Improvements on the k-center problem for uncertain data

In real applications, there are situations where we need to model some problems based on uncertain data. This leads us to define an uncertain model for some classical geometric optimization problems and propose algorithms to solve them. In this paper, we study the $k$-center problem, for uncertain input. In our setting, each uncertain point $P_i$ is located independently from other points in one of several possible locations $\{P_{i,1},\dots, P_{i,z_i}\}$ in a metric space with metric $d$, with specified probabilities and the goal is to compute $k$-centers $\{c_1,\dots, c_k\}$ that minimize the following expected cost $$Ecost(c_1,\dots, c_k)=\sum_{R\in \Omega} prob(R)\max_{i=1,\dots, n}\min_{j=1,\dots k} d(\hat{P}_i,c_j)$$ here $\Omega$ is the probability space of all realizations $$R=\{\hat{P}_1,\dots, \hat{P}_n\}$$ of given uncertain points and $$prob(R)=\prod_{i=1}^n prob(\hat{P}_i).$$ In restricted assigned version of this problem, an assignment $A:\{P_1,\dots, P_n\}\rightarrow \{c_1,\dots, c_k\}$ is given for any choice of centers and the goal is to minimize $$Ecost_A(c_1,\dots, c_k)=\sum_{R\in \Omega} prob(R)\max_{i=1,\dots, n} d(\hat{P}_i,A(P_i)).$$ In unrestricted version, the assignment is not specified and the goal is to compute $k$ centers $\{c_1,\dots, c_k\}$ and an assignment $A$ that minimize the above expected cost. We give several improved constant approximation factor algorithms for the assigned versions of this problem in a Euclidean space and in a general metric space. Our results significantly improve the results of \cite{guh} and generalize the results of \cite{wang} to any dimension. Our approach is to replace a certain center point for each uncertain point and study the properties of these certain points. The proposed algorithms are efficient and simple to implement

Data-Collection for the Sloan Digital Sky Survey: a Network-Flow Heuristic

The goal of the Sloan Digital Sky Survey is ``to map in detail one-quarter of the entire sky, determining the positions and absolute brightnesses of more than 100 million celestial objects''. The survey will be performed by taking ``snapshots'' through a large telescope. Each snapshot can capture up to 600 objects from a small circle of the sky. This paper describes the design and implementation of the algorithm that is being used to determine the snapshots so as to minimize their number. The problem is NP-hard in general; the algorithm described is a heuristic, based on Lagriangian-relaxation and min-cost network flow. It gets within 5-15% of a naive lower bound, whereas using a ``uniform'' cover only gets within 25-35%.Comment: proceedings version appeared in ACM-SIAM Symposium on Discrete Algorithms (1998

The pp-Center Problem in Tree Networks Revisited

We present two improved algorithms for weighted discrete $p$-center problem for tree networks with $n$ vertices. One of our proposed algorithms runs in $O(n \log n + p \log^2 n \log(n/p))$ time. For all values of $p$, our algorithm thus runs as fast as or faster than the most efficient $O(n\log^2 n)$ time algorithm obtained by applying Cole's speed-up technique [cole1987] to the algorithm due to Megiddo and Tamir [megiddo1983], which has remained unchallenged for nearly 30 years. Our other algorithm, which is more practical, runs in $O(n \log n + p^2 \log^2(n/p))$ time, and when $p=O(\sqrt{n})$ it is faster than Megiddo and Tamir's $O(n \log^2n \log\log n)$ time algorithm [megiddo1983]

Complexity of fuzzy answer set programming under Łukasiewicz semantics

Fuzzy answer set programming (FASP) is a generalization of answer set programming (ASP) in which propositions are allowed to be graded. Little is known about the computational complexity of FASP and almost no techniques are available to compute the answer sets of a FASP program. In this paper, we analyze the computational complexity of FASP under Łukasiewicz semantics. In particular we show that the complexity of the main reasoning tasks is located at the first level of the polynomial hierarchy, even for disjunctive FASP programs for which reasoning is classically located at the second level. Moreover, we show a reduction from reasoning with such FASP programs to bilevel linear programming, thus opening the door to practical applications. For definite FASP programs we can show P-membership. Surprisingly, when allowing disjunctions to occur in the body of rules – a syntactic generalization which does not affect the expressivity of ASP in the classical case – the picture changes drastically. In particular, reasoning tasks are then located at the second level of the polynomial hierarchy, while for simple FASP programs, we can only show that the unique answer set can be found in pseudo-polynomial time. Moreover, the connection to an existing open problem about integer equations suggests that the problem of fully characterizing the complexity of FASP in this more general setting is not likely to have an easy solution

Dispersion on Trees

In the k-dispersion problem, we need to select k nodes of a given graph so as to maximize the minimum distance between any two chosen nodes. This can be seen as a generalization of the independent set problem, where the goal is to select nodes so that the minimum distance is larger than 1. We design an optimal O(n) time algorithm for the dispersion problem on trees consisting of n nodes, thus improving the previous O(n log n) time solution from 1997. We also consider the weighted case, where the goal is to choose a set of nodes of total weight at least W. We present an O(n log^2n) algorithm improving the previous O(n log^4 n) solution. Our solution builds on the search version (where we know the minimum distance lambda between the chosen nodes) for which we present tight Theta(n log n) upper and lower bounds

Center location problems on tree graphs with subtree-shaped customers

We consider the p-center problem on tree graphs where the customers are modeled as continua subtrees. We address unweighted and weighted models as well as distances with and without addends. We prove that a relatively simple modification of Handler’s classical linear time algorithms for unweighted 1- and 2-center problems with respect to point customers, linearly solves the unweighted 1- and 2-center problems with addends of the above subtree customer model. We also develop polynomial time algorithms for the p-center problems based on solving covering problems and searching over special domains