EuroCG 2011, Morschach, Switzerland, March 28{30, 2011 The Dawn of an Algebraic Era in Discrete Geometry? Jiˇr´ıMatouˇsek∗ Section One, Where the Author Is Browsing the that for systems such as halfspaces, balls, sim- d 1 1 2010 ArXiv plices in R , for d fixed, "-nets of size O( " log " ) exist, for every finite set X and every ". The To me, 2010 looks as annus mirabilis, a miracu- proof was based on a very general combinato- lous year, in several areas of my mathematical inter- rial property of the considered set system, called ests. Below I list seven highlights and breakthroughs, bounded VC-dimension, and there was hope that mostly in discrete geometry, hoping to share some of simple geometrically defined systems might ad- my wonder and pleasure with the readers. 1 mit even smaller "-nets, perhaps of size O( " ). In- Of course, hardly any of these great results have deed, there were some reasons for optimism, since come out of the blue: usually the paper I refer to 1 3 O( " ) was known for halfspaces in R , and more adds the last step to earlier ideas. Since this is an recently, Aronov, Ezra, and Sharir [2] proved extended abstract (of a nonexistent paper), I will be 1 1 an O( " log log " ) upper bound for axis-parallel rather brief, or sometimes completely silent, about the rectangles in R2. However, Alon [1] established history, with apologies to the unmentioned giants on the first superlinear lower bound in a geomet- whose shoulders the authors I do mention have been ric setting (for lines in the plane), and Pach and standing.1 A careful reader may notice that together 1 1 Tardos got the tight lower bound of Ω( " log " ) with these great results, I will also advertise some 4 1 1 for halfspaces in R , as well as Ω( " log log " ) for smaller results of mine. axis-parallel rectangles in R2. These results may perhaps not look as significant to others, but • Larry Guth and Nets Hawk Katz [16] completed for me, they close a long open and tantalizing a bold project of Gy¨orgyElekes (whose previous problem, which for myself I considered almost stage is reported in [10]) and obtained a near- hopeless. Moreover, I think that the proofs con- tight bound for the Erd}osdistinct distances tain (reinforce?) a general lesson: in order to problem: they proved that every n points in prove an \irregularity" result, in the sense that the plane determine at least Ω(n= log n) distinct something cannot be very uniform, it may of- distances. This almostp matches the best known ten be good to strive for a Ramsey-type result, pupperp bound of O(n= log n), attained for the showing that there has to be a completely non- n × n grid. Their proof and some related uniform, \monochromatic" spot|in the case of results and methods constitute the main topic of "-nets, this way even gives a tight quantitative this note, and will be discussed later. bound! • J´anosPach and G´abor Tardos [27] found tight • Mikhail Gromov [14] invented a new topologi- lower bounds for the size of "-nets for geometric cal proof of the first selection lemma. The set systems.2 It has been known for a long time lemma states that for every n-point set P ⊂ Rd there exists a point a (not necessarily in P ) con- ∗Department of Applied Mathematics and Institute of Theo- n tained in at least cd of the d-dimensional retical Computer Science (ITI), Charles University, Malostran- d+1 sk´en´am. 25, 118 00 Praha 1, Czech Republic, and Institute simplices with vertices in P , where cd is a pos- of Theoretical Computer Science, ETH Zurich, 8092 Zurich, itive constant depending only on d. (There are Switzerland n d-simplices spanned by P in total, so a is 1 d+1 I should also add that my selection is entirely personal and in a positive fraction of them.) Gromov's proof subjective, shall not indicate or imply any ranking of the results listed, any judgment of results not mentioned, or any compar- yields significantly better value of cd than all pre- ison thereof, and shall not encompass any warranty of sound vious ones, provides a far-reaching generalization, and safe conditions of the cited papers or any liability on my and hopefully opens a way towards deeper under- side, in particular, no rights of readers or third parties to be standing of many related problems. By including indemnified in case of loss or damage directly or indirectly re- lated to the aforementioned papers. But I do not want this Gromov's paper in my 2010 list I am cheating abstract to look like a car rental contract. slightly, since a preprint was circulated one or 2 d Let F be some system of subsets of R , say all closed half- two years