Improved bounds on the Hadwiger-Debrunner numbers
نویسندگان
چکیده
We prove that any family of compact convex sets in R which satisfy the (p, q)-property (p ≥ q ≥ d+1) can be pierced with Õ((pq) q−1 q−d ) points for d ≥ 3 and O((pq) q−1 q−2 ) for d = 2. This improves (already for d = 2 and q = 3) the previously best known bound (of Õ(p 2+d)) provided in Alon and Kleitman’s celebrated proof of the Hadwiger Debrunner conjecture (for the case q = d+ 1). We also prove a (p, 2)-Theorem for families with sub-quadratic union complexity in R. Based on this, we also introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets with sub-quadratic union complexity.
منابع مشابه
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تاریخ انتشار 2015