Non-uniform Geometric Set Cover and Scheduling on Multiple Machines

We consider the following general scheduling problem studied recently by Moseley. There are $n$ jobs, all released at time $0$, where job $j$ has size $p_j$ and an associated arbitrary non-decreasing cost function $f_j$ of its completion time. The goal is to find a schedule on $m$ machines with minimum total cost. We give an $O(1)$ approximation for the problem, improving upon the previous $O(\log \log nP)$ bound ($P$ is the maximum to minimum size ratio), and resolving the open question of Moseley. We first note that the scheduling problem can be reduced to a clean geometric set cover problem where points on a line with arbitrary demands, must be covered by a minimum cost collection of given intervals with non-uniform capacity profiles. Unfortunately, current techniques for such problems based on knapsack cover inequalities and low union complexity, completely lose the geometric structure in the non-uniform capacity profiles and incur at least an $\Omega(\log\log P)$ loss. To this end, we consider general covering problems with non-uniform capacities, and give a new method to handle capacities in a way that completely preserves their geometric structure. This allows us to use sophisticated geometric ideas in a black-box way to avoid the $\Omega(\log \log P)$ loss in previous approaches. In addition to the scheduling problem above, we use this approach to obtain $O(1)$ or inverse Ackermann type bounds for several basic capacitated covering problems

Subsampling in Smoothed Range Spaces

We consider smoothed versions of geometric range spaces, so an element of the ground set (e.g. a point) can be contained in a range with a non-binary value in $[0,1]$. Similar notions have been considered for kernels; we extend them to more general types of ranges. We then consider approximations of these range spaces through $\varepsilon$-nets and $\varepsilon$-samples (aka $\varepsilon$-approximations). We characterize when size bounds for $\varepsilon$-samples on kernels can be extended to these more general smoothed range spaces. We also describe new generalizations for $\varepsilon$-nets to these range spaces and show when results from binary range spaces can carry over to these smoothed ones.Comment: This is the full version of the paper which appeared in ALT 2015. 16 pages, 3 figures. In Algorithmic Learning Theory, pp. 224-238. Springer International Publishing, 201

Small Candidate Set for Translational Pattern Search

In this paper, we study the following pattern search problem: Given a pair of point sets A and B in fixed dimensional space R^d, with |B| = n, |A| = m and n >= m, the pattern search problem is to find the translations T\u27s of A such that each of the identified translations induces a matching between T(A) and a subset B\u27 of B with cost no more than some given threshold, where the cost is defined as the minimum bipartite matching cost of T(A) and B\u27. We present a novel algorithm to produce a small set of candidate translations for the pattern search problem. For any B\u27 subseteq B with |B\u27| = |A|, there exists at least one translation T in the candidate set such that the minimum bipartite matching cost between T(A) and B\u27 is no larger than (1+epsilon) times the minimum bipartite matching cost between A and B\u27 under any translation (i.e., the optimal translational matching cost). We also show that there exists an alternative solution to this problem, which constructs a candidate set of size O(n log^2 n) in O(n log^2 n) time with high probability of success. As a by-product of our construction, we obtain a weak epsilon-net for hypercube ranges, which significantly improves the construction time and the size of the candidate set. Our technique can be applied to a number of applications, including the translational pattern matching problem

Non-uniform geometric set cover and scheduling on multiple machines

We consider the following general scheduling problem studied recently by Moseley [27]. There are n jobs, all released at time 0, where job j has size pj and an associated arbitrary non-decreasing cost function fj of its completion time.

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum