Discrete and Lexicographic Helly Theorems and their Relations to LP-Type Problems

Transcription

1 TEL AVIV UNIVERSITY The Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Discrete and Lexicographic Helly Theorems and their Relations to LP-Type Problems Thesis submitted for the degree of Doctor of Philosophy by Nir Halman Under the supervision of Professor Arie Tamir Submitted to the senate of Tel-Aviv University June 2004

2 The work on this thesis was carried out at the Department of Statistics and Operations Research Tel-Aviv University under the supervision of Professor Arie Tamir ii

3 Abstract Helly s theorem says that if every d + 1 elements of a given finite set of convex objects in IR d have a common point, then there is a point common to all of the objects in the set. We define three new types of Helly theorems: discrete Helly theorems - where the common point should belong to an a-priori given set, lexicographic Helly theorems - where the common point should not be lexicographically greater than a given point, and lexicographic-discrete Helly theorems. We show the relations between these new Helly theorems and their corresponding (standard) Helly theorems. We obtain several new discrete and lexicographic Helly numbers. Using these new types of Helly theorems we get linear time solutions for various optimization problems. For this, we define a new framework, discrete LP-type (LP stands for Linear Programming), and provide new algorithms that solve in randomized linear time fixed-dimensional discrete LP-type problems. We show that the complexity of the discrete LP-type class stands somewhere between Linear Programming (LP) and Integer Programming (IP). Finally, we use our results in order to solve in randomized linear time problems such as the discrete p- center on the real line, the discrete weighted 1-center problem in IR d with l norm, the standard (continuous) and a discrete version of the optimization problem of finding a line transversal for a finite set of planar axisparallel rectangles, and the (planar) lexicographic rectilinear p-center problem for p = 1, 2, 3. These are the first known linear time algorithms for these problems. iii

4 Acknowledgments I would like to thank first of all my advisor, Professor Arie Tamir, for his invaluable help and never-ending patience. Whenever I knocked on his door, I knew he would be genuinely glad to see me - this made a tremendous difference. I would like to thank Professor Nina Amenta whose excellent Ph.D. dissertation inspired me throughout my research. Special thanks to Professor Micha Sharir for teaching me computational geometry and being available when I needed his help. I thank Professor Nimrod Megiddo and Professor Noga Alon for their good advices and profound questions, which influenced my work greatly. I thank as well Professors Uri Zwick, Eilon Solan and Michael Krivelevich for inspiring conversations. To my amazing Tali, thank you for your help, support, understanding and patience. It was a sheer pleasure to be with you in the same room in the university, during the second half of my Ph.D. studies! I would like to thank my parents, who accompanied me from the beginning of my academic life with love and support. Nir Halman June 2004 iv

5 Contents 1 Introduction Helly-type theorems Optimization problems The relations between Helly-type theorems and optimization problems Applications I Theory 9 2 Helly theorems Continuous Helly theorems Definitions Examples Helly systems Discrete Helly theorems Definitions Examples Discrete Helly systems Lexicographic Helly theorems Definitions Examples Lexicographic Helly systems Lexicographic-discrete Helly theorems LP-type and discrete LP-type problems LP-type problems The LP-type framework Computational assumptions LP-type problems specialized to mathematical programming Dual LP-type problems Discrete LP-type problems v

6 3.3.1 The discrete LP-type framework Some properties of bases Computational assumptions Discrete LP-type problems specialized to mathematical programming From LP-type problems to Helly theorems Continuous case Discrete case From Helly theorems to LP-type problems (Non-lexicographic) continuous case (Non-lexicographic) discrete case Lexicographic (continuous) case Previous results New results Lexicographic-discrete case Discrete LP-type algorithms Clarkson s algorithm Sharir and Welzl s algorithm Discrete LP-type algorithms Algorithm I Algorithm II Algorithm III More algorithms II Applications 77 7 Discrete linear programming Definitions Discrete linear programming on the real line Discrete linear programming in the plane dimensional discrete linear programming in the plane dimensional discrete linear programming in the plane Discrete linear programming in IR k (k 3) Recovery points problems Placing recovery points on graphs Placing Recovery points on trees A formulation as an abstract problem (H, ω p ) (H, ω p ) is a p-dimensional dual LP-type problem vi

7 8.2.3 Complexity and data structures Placing recovery points on the real line A formulation as a p-dimensional dual LP-type problem An application of the p-recovery Points problem on the real line: p-axis-parallel guillotine cuts of a d-dimensional axis-parallel box Placing recovery points on circles p-center problems p-center problems on the line A formulation as a discrete abstract problem (D, S, ω) (D, S, ω) is a (p + 1, p)-dimensional discrete LP-type problem A linear time algorithm p-center problems on circles Continuous case Discrete case p-center problems with l norm Continuous case Lexicographic planar case General discrete weighted 1-center problem in IR d Discrete planar rectilinear p-center problem (p > 1) Euclidean p-center problems Continuous Euclidean p-center problems Discrete planar Euclidean 1-center problem A summary of Helly numbers for p-center problems in different spaces Line transversals in the plane Continuous case Discrete case I - A finite number of allowed directions of line transversals A formulation as a discrete LP-type problem A linear time algorithm Discrete case II - A finite number of allowed line transversals The simple stochastic game Introduction Definitions and previous results Simple Stochastic Games Parity Games, Mean Payoff Games and Discounted Payoff Games Formulating the SSG as an LP-type problem Solving SSG s, PG s, MPG s and DPG s in strongly e O( nlog n) time Solving binary SSG s in strongly e O( n) time Concluding remarks vii

