arXiv subjects

Björn Feldkord

Publications and source records attributed to Björn Feldkord.

7 recordsLinked to original sources

arXiv2022

The k-Server with Preferences Problem

The $k$-Server Problem covers plenty of resource allocation scenarios, and several variations have been studied extensively for decades. We present a model generalizing the $k$-Server Problem by preferences of the requests, where the servers are not identical and requests can express which specific servers should serve them. In our model, requests can either be answered by any server (general requests) or by a specific one (specific requests). If only general requests appear, the instance is one of the original $k$-Server Problem, and a lower bound for the competitive ratio of $k$ applies. If only specific requests appear, a solution with a competitive ratio of $1$ becomes trivial. We show that if both kinds of requests appear, the lower bound raises to $2k-1$. We study deterministic online algorithms and present two algorithms for uniform metrics. The first one has a competitive ratio dependent on the frequency of specific requests. It achieves a worst-case competitive ratio of $3k-2$ while it is optimal when only general requests appear or when specific requests dominate the input sequence. The second has a worst-case competitive ratio of $2k+14$. For the first algorithm, we show a lower bound of $3k-2$, while the second algorithm has a lower bound of $2k-1$ when only general requests appear. The two algorithms differ in only one behavioral rule that significantly influences the competitive ratio. We show that there is a trade-off between performing well against instances of the $k$-Server Problem and mixed instances based on the rule. Additionally, no deterministic online algorithm can be optimal for both kinds of instances simultaneously. Regarding non-uniform metrics, we present an adaption of the Double Coverage algorithm for $2$ servers on the line achieving a competitive ratio of $6$, and an adaption of the Work-Function-Algorithm achieving a competitive ratio of $4k$.

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arXiv2021

A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location

In the online non-metric variant of the facility location problem, there is a given graph consisting of a set $F$ of facilities (each with a certain opening cost), a set $C$ of potential clients, and weighted connections between them. The online part of the input is a sequence of clients from $C$, and in response to any requested client, an online algorithm may open an additional subset of facilities and must connect the given client to an open facility. We give an online, polynomial-time deterministic algorithm for this problem, with a competitive ratio of $O(\log |F| \cdot (\log |C| + \log \log |F|))$. The result is optimal up to loglog factors. Our algorithm improves over the $O((\log |C| + \log |F|) \cdot (\log |C| + \log \log |F|))$-competitive construction that first reduces the facility location instance to a set cover one and then later solves such instance using the deterministic algorithm by Alon et al. [TALG 2006]. This is an asymptotic improvement in a typical scenario where $|F| \ll |C|$. We achieve this by a more direct approach: we design an algorithm for a fractional relaxation of the non-metric facility location problem with clustered facilities. To handle the constraints of such non-covering LP, we combine the dual fitting and multiplicative weight updates approach. By maintaining certain additional monotonicity properties of the created fractional solution, we can handle the dependencies between facilities and connections in a rounding routine. Our result, combined with the algorithm by Naor et al. [FOCS 2011] yields the first deterministic algorithm for the online node-weighted Steiner tree problem. The resulting competitive ratio is $O(\log k \cdot \log^2 \ell)$ on graphs of $\ell$ nodes and $k$ terminals.

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arXiv2020

The Online Multi-Commodity Facility Location Problem

We consider a natural extension to the metric uncapacitated Facility Location Problem (FLP) in which requests ask for different commodities out of a finite set $S$ of commodities. Ravi and Sinha (SODA'04) introduced the model as the Multi-Commodity Facility Location Problem (MFLP) and considered it an offline optimization problem. The model itself is similar to the FLP: i.e., requests are located at points of a finite metric space and the task of an algorithm is to construct facilities and assign requests to facilities while minimizing the construction cost and the sum over all assignment distances. In addition, requests and facilities are heterogeneous; they request or offer multiple commodities out of $S$. A request has to be connected to a set of facilities jointly offering the commodities demanded by it. In comparison to the FLP, an algorithm has to decide not only if and where to place facilities, but also which commodities to offer at each. To the best of our knowledge we are the first to study the problem in its online variant in which requests, their positions and their commodities are not known beforehand but revealed over time. We present results regarding the competitive ratio. On the one hand, we show that heterogeneity influences the competitive ratio by developing a lower bound on the competitive ratio for any randomized online algorithm of $Ω(\sqrt{|S|}+\frac{\log n}{\log\log n})$ that already holds for simple line metrics. Here, $n$ is the number of requests. On the other side, we establish a deterministic $O(\sqrt{|S|}\cdot\log n)$-competitive algorithm and a randomized $O(\sqrt{|S|}\cdot\frac{\log n}{\log\log n})$-competitive algorithm. Further, we show that when considering a more special class of cost functions for the construction cost of a facility, the competitive ratio decreases given by our deterministic algorithm depending on the function.

