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

Helly theorems Helly’s theorem (1911): Given a finite set H of convex objects in R d, H has a non empty intersection if every k=d+1 of its elements have a common point Radius theorem: Given a finite set H of points in R d, H is contained in a unit ball if every k=d+1 of its elements are contained in a unit ball Observation: radius theorem follows from Helly’s 

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

Discrete Helly theorems Doignon (1973): Given a finite set D of convex objects in R d, the objects in D have a common point in S=the integer lattice if every k=2 d of its elements do Theorem 1: S arbitrary set of points  unbounded Helly number

Example: Intersecting axis parallel boxes in R d Theorem 2: Given a finite set D of axis parallel boxes in R d, and a finite set S of points, the objects in D have a common point in S if every k=2d of its elements do Proof sketch:

Lexicographic Helly theorems Lex Helly’s theorem: Given a finite set H of convex objects in R d and a point x  R d, H has a non empty intersection in a point not lex greater than x if every k=d+1 of its elements do lexdiscretecontinuous type objects d+1  convex max {2,d}2d2d 2 axis-parallel boxes

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 

Helly theorems and Optimization  = radius of square Example: discrete smallest enclosing cube Input: n green points and m red cube centers No linear time algorithm known  = 0  = 2  = 1

LP-type problems [SW92] Def: a pair (H,  ), H: constraints  : objective function satisfying: monotonicity (  F  G  H,  (F)   (G)) locality Interpretation:  (G)=minimum value s.t. constraints in G Dual LP-type problems : (H,  ) with inequality signs reversed Goal: calculate  (H)

An example Wanted: A basis of H The optimal value r =  (H) r Linear time (randomized) algorithms [Cl88], [Ka92],[SW92] Smallest Enclosing Ball H: points  (H): radius of the smallest enclosing ball of H combinatorial dimension: d+1

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

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 ?

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

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

parameterized Helly system (PHS) Theorem 6: discrete opt. problems with discrete PHS s.t. UMC, are fixed-dimensional DLP Discrete Helly theorems  DLP Corollary: discrete opt. problems with discrete PHS s.t. UMC, are solvable in linear time Extends [A94] unique minimum condition (UMC)

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  

Theorem 7: lex PHS are fixed-dimensional LP-type problems Lex Helly theorems  linear time alg Corollary: existence of a finite lex Helly number  solvability of the corresponding optimization problem by a linear LP-type algorithm

Future research Find more discrete/lex Helly theorems Develop more algorithms for DLP model Find more applications for DLP model

Thank you !