earlier. But a completely honest 2010 d spaces or all axis-parallel boxes, and let X be a finite set in R . item was supplied by Karasev [21], who found a A subset N ⊆ X is called an "-net for X with respect to F, where " 2 [0; 1] is a given real number, if N intersects every greatly simplified and fairly elementary version \large" set, i.e., every F 2 F for which jF \ Xj ≥ "jXj. of the argument. Readers interested in the com- 27th European Workshop on Computational Geometry, 2011 binatorial issues involved and in attempts at ex- walk. The algorithm is not an approximation al- plaining Gromov's method may also reach for a gorithm for discrepancy in general (indeed, ap- paper of Wagner and mine [25]. proximating discrepancy is pretty much hopeless [5]), but it makes several existential bounds for • Francisco Santos [28] disproved the 1957 Hirsch the discrepancy of certain set systems, such as conjecture, which states that the graph of a d- all arithmetic progressions on f1; 2; : : : ; ng, con- dimensional convex polytope with n facets has structive, which has been a major open problem diameter at most n−d. The conjecture was moti- in discrepancy theory. It also has structural con- vated by linear programming, more precisely, by sequences; one of the quick spinoffs is a near-tight the analysis of simplex-type algorithms. Santos' answer [24] to an old questions of S´osconcern- ingenious examples have diameter (1 + ")(n − d) ing the discrepancy of a union of two set sys- for a small positive ", and the fascinating ques- tems on the same ground set (besides Bansal's tion of maximum diameter of a d-polytope with algorithm, the answer also relies on a beautiful n facets has become even more interesting (and linear-algebraic lower bound for discrepancy of the subject of Gil Kalai's polymath project), the Lov´asz,Spencer, and Vesztergombi [22]). best upper bound being nO(log d) [18]. • Oliver Friedmann, Thomas Dueholm Hansen, • June Huh [17] proved log-concavity of the and Uri Zwick [13] proved very strong lower sequence of coefficients of the chromatic polynomial.4 More precisely, for the chromatic bounds for various randomized simplex al- n polynomial written as a0 +a1x+···+anx , Huh's gorithms. 2 result asserts that ai−1ai+1 ≤ ai for every i (and The simplex method from the late 1940s remains for an arbitrary graph G, of course). This im- one of the best linear programming algorithms in plies that the sequence (ja0j; ja1j;:::; janj) is uni- practice, but a construction known as the Klee{ modal, a 1968 conjecture of Read. Minty cube showed in the 1970s that some of the widely used variants are exponential in the worst The proof relies on connections of the problem case. This started a quest for a polynomial-time to singularities of local analytic functions and ul- timately to mixed multiplicities of certain ide- version, and on the optimistic side, a subexpo-p nential upper bound, of roughly exp(O( n )), als. This result does not entirely fit my list since was proved in 1992 for an algorithm known as it cannot be passed for discrete geometry even RANDOM FACET. There was hope that the with considerable indulgence, but it looks beau- analysis might be further improved, or that some tiful and it does rest on geometric ideas. I do not other randomized variant could be shown to be understand much of it, but perhaps some day I polynomial. will have enough time and energy to learn the necessary background or someone will explain it Friedmann et al. shattered these great expecta- to me|at least it does not look as intimidating tions almost completely, by proving an almost as some other papers. matching lower bound for RANDOM FACET, as well as a similar lower bound for another promising-looking algorithm known as RAN- Section Two, On Distinct Distances and Other Al- DOM EDGE (lower bounds of this kind were gebraic Magic known before, but only for the performance of The following three problems were raised by Erd}os these algorithms on certain \generalized linear [12] in 1946: programs", while the possibility of polynomial bounds for actual linear programs was still open). • Estimate the maximum possible number of inci- They use a powerful new way of constructing dences between a set P of m points and a set L “difficult” linear programs, based on randomized of n lines in the plane (where an incidence is a parity games. The potential of this approach ap- pair (p; `) with p 2 P , ` 2 L, and p 2 `). parently has not yet been exhausted. • Estimate the maximum number of unit dis- • Nikhil Bansal [3] found a polynomial-time algo- tances among n points in the plane.