8 12 The weighted lexicographic rectilinear 1-center problem in the plane Introduction The restricted lex1-center problem vs. the min rays problem A linear time solution algorithm for the min rays problem A linear time solution algorithm for the lex1-center problem Concluding remarks Concluding remarks 154 A Proving theorems related to line transversals 156 A.1 Discrete and lex variants of Theorem (Theorems 2.2.7, and 2.4.3) A.2 Discrete and lex variants of Theorem (Theorems , and 2.4.6) A.3 Discrete and lex variants of Theorem (Theorems 2.2.9, and 2.4.5) B Proving Theorem B.1 Proving Cases (i), (ii) and (v) B.2 Some useful definitions and lemmas B.3 Proving Case (iii) B.4 Proving Case (iv) viii

9 List of Figures 2.1 The 3 points on the left side are contained in a unit ball if and only if the 3 unit balls on the right side intersect Every 2 rectangles intersect but all 3 don t Every 3 rectangles intersect in a black point but all 4 don t Every 6 intervals admit a line transversal but not all The discrete Helly system corresponding to an instance of Theorem (for p = 2) The intersection of each pair of strips intersects a thick line, but all strips do not intersect in a common point An instance of the 2-piercing problem which demonstrates that h (2, 2) > h(2, 2) A basis for LP Adding h increases the diameter of G but not of F An instance of the Sum of Two Smallest Numbers problem An instance of the general 1-center problem A parameterized Helly system A discrete parameterized Helly system h and h (1,0,0) An instance of a 0-dimensional DLP problem in the plane Illustration of the mapping for A = B = { 1 3, 1, 3} An instance of the 5-Recovery Points problem on trees recovery points and their corresponding C state variables. Here n h = n l = An instance of the 3-center problem and the corresponding state variables Transversal not tangent to every basis element An instance of the planar 1-center problem with l norm An instance of the min rays problem The 4 types of mixed pairs ix

10 12.4 The 4 types of increasing pairs A.1 Two rectangles and their corresponding line transversals A.2 An instance of Theorem (i) A.3 4 squares admitting line transversals with slopes between 1 5 to A.4 8 rectangles and their corresponding line transversals slope ranges B.1 An instance of the 2-piercing problem where no axis parallel line traverses all the rectangles. 170 B.2 An instance of the 3-piercing problem where an horizontal line traverses all the rectangles B.3 An instance of the 2-piercing problem where a vertical line traverses all the rectangles B.4 An instance of the 3-piercing problem where a vertical line traverses all the rectangles x

11 Chapter 1 Introduction In this thesis we study the relations between Helly-type theorems, a collection of results from combinatorial geometry, and optimization problems. 1.1 Helly-type theorems The classical theorem of Helly stands at the origin of what is known today as the combinatorial geometry of convex sets. It was discovered in 1913 and may be formulated as Theorem (Helly s Theorem) Let H be a family of convex sets in IR d, and suppose H is finite or each member of H is compact. If every d + 1 or fewer members of H have a common point, then there is a point common to all members of H. A possible generalization is as follows. Let H be a family of objects, and let P be a predicate on subsets of H. A Helly-type theorem for H is a result of the form: There is a constant k such that for every finite set G, G H, P(G), if and only if, for every F G with F k, P(F). The least such constant k is called the Helly number of H with respect to the predicate P. If no such constant exists we say that the Helly number of H with respect to P is unbounded or infinite ( ). In Helly s Theorem, the Helly number is d + 1 and P is the predicate of having a non-empty intersection. Over the years, a vast body of applications analogues and far reaching generalizations of Helly s Theorem has been assembled in the literature (see for instance the excellent surveys of [DGK63], [Ec93] and [GPW93]). In Chapter 2 we present three ways to generalize Helly-types theorems. For instance, suppose H is a family of objects in IR d, which has a finite Helly number with respect to the intersection predicate. We ask whether there exists for it a lexicographic 1 version, that is 1 For any x = (x 1,...,x d ), y = (y 1,...,y d ) IR d we say that x < L y (x is lexicographically smaller than y) if either x 1 < y 1, or there exists d k > 1 such that x i = y i for i = 1, 2,...,k 1, and x k < y k. 1

