Transversals of d-Intervals

Abstract

Abstract. We present a method which reduces a family of problems in combinatorial geometry (concerning multiple intervals) to purely combinatorial questions about hypergraphs. The main tool is the Borsuk-Ulam theorem together with one of its extensions.For a positive integer d, a homogeneous d-interval is a union of at most d closed intervals on a fixed line . Let H be a system of homogeneous d-intervals such that no k + 1 of its members are pairwise disjoint. It has been known that its transversal number τ (H) can then be bounded in terms of k and d. Tardos [9] proved that for d = 2, one has τ (H) ≤ 8k. In particular, the bound is linear in k. We show that the latter holds for any d, and prove the tight bound τ (H) ≤ 3k for d = 2.We obtain similar results in the case of nonhomogeneous d-intervals whose definition appears below.

A Helly Type Conjecture

“…Tardos's theorem improves earlier results of Gyárfás and Lehel [6], and has been generalized to m-intervals, m ≥ 3, by Kaiser [7]. (However, Kaiser's result is not best posible whereas Tardos's result is.)…”

The Maximum Clique Problem in Multiple Interval Graphs (Extended Abstract)

“…a set of points touching at least one interval of each vertex) of size τ = t 2 − t + 1 (resp. τ = t 2 −t) [17]. Scanning the representation of a graph G from left to right (in time O(tn)) one passes through the points of the transversal of a maximum clique K of G. At some of those points there are at least |K|/τ intervals forming a subclique of K. Thus, this gives an O(tn)-time τ -approximation.…”

Lower Bounds on the Transversal Numbers of d-Intervals

“…for a fixed d. Károlyi and Tardos [6] obtained a bound of O(k d ) for every fixed d. A major breakthrough was achieved by Tardos [9], who proved τ (F) ≤ 2ν(F) for any family of 2-intervals (which is tight in the worst case) by an ingenious topological method. By a somewhat different topological approach, Kaiser [4] proved that for any system of d intervals with ν(F) ≤ k, we have τ (F) ≤ (d 2 − d)k. Later Alon [1] found a short nontopological proof of the slightly weaker bound τ (F) ≤ 2d 2 k.…”

“…Homogeneous d-intervals are strictly more general than d-intervals, and the upper bounds obtained for them are slightly worse: Kaiser [4] showed τ (F) ≤ 3k for any family of homogeneous 2-intervals with ν(F) ≤ k, which is the best possible, and τ (F) ≤ (d 2 −d +1)k for an arbitrary family of homogeneous d-intervals with ν(F) ≤ k, for any d ≥ 3. Alon's upper bound applies for homogeneous d-intervals as well.…”