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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