Online clustering on the line with square cost variable sized clusters

In the online clustering problems, the classification of points into sets (called clusters) is done in an online fashion. Points arrive one by one at arbitrary locations, to be assigned to clusters at the time of arrival without any information about the further points. A point can be assigned to an existing cluster, or a new cluster can be opened for it. Existing clusters cannot be merged or split. We study one-dimensional variants. The cost of a cluster is the sum of a fixed setup cost scaled to 1 and the square of the diameter of the cluster. The goal is to minimize the sum of costs of the clusters used by the algorithm. In this paper we investigate the problem on the line. We examine two versions, both maintaining the properties that a point which was assigned to a given cluster must remain assigned to this cluster, and clusters cannot be merged. In the strict variant, the size and the exact location of the cluster must be fixed when it is initialized. In the flexible variant, the algorithm can shift the cluster or expand it, as long as it contains all points assigned to it. We consider the online and the semi-online (the input is sorted according to their coordinates from smallest to largest i.e., from left to right) versions of the above two variants. We present the first online algorithms for the solution of the problem. We describe algorithms for the strict and the flexible variant both for the online and semi-online versions. We also give lower bounds on the possible competitive ratio in all of the cases

Nothing is free: a survey of the social cost of the main payment instruments in Hungary

The study applies two approaches for the estimation of the social costs of main payment instruments (cash, debit card and credit card transactions, credit transfers, direct debits, business-to-business direct debits, postal inpayment money orders, postal outpayment money orders for pensions) used in Hungary in 2009. The first approach is based on the current payment structure, while the second approach is based on a more modern, hypothetical payment structure involving less cash, with no use of paper-based methods. In the first approach, the social cost amounts to HUF 388 billion, i.e. 1.49% of the GDP, while in the second approach, such cost amounts to HUF 285 billion, i.e. 1.09% of the GDP. In this context, social cost means the use of all resources (time, materials and money) necessary for the execution of payments, calculated as a net value (i.e. exclusive of fees paid for payment services). Thus, HUF 103 billion could be saved in social costs if the use of payment instruments were to be modified.private cost, social cost, net private cost, unit cost, social savings, cash transactions, debit card transactions, credit card transactions, paper-based credit transfers, electronic credit transfers, direct debits, business-to-business direct debits, postal inpayment money orders, postal outpayment money orders for pensions

Nothing is free: A survey of the social cost of the main payment instruments in Hungary

The study applies two approaches for the estimation of the social costs of main payment instruments (cash, debit card and credit card transactions, credit transfers, direct debits, business-to-business direct debits, postal inpayment money orders, postal outpayment money orders for pensions) used in Hungary in 2009. The first approach is based on the current payment structure, while the second approach is based on a more modern, hypothetical payment structure involving less cash, with no use of paper-based methods. In the first approach, the social cost amounts to HUF 388 billion, i.e. 1.49% of the GDP, while in the second approach, such cost amounts to HUF 85 billion, i.e. 1.09% of the GDP. In this context, social cost means the use of all resources (time, materials and money) necessary for the execution of payments, calculated as a net value (i.e. exclusive of fees paid for payment services). Thus, HUF 103 billion could be saved in social costs if the use of payment instruments were to be modified

Facility Location in Evolving Metrics

Understanding the dynamics of evolving social or infrastructure networks is a challenge in applied areas such as epidemiology, viral marketing, or urban planning. During the past decade, data has been collected on such networks but has yet to be fully analyzed. We propose to use information on the dynamics of the data to find stable partitions of the network into groups. For that purpose, we introduce a time-dependent, dynamic version of the facility location problem, that includes a switching cost when a client's assignment changes from one facility to another. This might provide a better representation of an evolving network, emphasizing the abrupt change of relationships between subjects rather than the continuous evolution of the underlying network. We show that in realistic examples this model yields indeed better fitting solutions than optimizing every snapshot independently. We present an $O(\log nT)$-approximation algorithm and a matching hardness result, where $n$ is the number of clients and $T$ the number of time steps. We also give an other algorithms with approximation ratio $O(\log nT)$ for the variant where one pays at each time step (leasing) for each open facility

Online unit clustering in higher dimensions

We revisit the online Unit Clustering and Unit Covering problems in higher dimensions: Given a set of $n$ points in a metric space, that arrive one by one, Unit Clustering asks to partition the points into the minimum number of clusters (subsets) of diameter at most one; while Unit Covering asks to cover all points by the minimum number of balls of unit radius. In this paper, we work in $\mathbb{R}^d$ using the $L_\infty$ norm. We show that the competitive ratio of any online algorithm (deterministic or randomized) for Unit Clustering must depend on the dimension $d$. We also give a randomized online algorithm with competitive ratio $O(d^2)$ for Unit Clustering}of integer points (i.e., points in $\mathbb{Z}^d$, $d\in \mathbb{N}$, under $L_{\infty}$ norm). We show that the competitive ratio of any deterministic online algorithm for Unit Covering is at least $2^d$. This ratio is the best possible, as it can be attained by a simple deterministic algorithm that assigns points to a predefined set of unit cubes. We complement these results with some additional lower bounds for related problems in higher dimensions.Comment: 15 pages, 4 figures. A preliminary version appeared in the Proceedings of the 15th Workshop on Approximation and Online Algorithms (WAOA 2017

Possibility of a photometric detection of "exomoons"

We examined which exo-systems contain moons that may be detected in transit. We numerically modeled transit light curves of Earth-like and giant planets that cointain moons with 0.005--0.4 Earth-mass. The orbital parameters were randomly selected, but the entire system fulfilled Hill-stability. We conclude that the timing effect is caused by two scenarios: the motion of the planet and the moon around the barycenter. Which one dominates depends on the parameters of the system. Already planned missions (Kepler, COROT) may be able to detect the moon in transiting extrasolar Earth-Moon-like systems with a 20% probability. From our sample of 500 free-designed systems, 8 could be detected with the photometric accuracy of 0.1 mmag and a 1 minute sampling, and one contains a stony planet. With ten times better accuracy, 51 detections are expected. All such systems orbit far from the central star, with the orbital periods at least 200 and 10 days for the planet and the moon, while they contain K- and M-dwarf stars. Finally we estimate that a few number of real detections can be expected by the end of the COROT and the Kepler missions.Comment: 5 pages, 4 figures, accepted by Astronomy and Astrophysic