12 Existence of lexicographic Helly-type theorem: Does there exist a number k(d) such that for any vector x IR d, and for any (sufficiently large) family G H, if every k(d) members of G have a common point which is lexicographically at most x, then all the members of G do? We call such k(d) the lexicographic Helly number (lex Helly number, in short) of H. We also ask about a discrete version Existence of discrete Helly-type theorem: Does there exist a number k(d) such that for any set S of points in IR d, and for any (sufficiently large) family D H, if every k(d) members of D have a common point in S, then all the members of D do? We call such k(d) the discrete Helly number of H. We ask also about the existence of a combined lexicographicdiscrete version, that is, of a finite lexicographic-discrete Helly number (lex-discrete Helly number, in short) 2. We obtain more than a dozen new such discrete, lexicographic and lex-discrete Helly numbers. 1.2 Optimization problems A fundamental optimization problem is Problem: Linear Programming (LP) Input: A finite family of closed linear halfspaces in d-dimensional real linear space IR d, called the constraints, and a linear objective function on IR d. Output: The minimum of the objective function over the intersection of the constraints. The earliest algorithm for this problem was the simplex algorithm, introduced by Dantzig in 1951 ([Dant51]). The simplex algorithm is essentially combinatorial, in that it searches the d element subfamilies of constraints for one which determines the minimum. Although it is arguably still the most efficient algorithm in practice, most of its variants require exponential time in the worst case ([KK87]). There is a well developed theory of linear programming, but problems in which the constraints or objective function are non-linear are less well understood. An extension of linear programming is Problem: Convex Programming Input: A finite family of closed convex sets in IR d and a convex objective function on IR d. Output: The minimum of the objective function over the intersection of the constraints. (An objective function is convex when f(λa + (1 λ)b) λf(a) + (1 λ)f(b), a, b IR d ; 0 λ 1). More general classes of problems are called mathematical programming or nonlinear programming. Combinatorial approaches analogous to the simplex method are often applied to convex and other nonlinear problems ([Fl87]). 2 The lex Helly number of Helly s theorem is k(d) = d + 1 ([Ma02]), its discrete Helly number is infinite (Theorem 2.2.3), and consequently its lex-discrete Helly number is infinite. By limiting the sets H and S we can get finite Helly numbers for the discrete and lex-discrete versions of Helly s Theorem (Theorem 2.2.1). 2

13 The problems mentioned so far belong to the field of continuous optimization. In continuous optimization models related to LP the feasible set is defined by a finite set of constraints. In the discrete versions, in addition to the above, there is also a pre-specified set of relaxations (we will explain this term shortly). A feasible solution is restricted to be in the set of relaxations as well as to satisfy the constraints. An example of a discretization of linear programming, where the set of relaxations is the integer lattice, is Problem: Integer Programming (IP) Input: A finite family of closed linear halfspaces in IR d, called the constraints, and a linear objective function on IR d. Output: The minimum of the objective function over the intersection of the constraints and the integer lattice ZZ d. More generally, we can define a set S of points in IR d, and ask that the solution belongs to this set Problem: Discrete Linear Programming (DLP) Input: A finite family D of closed linear halfspaces in IR d, a finite family S of k-flats in IR d (0 k < d) and a linear objective function ω on IR d. Output: The minimum of the objective function over the intersection of the constraints in D and the union of the relaxations in S. (A k-flat is a k-dimensional affine space, e.g., a 0-flat is a point and a 1-flat is a line). Integer programming can be formulated as a special case of DLP where S = ZZ d. There is one problem with this formulation: the set S is not finite. We can overcome this by noting that when given an instance of an IP problem, it is always possible to bound the integer lattice by a big box such that the solution of the IP problem, if exists, is found inside the bounding box (see for example Theorem 17.2 in [Sc86]). We consider again for a moment the terms constraints and relaxations. We call the elements in D constraints since adding to D a constraint can only increase the value of the objective function. We call the elements in S relaxations since adding to S a relaxation can only decrease the value of the objective function. We say that a DLP (D 1, S 1, ω 1 ) is a relaxation of a DLP (D 2, S 2, ω 2 ) if D 1 = D 2, ω 1 = ω 2 and S 1 S 2. d-dimensional LP can be viewed as DLP with the set of relaxations be S = IR d (at least if we permit S to be an infinite set). In this way, one can view LP as a relaxation of IP. Continuous optimization vs. discrete optimization: The field of discrete optimization is sometimes considered more difficult than continuous optimization because many of the methods applied to solve continuous optimization problems are not valid for discrete optimization problems. In fact, many discrete optimization problems are proved to be more time consuming to solve than their corresponding continuous versions. To illustrate this we give two examples. The planar Euclidean 1-center problem, where one needs to find a center which minimizes the maximal distance to n given points, can be solved in linear time ([Me83a]). The discrete version where the center needs to be one of the given n points is known to have a lower bound of Ω(n log n) ([LW86]). As a second example, let us consider LP vs. IP. LP as a function of, n, the number of constraints, d, the number of variables (the dimension) and l, the binary coding size of the input numbers, can be solved in 3

