On the Power of Discrete and of Lexicographic Helly Theorems

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On the Power of Discrete and of Lexicographic
Helly Theorems
Nir Halman, Technion, Israel
This work is part of my Ph.D. thesis, held in Tel
Aviv University, under the supervision of
Professor Arie Tamir, and appeared in FOCS 2004
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Helly theorems
Hellys theorem (1911) Given a finite set H of
convex objects in Rd, H has a non empty
intersection if every kd1 of its elements have
a common point
Radius theorem Given a finite set H of points in
Rd, H is contained in a unit ball if every kd1
of its elements are contained in a unit ball
Observation radius theorem follows from Hellys
?
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Our results
Discrete/lex Helly theorems Useful for
solving discrete optimization problems in linear
time (extending LP-type) Characterization of
Helly theorems that yield linear time algorithms
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Discrete Helly theorems
Doignon (1973) Given a finite set D of convex
objects in Rd, the objects in D have a common
point in Sthe integer lattice if every k2d of
its elements do
Theorem 1 S arbitrary set of points ? unbounded
Helly number
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Example Intersecting axis parallel boxes in Rd
Theorem 2 Given a finite set D of axis parallel
boxes in Rd, and a finite set S of points, the
objects in D have a common point in S if every
k2d of its elements do
Proof sketch
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Lexicographic Helly theorems
Lex Hellys theorem Given a finite set H of
convex objects in Rd and a point x ? Rd, H has a
non empty intersection in a point not lex greater
than x if every kd1 of its elements do
lex discrete continuous type objects
d1 ? d1 convex
max 2,d 2d 2 axis-parallel boxes
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Our results
Discrete/lex Helly theorems Useful for
solving discrete optimization problems in linear
time (extending LP-type) Characterization of
Helly theorems that yield linear time algorithms
?
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Helly theorems and Optimization
Example discrete smallest enclosing cube
Input n green points and m red cube centers
No linear time algorithm known
? 0
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LP-type problems SW92
Def a pair (H, ?), H constraints ?
objective function satisfying monotonicity
(? F?G?H, ?(F) ? ?(G)) locality
Goal calculate ?(H)
Interpretation ?(G)minimum value s.t.
constraints in G
Dual LP-type problems (H, ?) with inequality
signs reversed
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An example
Smallest Enclosing Ball
H points ?(H) radius of the smallest
enclosing ball of H
Wanted
The optimal value r ?(H)
A basis of H
combinatorial dimension d1
Linear time (randomized) algorithms Cl88,
Ka92,SW92
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Usage of LP-type framework
Mostly in computational geometry / location
theory distance between polytopes
smallest enclosing ball/ellipsoid
largest ball/ellipsoid in polytope
angle-optimal placement of point in
polygon line transversal of translates
p-center on the line/in the plane
with rectilinear norm convex Hausdorff
distance etc.
p-recovery points on the line and on directed
trees simple stochastic game
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Discretization of
Center of ball (cube) must be an input point We
lose monotonicity !
Lower bound ?(n log n) even for circles LW86
Lower bound for cubes is ?(n)
Can we solve discrete smallest enclosing
(d-dimensional) cube in linear time ?
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Discretization of smallest enclosing cube problem
Input a set D of demand points and
a set S of cube center locations (supply) Output
the center and radius of a smallest
enclosing cube
Observation problem obeys double monotonicity
Adding a demand point cannot decrease the
value Adding a supply point cannot increase the
value
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Discrete LP-type problems (DLP)
Triple (D,S,?). D demand set, S supply set, ?
objective function s.t. for any D? D and S? S
?(D)?(D,S) ? (D, ?) is LP-type
?(S)?(D,S) ? (S, ?) is dual LP-type
Theorem 5 fixed-dimensional DLP problems are
solvable in linear time
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Discrete Helly theorems ? DLP
parameterized Helly system (PHS)
unique minimum condition (UMC)
Theorem 6 discrete opt. problems with discrete
PHS s.t. UMC, are fixed-dimensional DLP
Extends A94
Corollary discrete opt. problems with discrete
PHS s.t. UMC, are solvable in linear time
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Our results
Discrete/lex Helly theorems Useful for
solving discrete optimization problems in linear
time (extending LP-type) Characterization of
Helly theorems that yield linear time algorithms
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Lex Helly theorems ? linear time alg
Theorem 7 lex PHS are fixed-dimensional LP-type
problems
Corollary existence of a finite lex Helly number
? solvability of the corresponding optimization
problem by a linear LP-type algorithm
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Future research
Find more discrete/lex Helly theorems Develop
more algorithms for DLP model Find more
applications for DLP model
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Thank you !