev emen t I m proud to b e an Amen ta
iii
Con ten ts
In tro duction
Helly theorems
Denitions
Helly systems
More general Helly theorems
Helly system witness problems
Helly theorems and V Cdimension
Generalized Linear Programming
GLP framew ork
Computational assumptions
GLP sp ecialized to mathematical programming
Necessit y of the GLP framew ork
Pro ving Helly Theorems with GLP
A short pro of of Morris theorem
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