14 polynomial time ([Kh79]). There is no known algorithm for LP with polynomial dependence only on n and d. A polynomial time algorithm (in n, d and l) for IP is not likely to exist, since the problem in question is NP-complete ([GJ79]). Let us now concentrate in the fixed-dimensional case of LP, IP and DLP, in which the number of variables, d, is assumed to be constant, and the goal is to optimize the running time of a solution algorithm with respect to the number of constraints, n (and in DLP - also with respect to the size of the set S, m). We prove in this dissertation that DLP has a lower bound of Ω(min{n, m} logmin{n, m}), for every d > 1. Fixed-dimensional IP is polynomially solvable ([Le83]). The corresponding continuous problem, i.e., fixed-dimensional LP, is solvable in linear time. The first linear time algorithms for fixed-dimensional LP were invented in the mid 80 by Megiddo [Me84], Dyer [Dy86] and Clarkson [Cl86]. Their running time is O(2 2d n), O(3 d2 n) and O(3 d2 n), respectively, that is exponentially in d. The currently best known deterministic algorithm is of Chazelle and Matoušek [CM96], it runs in d O(d) n time. The algorithms of [Cl95], [Sei91] and [SW92] solve fixed-dimensional LP in randomized linear time with a better dependence on d. These algorithms are combinatorial in the sense given below, and are closely related to the simplex algorithm. The simplex algorithm in [Kal92] and the re-analysis of [SW92] in [MSW96], give rise to linear time algorithms with subexponential dependence on d. We will survey these algorithms in detail in Chapter 6. The LP-type and discrete LP-type models: An important feature of the randomized algorithms in [Cl95] and [SW92] is that they can be applied to certain nonlinear problems as well, as all the authors observed. In [SW92], Sharir and Welzl formalize this idea by giving an abstract framework, that is, a list of combinatorial conditions on the family of constraints and the objective function, under which these algorithms can be applied. This framework defines a class of problems, which they call LP-type and which Amenta [A94] calls Generalized Linear Programming (GLP). In [SW92], Sharir and Welzl formulate many problems as LPtype problems, almost all of which can also be formulated as special cases of convex programming. In [A94], Amenta formulates more problems, some of which are non-convex, as LP-type problems. The LP-type algorithms handle the combinatorial aspects of these problems. Their running times are measured by the number of calls to primitive operations, which are required to solve subproblems of some fixed size d. When the time required for a primitive operation is independent on n, the number of constraints, the LP-type algorithms run in expected time linear in n. This applies, for instance, to the problem of placing recovery points listed below. When we can show that the primitive operations require at most subexponential time in d, the algorithm of [MSW96] is subexponential in the general case as well. The Simple Stochastic Game decision problem defined below is an application of this kind. In Chapter 3 we define a special class of discrete optimization problems which we call discrete LP-type problems. We develop in Chapter 6 randomized algorithms to solve discrete LP-type problems satisfying a condition we call the Violation Condition (VC). These algorithms are based upon LP-type algorithms. Similarly to LP-type algorithms, the running time of our algorithms are measured by the number of calls to primitive operations, which are required to solve subproblems of some fixed size. When the dimension of these discrete LP-type problems is fixed, our algorithms run in linear time. This applies, for instance, to the Discrete p-center problem on the real line, as well as to the Line Transversal of Axis-parallel Rectangles problem, listed below. When we can show that the primitive operations require at most subexponential time 4

15 in d, our algorithms run in subexponential time. 1.3 The relations between Helly-type theorems and optimization problems Amenta ([A94]) observes that there is a Helly theorem about the constraint set of every LP-type problem. For example, Helly s theorem is about the constraint family of convex programming. One consequence of this observation is that we can prove a Helly theorem by showing that the set family in question is the constraint family of an LP-type problem. A natural question is, can we go in the other direction? Given a family of constraints about which there is a Helly theorem, is it always possible to construct an objective function which gives an LP-type problem? Amenta ([A94]) shows that the answer is no by giving an example for which there is no such function (Theorem 5.1.5). But she also gives a paradigm which does yield an appropriate objective function in many interesting and important cases, where a condition she calls the Unique Minimum Condition (UMC) is satisfied. Applying this paradigm to the collections of Helly theorems, [A94] gives new algorithms for a variety of geometric optimization problems. In other cases, the UMC is not satisfied, and in order to satisfy it, it is common to assume general position on the input and even change and extend the objective function itself. We suggest a different approach. We consider a lexicographic version of the original LP-type problem, which by its definition always satisfies the UMC. We note that the solution of the lexicographic problem is a solution for the original problem. We solve the lexicographic problem as follows. We show that given a family of constraints about which there exists a lexicographic Helly theorem, it is always possible to construct an objective function which gives a fixed-dimensional LP-type problem (unlike in [A94], no additional conditions are needed). In this we solve (at least partially) the main open problem raised by Amenta (Section 10 in [A94]). Similarly to [A94], this provides a framework for obtaining linear time algorithms (i.e., the LP-type algorithms mentioned above), for the optimization problems related to these lexicographic Helly numbers. In this way the existence of finite lexicographic Helly numbers implies the solvability of their corresponding optimization problems by the linear time LP-type algorithms. To conclude, given a continuous optimization problem, we show that if one can find a finite lex-helly number related to that problem, then the problem can be solved in linear time by LP-type algorithms (such as [Cl95] and [MSW96]). For the discrete case, we define a new discrete optimization model (discrete LP-type). We present linear time algorithms for fixed-dimensional discrete LP-type problems in many cases (i.e., when the Violation Condition applies). Given a discrete optimization problem, we show that if one can find a finite lex-discrete Helly number related to that problem, then this problem can be formulated as a fixed-dimensional discrete LP-type problem. 1.4 Applications We solve the following problems by formulating them as LP-type problems 5