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arXiv2019

Managing Multiple Mobile Resources

We extend the Mobile Server Problem, introduced in SPAA'17, to a model where k identical mobile resources, here named servers, answer requests appearing at points in the Euclidean space. In order to reduce communication costs, the positions of the servers can be adapted by a limited distance m_s per round for each server. The costs are measured similar to the classical Page Migration Problem, i.e., answering a request induces costs proportional to the distance to the nearest server, and moving a server induces costs proportional to the distance multiplied with a weight D. We show that, in our model, no online algorithm can have a constant competitive ratio, i.e., one which is independent of the input length n, even if an augmented moving distance of (1+δ)m_s is allowed for the online algorithm. Therefore we investigate a restriction of the power of the adversary dictating the sequence of requests: We demand locality of requests, i.e., that consecutive requests come from points in the Euclidean space with distance bounded by some constant m_c. We show constant lower bounds on the competitiveness in this setting (independent of n, but dependent on k, m_s and m_c). On the positive side, we present a deterministic online algorithm with bounded competitiveness when augmented moving distance and locality of requests is assumed. Our algorithm simulates any given algorithm for the classical k-Page Migration problem as guidance for its servers and extends it by a greedy move of one server in every round. The resulting competitive ratio is polynomial in the number of servers k, the ratio between m_c and m_s, the inverse of the augmentation factor 1/δand the competitive ratio of the simulated k-Page Migration algorithm.

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arXiv2019

The Mobile Server Problem

We introduce the mobile server problem, inspired by current trends to move computational tasks from cloud structures to multiple devices close to the end user. An example for this are embedded systems in autonomous cars that communicate in order to coordinate their actions. Our model is a variant of the classical Page Migration Problem. More formally, we consider a mobile server holding a data page. The server can move in the Euclidean space (of arbitrary dimension). In every round, requests for data items from the page pop up at arbitrary points in the space. The requests are served, each at a cost of the distance from the requesting point and the server, and the mobile server may move, at a cost $D$ times the distance traveled for some constant $D$. We assume a maximum distance $m$ the server is allowed to move per round. We show that no online algorithm can achieve a competitive ratio independent of the length of the input sequence in this setting. Hence we augment the maximum movement distance of the online algorithms to $(1+δ)$ times the maximum distance of the offline solution. We provide a deterministic algorithm which is simple to describe and works for multiple variants of our problem. The algorithm achieves almost tight competitive ratios independent of the length of the input sequence. Our Algorithm also achieves a constant competitive ratio without resource augmentation in a variant where the distance between two consecutive requests is restricted to a constant smaller than the limit for the server.

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arXiv2018

A Tight Approximation for Fully Dynamic Bin Packing without Bundling

We consider a variant of the classical Bin Packing Problem, called Fully Dynamic Bin Packing. In this variant, items of a size in $(0,1]$ must be packed in bins of unit size. In each time step, an item either arrives or departs from the packing. An algorithm for this problem must maintain a feasible packing while only repacking a bounded number of items in each time step. We develop an algorithm which repacks only a constant number of items per time step and, unlike previous work, does not rely on bundling of small items which allowed those solutions to move an unbounded number of small items as one. Our algorithm has an asymptotic approximation ratio of roughly $1.3871$ which is complemented by a lower bound of Balogh et al., resulting in a tight approximation ratio for this problem. As a direct corollary, we also close the gap to the lower bound of the Relaxed Online Bin Packing Problem in which only insertions of items occur.

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arXiv2017

A Communication-Efficient Distributed Data Structure for Top-k and k-Select Queries

We consider the scenario of $n$ sensor nodes observing streams of data. The nodes are connected to a central server whose task it is to compute some function over all data items observed by the nodes. In our case, there exists a total order on the data items observed by the nodes. Our goal is to compute the $k$ currently lowest observed values or a value with rank in $[(1-\varepsilon)k,(1+\varepsilon)k]$ with probability $(1-δ)$. We propose solutions for these problems in an extension of the distributed monitoring model where the server can send broadcast messages to all nodes for unit cost. We want to minimize communication over multiple time steps where there are $m$ updates to a node's value in between queries. The result is composed of two main parts, which each may be of independent interest: (1) Protocols which answer Top-k and k-Select queries. These protocols are memoryless in the sense that they gather all information at the time of the request. (2) A dynamic data structure which tracks for every $k$ an element close to $k$. We describe how to combine the two parts to receive a protocol answering the stated queries over multiple time steps. Overall, for Top-$k$ queries we use $O(k + \log m + \log \log n)$ and for $k$-Select queries $O(\frac{1}{\varepsilon^2} \log \frac{1}δ + \log m + \log^2 \log n)$ messages in expectation. These results are shown to be asymptotically tight if $m$ is not too small.

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