New Approximability Results for the Robust k-Median Problem

We consider a robust variant of the classical $k$-median problem, introduced by Anthony et al. \cite{AnthonyGGN10}. In the \emph{Robust $k$-Median problem}, we are given an $n$-vertex metric space $(V,d)$ and $m$ client sets $\set{S_i \subseteq V}_{i=1}^m$. The objective is to open a set $F \subseteq V$ of $k$ facilities such that the worst case connection cost over all client sets is minimized; in other words, minimize $\max_{i} \sum_{v \in S_i} d(F,v)$. Anthony et al.\ showed an $O(\log m)$ approximation algorithm for any metric and APX-hardness even in the case of uniform metric. In this paper, we show that their algorithm is nearly tight by providing $\Omega(\log m/ \log \log m)$ approximation hardness, unless ${\sf NP} \subseteq \bigcap_{\delta >0} {\sf DTIME}(2^{n^{\delta}})$. This hardness result holds even for uniform and line metrics. To our knowledge, this is one of the rare cases in which a problem on a line metric is hard to approximate to within logarithmic factor. We complement the hardness result by an experimental evaluation of different heuristics that shows that very simple heuristics achieve good approximations for realistic classes of instances.Comment: 19 page

On k-Column Sparse Packing Programs

We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an (ek+o(k))-approximation algorithm for k-column sparse PIPs, improving on recent results of $k^2\cdot 2^k$ and $O(k^2)$. We also show that the integrality gap of our linear programming relaxation is at least 2k-1; it is known that k-column sparse PIPs are $\Omega(k/ \log k)$-hard to approximate. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over k-column sparse packing constraints.Comment: 19 pages, v3: additional detail

The problems of selecting problems

We face several teaching problems where a set of exercises has to be selected based on their capability to make students discover typical misconceptions or their capability to evaluate the knowledge of the students. We consider four different optimization problems, developed from two basic decision problems. The first two optimization problems consist in selecting a set of exercises reaching some required levels of coverage for each topic. In the first problem we minimize the total time required to present the selected exercises, whereas the surplus coverage of topics is maximized in the second problem. The other two optimization problems consist in composing an exam in such a way that each student misconception reduces the overall mark of the exam to some specific required extent. In particular, we consider the problem of minimizing the size of the exam fulfilling these mark reduction constraints, and the problem of minimizing the differences between the required marks losses due to each misconception and the actual ones in the composed exam. For each optimization problem, we formally identify its approximation hardness and we heuristically solve it by using a genetic algorithm. We report experimental results for a case study based on a set of real exercises of Discrete Mathematics, a Computer Science degree subject

Greedy D-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

This paper describes a simple greedy D-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most D variables of the problem. (A simple example is Vertex Cover, with D = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems

A nearly linear-time approximation scheme for the Euclidean k-median problem

This paper provides a randomized approximation scheme for the k-median problem when the input points lie in the d-dimensional Euclidean space. The worst-case running time is O(2O((log(1/∈)/ε)d-1) n log d+6n), which is nearly linear for any fixed ε and d. Moreover, our method provides the first polynomial-time approximation scheme for k-median and uncapacitated facility location instances in d-dimensional Euclidean space for any fixed d > 2. Our work extends techniques introduced originally by Arora for the Euclidean traveling salesman problem (TSP). To obtain the improvement we develop a structure theorem to describe hierarchical decomposition of solutions. The theorem is based on an adaptive decomposition scheme, which guesses at every level of the hierarchy the structure of the optimal solution and accordingly modifies the parameters of the decomposition. We believe that our methodology is of independent interest and may find applications to further geometric problems. ©2007 Society for Industrial and Applied Mathematics

Single-item lot-sizing with quantity discount and bounded inventory

In this paper, an efficient O(n2) algorithm is proposed to solve a special case of single-item lot-sizing problems (SILSP) in which both the production and holding costs are piecewise linear, there is an all-unit discount with one breakpoint for the production cost, and the inventory is bounded. The algorithm is based on a key structural property that may be of more general interest, that of a just-in-time ordering policy. Finally, we show that when the problem is extended to two items, it is NP-complete. © 2021 Elsevier B.V

Exact Algorithms for Set Multicover and Multiset Multicover Problems

Abstract. Given a universe N containing n elements and a collection of multisets or sets over N, the multiset multicover (MSMC) or the set multicover (SMC) problem is to cover all elements at least a number of times as specified in their coverage requirements with the minimum number of multisets or sets. In this paper, we give various exact algorithms for these two problems, with or without constraints on the number of times a multiset or set may be picked. First, we can exactly solve the MSMC without multiplicity constraints problem in O(((b +1)(c +1)) n) time where b and c (c ≤ b and b ≥ 2) respectively are the maximum coverage requirement and the maximum number of times that each element can appear in a multiset. To our knowledge, this is the first known exact algorithm for the MSMC without multiplicity constraints problem. Second, we can solve the SMC without multiplicity constraints problem in O((b +2) n) time. Compared with the two recent results in [Hua et al., Set Multi-covering via inclusion-exclusion, Theoretical Computer Science, 410(38-40):3882-3892 (2009)] and [Nederlof, J.: Inclusion Exclusio