16 Simple Stochastic Game (SSG) [Chapter 11] A Simple Stochastic Game (SSG) is defined on a directed graph with three types of vertices, min, max, and average, along with two sink vertices, the 0-sink and the 1-sink. The sink vertices have no outgoing edges. For every average vertex a k, the outgoing edges from a k have positive rational weights such that the sum of their weights is 1. The outgoing edges from the min and max vertices are unweighed. One of the vertices is a start vertex. The game is a contest between two players, 0 and 1. It is played in the following way. Begin by placing a token on the start vertex. When the token is on a min vertex y j, player 0 moves it along one of the outgoing edges of y j. When the token is on a max vertex x i, player 1 moves it along one of the outgoing edges of x i. When the token is on an average vertex a k, the edge along which the token is moved is determined randomly, in proportion to the weights of the edges outgoing from a k. The game ends when one of the sink vertices is reached. The goal of player 1 is to reach the 1-sink. The goal of player 0 is to avoid the 1-sink. Before the game begins, player 1 chooses an outgoing edge from each max vertex. These edges will define a strategy for player 1. During the game, whenever the token is on a max vertex, player 1 will move the token along the edge that is included in his strategy. Similarly, a choice of an outgoing edge for each min vertex is a strategy for player 0. Given an SSG, we would like to determine whether player 1 wins with probability greater than 1 2 when both players use their best strategies. A binary SSG is a special case of a SSG, where the outgoing degrees of the min and max edges are bounded by 2, and where all average vertices have exactly two outgoing edges of weight 1 2 each. Ludwig [Lu95] gave the first subexponential algorithm for binary SSG by developing an ad-hoc algorithm. We give the first strongly subexponential (in the number of vertices, edges and the coding size of the input numbers) solution for (general) SSG by formulating this problem as a variable dimensional basis-regular LP-type problem. Placing Recovery Points on Trees and on the Real Line [Chapter 8] Given a directed tree (possibly a path) with n nodes and an integer p < n, choose a subset H of p nodes, such that the maximal directed distance from any node in the tree to its closest neighbor in H is minimal. This problem is solvable in linear time by the fairly involved technique of Frederickson [Fr91]. Assuming p is fixed, we solve this problem in linear time using LP-type algorithms. Our solution applies in the same running time for the case where we are given a (not necessarily sorted) sequence of points on the real line. We are unaware for any other linear time solution for the real line case. We solve the following problems in randomized linear time, by formulating them as fixed-dimensional discrete LP-type problems. (To the best of our knowledge no other linear time algorithm exists for these problems). Discrete p-center on the Real Line [Section 9.1] Given a finite set D of real numbers (points) and a finite set S of real numbers (center locations), find a subset C S of p points (centers), which minimizes the optimal radius r p (D, S) = min C S s.t. C p max h D dist(h, C). (For every finite set of real numbers C, and real h, dist(h, C) = min c C h c ). 6

17 Assuming the order of the points on the line is given, the discrete p-center problem on the real line is solvable in linear time by the fairly involved technique of Frederickson [Fr91]. General Discrete Weighted 1-Center in IR d with l Norm [Subsection 9.3.3] Given are sets D = {d 1,..., d n } and S = {s 1,..., s m } of points in IR d, and a set W = {w 1,..., w n } of weights in IR +. Find a point s S (center) which minimizes the optimal radius r(d, S) = min s S max i w i s d i It is a folklore that the latter problem restricted to the case S = D is solvable in O(n log n) time. Line Transversal of Axis-parallel Rectangles [Section 10.2] Given is a finite set D of axis-parallel rectangles in the plane, and a set S of line directions. Find a line transversal for D with direction in S, i.e., a line with direction in S, which meets all the rectangles in D. By proving lexicographic Helly theorems and using the fact that every lexicographic Helly theorem yields a fixed-dimensional LP-type problem (Theorem 5.3.7) we solve the following problems in randomized linear time. (To the best of our knowledge no other linear time algorithm exists for these problems). Planar Lexicographic Rectilinear p-center, p = 1, 2, 3 [Subsection 9.3.2] Given a finite set H of axis-parallel rectangles, find the lexicographically smallest vector (λ 1, x 1, y 1, x 2, y 2,..., x p, y p ) IR + IR 2p such that for every scaled rectangle λ 1 h, h H, there exists i such that λ 1 h contains (i.e., is pierced by) point (x i, y i ). For p > 3 [SW96] showed a lower bound of Ω(n log n). [SW96] and [Hof99] solve the corresponding non-lexicographic problem in linear time. Line Transversal of a Totally Separable Set of Convex Objects [Section 10.1] Given is a set H of convex simple objects in the plane, for which there exists a direction such that each line in this direction intersects at most one convex object from H. Find a line transversal for H, which meets all the objects in H. We also solve the following location problem in linear time by using a different technique Weighted Lexicographic Rectilinear 1-Center in the Plane [Chapter 12] Given is a set D of n points (customers) in the plane. In the (standard) rectilinear 1-center problem we need to locate a service center c in the plane such that the maximal distance (in l 1 norm) between any point in D and c is minimal. We call the monotone non-decreasing vector of the n distances (in l 1 norm) between c and the customers the distances vector. In the lexicographic rectilinear 1-center problem we need to locate a service center c in the plane such that the resulted distances vector is lexicographically minimal. In this way not only the maximal distance between c and the customers is of our concern, but also all of the remaining n 1 distances. Organization of the thesis: In Chapter 2 we review Helly theorems, introduce new types of Helly theorems, and show the connections between these types. In Chapter 3 we review the LP-type model and introduce 7

18 the discrete LP-type model. We show in Chapter 4 that the constraint set of every discrete LP-type problem results a discrete Helly theorem. In Chapter 5 we also give a paradigm which shows that in many interesting and important cases, discrete Helly theorems yield discrete LP-type problems. Using our discrete LP-type algorithms, which we describe in Chapter 6, we give in the following chapters linear time solutions for several discrete optimization problems. Remark: Parts of this thesis were published in [Hal03] and [Hal04]. 8

19 Part I Theory 9

20 Chapter 2 Helly theorems In this chapter we introduce 3 new types of Helly theorems: discrete Helly theorems, lexicographic Helly theorems, and lexicographic-discrete Helly theorems. We review Helly theorems, give examples and introduce new notation. 2.1 Continuous Helly theorems In this section we review the definitions related to Helly theorems from [A94] Definitions Let H be a family of objects, and P a predicate on subsets of H. A Helly theorem for H is a result of the form There is a constant k IN such that for all G H, P(G), if and only if, for every F G with F k, P(F). The least such constant k is called the Helly number of H with respect to the predicate P. We say that the Helly number of H with respect to P is unbounded ( ) when For every constant k there is G k H, such that for every proper subset F G k with F k, P(F) but G k does not satisfy P. (Throughout this dissertation we use the symbol to denote a proper subset.) We sometimes call this type of theorems standard or continuous Helly theorems Examples We give here several examples of well known Helly theorems which will serve us later. All the Helly theorems in this subsection are about (not necessarily finite) families of compact sets, unless otherwise indicated. Theorem (Radius Theorem) A family H of points in the Euclidean d-dimensional space E d, is contained in a unit ball if and only if every d + 1 or fewer points from H are contained in a unit ball. 10

21 Here the family of objects is the set of points in E d, the predicate is that a subfamily is contained in a (closed) unit ball, and k = d + 1. In fact, the Radius Theorem is a corollary of Helly s theorem proper. To see that we apply the following duality transformation. We transform every point h H into the set D(h) of centers of unit balls containing h. In this way D(h) is a unit ball centered at h. Let D(H) = {D(h) h H}. From the definition of this duality transformation we get that the points in H are contained in a unit ball if and only if the unit balls in D(H) have a non empty intersection (see Figure 2.1). Since the objects in D(H) are convex sets, and by using Helly s Theorem, the Radius Theorem follows. We give a formal description of this transformation in the next subsection. Figure 2.1: The 3 points on the left side are contained in a unit ball if and only if the 3 unit balls on the right side intersect. In the rectilinear p-piercing problem (p IN) we are given a finite set B of compact boxes in IR d with edges parallel to the coordinate axes and decide whether there exists an ordered p-tuple of points A = (a 1, a 2,..., a p ) (a 1, a 2,..., a p are points in IR d ), such that for every box b B there exists i such that box b is pierced by (i.e., contains) point a i. If such a p-tuple A exists we say that B is p-pierceable and that A p-pierces B. We note that B is 1-pierceable if and only if all the boxes in B intersect. The related Helly problem is to find the least h = h(d, p) such that B is p-pierceable if each B B with B h is p-pierceable. Danzer and Grünbaum proved the following theorem for a (not necessarily finite) family of compact axis-parallel boxes Theorem ([DG82]) (i) h(d, 1) = 2 for all d IN; (ii) h(1, p) = p { + 1 for all p IN; (iii) h(d, 2) = 3d for odd d; 3d 1 for even d; (iv) h(2, 3) = 16; (v) h(d, p) = for d 2, p 3, and (d, p) (2, 3). It is easy to change their proof slightly such that the theorem holds also for finite families of open axis-parallel boxes. If all the boxes are convex, but some of the boxes are neither closed nor open the theorem does not hold. Consider for example case (i) for d = 2 with an instance of the following three boxes. Let b 1 be the closed rectangle with vertical length 1, horizontal length 3 and bottom-left corner at the origin, minus the interval ((1, 0); (2, 0)]. Clearly b 1 is convex. Let b 2 be a closed rectangle with the same lengths whose upper-left corner is the origin, minus the same interval (so b 1 and b 2 are both convex and intersect in the 11

22 b 1 b 3 b 2 Figure 2.2: Every 2 rectangles intersect but all 3 don t. closed interval [(0, 0); (1, 0)]. Let b 3 be a closed square as drawn in Figure 2.2. Every two rectangles intersect but all three don t. We note that case (i) in the theorem above is a special case of Helly s theorem where the Helly number is reduced from d + 1 to 2 since the convex sets are restricted to be axis-parallel boxes. We conclude this subsection by giving examples of important known Helly theorems in the field of geometric transversal theory (all these examples are exposed in the excellent surveys of [DGK63], [GPW93] and [Ec93]). In 1935 Vincensini [V35] observed that a point lying in the intersection of a family of sets is the special case k = 0 of a k transversal for the family, i.e., a k-flat which meets every member of the family. Hence Helly s theorem deals with 0-transversals. Vincensini wondered about k-transversals for positive k s Vincensini s Problem [V35]: For 0 < k < d, can we find a number r(d, k) such that for any (sufficiently large) family A of convex sets in IR d, if every r members have a common k-transversal, then all the members of A do? In the same paper he provided a proof that the answer is positive for d = 2 and k = 1, with r = 6, i.e., if any six (or fewer) members of a planar family of convex sets have a common line transversal, then all the sets do. Santaló was able to show 5 years later [Sa40] that r(2, 1) does not exist in general, i.e., that there is no Helly-type theorem for transversals of positive dimension. In this way he showed that the proof of [V35] was in error. Hadwiger, Debrunner and Klee [HDK64] showed subsequently that r(2, 1) does not exist even if the convex sets are pairwise disjoint. It soon became clear that Helly-type theorems with transversals cannot be expected unless rather severe restrictions are placed on the shapes and the mutual position of the sets considered. A typical condition, for instance, is that the sets be pairwise disjoint and translates of one another. When dealing with transversals of positive dimension, we will concentrate here solely on line transversal problems in the plane. Definition Let H be a family of sets in the plane. We say that H is totally separable if there exists a direction such that each line in this direction intersects at most one set from H. We note that if H is totally separable then pairwise disjoint parallel strips can be formed in the plane in such a way that each strip contains exactly one set from H, so the sets in H are pairwise disjoint. 12

23 Vincensini [V53] showed that if each 4 members of a totally separable set H of convex sets admit a line transversal, then also does H. This number was reduced by Klee in the following Theorem ([Kl54]) Let H be a totally separable family of convex sets. If every subset of at most 3 sets admits a line transversal then H does as well. It is easy to modify his proof slightly such that the theorem holds also for finite families of open convex sets. Another way of obtaining Helly numbers for line transversals is to restrict the shape of the convex sets rather than to restrict their relative position. A set H is said to be of parallel rectangles, if the edges of the rectangles in H are of only two different directions. Theorem ([Sa40]) Let H be a family of parallel rectangles in the plane. If every subset of at most 6 rectangles admits a line transversal then H does as well. Hadwiger and Debrunner deduce the above theorem by proving Theorem ([HD55]) Let H be a family of rectangles in the plane with edges parallel to the axes. If every subset of at most 3 rectangles admits a line transversal with a non-negative slope then H does as well. (A translation of their proof to English is given in [HDK64]). Their proof relies on a transformation to Helly s Theorem, and in fact is valid also when H is a finite family of open rectangles. When H is a family of pairwise disjoint translates of a disc, Danzer [Danz57] proved that the Helly number is 5. Grünbaum proved that if in addition to the conditions stated in Theorem 2.1.5, the rectangles in H are pairwise disjoint and translates of one another then the Helly number 6 can be replaced by 5. Theorem ([G58]) Let H be a family of pairwise disjoint translates of a rectangle in the plane. If every subset of at most 5 rectangles admit a line transversal then H does as well. Grünbaum pointed out, in addition, that the disjointness cannot be dropped from the theorem hypothesis, and that translates could not be weakened to homothets (i.e., scaled translates). He also strengthened Danzer s theorem on mutually disjoint unit disks by showing that the Helly number for families H of at least 6 discs, is only 4. More significantly, it was in this paper that Grünbaum conjectured that the Helly number 5 holds as well for a family of disjoint translates of any convex set, a conjecture which would take more than 30 years to prove Grünbaum s Conjecture [G58]: For a family H of pairwise disjoint translates of a compact convex set in IR 2, if every subset of at most five members of H has a line transversal then H has a line transversal. Katchalski [Kat86] was the first to show that a finite Helly number exists by obtaining a crude upper bound of 128. Finally, in 1989 Tverberg [Tv89] proved Grünbaum s conjecture (Theorem 2.1.8(v)). A brief outline of Tverberg s proof is given in [GPW93]. We summarize in 13

24 Theorem Let H be a set of at least k pairwise disjoint translates of a compact set O in the plane. The minimal k IN such that if any k translates from H are met by a common line, then some line meets all sets of H is: (i) ([Sa40]) 3 when O is a segment; (ii) ([Danz57]) 5 when O is a disc; (iii) ([G58]) 5 when O is a square; (iv) ([G58]) 5 when O is a rectangle; (v) ([Tv89]) 5 when O is convex. We will show in the following sections that many of these theorems can be restated such that the family of objects H, is a family of sets of points in IR d, for some integer d, and that the predicate P, is the property of having non-empty intersection, as was shown for the Radius Theorem Helly systems The term Helly-type theorem is often used to describe a larger class of theorems, including ones in which the fact that every subfamily has some property P implies that the whole family has some other property Q. We will not be concerned with this larger class. We are interested in a particular subclass of Helly-type theorems, in which the objects in H are subsets of a set X which we call a ground set, and P(G) is the intersection predicate, i.e., G H, P(G) is that the sets in G intersect in a common point. We call the subsets of H constraints. We write G for {x X x h, h G}, and we say that the family of sets intersects when G, that is, when P(G) is true. We need some more notation. A set system is a pair (X, H), where X is a set and H is a family of subsets of X. We say (X, H) is a Helly system if there exists a finite integer k such that H has Helly number k with respect to the intersection predicate P. The natural computational problem associated with a Helly system is, given a subset G H, return a point x G, or show that G =. Most Helly theorems can be restated in terms of the intersection predicate. For instance, let us consider a known restatement of Radius Theorem (over any set D of points in E d ) as a Helly system (see for example, [A93]). Instead of a set of points, we can state it as a theorem about the intersection of sets of unit balls, where each set consists of all the unit balls containing a particular point. A formal description is as follows. Let X be the set of all points in E d, and a let Y be the set of all possible positions for the center of a unit ball in E d (which of course is another copy of E d ). Let the predicate Q(x, y) mean that the point x X is contained in the unit ball with center y Y, let h x Y be the set of centers of unit balls containing x, and let H X = {h x x X}. By this construction we get that D and H D are equivalent in the sense that H D intersects if and only if the points in D are contained in a unit ball. This is true since any family D D of points corresponds to a family H D H D, such that there is a unit ball containing D if and only if H D intersects. Let us consider the ordered pair (Y, H D ). Clearly it is a set system. Since H D is a family of unit balls, we can apply Helly s theorem, and get that (Y, H D ) is a Helly system with Helly number d + 1. We have just showed that the Radius Theorem is a corollary of Helly s theorem. Our representation of the Radius Theorem as a Helly system is a sort of dual transformation. We can apply it to theorems of the form X has Helly number k with respect to P when X and P have the following 14

25 special form. There has to be a set Y, and a predicate Q on pairs in X Y, such that P is defined in terms of Q as follows. For A X, P(A) if and only if y Y such that x A Q(x, y). Then the theorem X has Helly number k with respect to P corresponds to the Helly system (Y, H), where h x = {y Y Q(x, y)}, for x X, and H = {h x x X}. 2.2 Discrete Helly theorems In this section we define discrete Helly theorems and systems and give examples Definitions Let D be a set of elements (d-elements) and S be a set of elements (s-elements). Let G and L be predicates on elements of 2 D 2 S and let k D, k S IN be finite constants. A discrete Helly theorem is a result of the form There is a constant k D such that for all finite S S, D D : L(D, S ) if and only if L(D, S ) holds for every D D with D k D, and There is a constant k S such that for all finite D D, S S : G(D, S ) if and only if G(D, S ) holds for every S S with S k S. The least such constant k D (k S ) is called the d-helly number (s-helly number) of (D, S) with respect to the predicate L (G), respectively. We say that (D, S) has Helly number (k D, k S ) with respect to L and G. We say that the d-helly number of (D, S) with respect to L is unbounded ( ) when For every S S and for every constant k there is a finite D k D, such that for every F D k with F k, L(F, S ) is true and L(G k, S ) is false. The definition of an unbounded s-helly number is analogous Examples As far as we know, the theorem of Doignon [Do73], which we state below, is the only example in the literature for a discrete Helly theorem. We present in this subsection several new discrete Helly theorems. We believe that many more such theorems exist. For the sake of presentation we give the full proofs of some of these theorems in the Appendix, and not in the main text. We first show that our (somewhat strange) definition of discrete Helly theorems is a natural generalization of continuous Helly theorems. By choosing G to be the trivial predicate which always returns true and L to satisfy D D S, S S L(D, S ) = L(D, S ) we get that the class of discrete Helly theorems contains the class of (standard) Helly theorems. The following theorem (which we prove in Subsection 9.3.3) is a discrete version of Theorem 2.1.2(i) 15