Reallocating Multiple Facilities on the Line

We study the multistage $K$-facility reallocation problem on the real line, where we maintain $K$ facility locations over $T$ stages, based on the stage-dependent locations of $n$ agents. Each agent is connected to the nearest facility at each stage, and the facilities may move from one stage to another, to accommodate different agent locations. The objective is to minimize the connection cost of the agents plus the total moving cost of the facilities, over all stages. $K$-facility reallocation was introduced by de Keijzer and Wojtczak, where they mostly focused on the special case of a single facility. Using an LP-based approach, we present a polynomial time algorithm that computes the optimal solution for any number of facilities. We also consider online $K$-facility reallocation, where the algorithm becomes aware of agent locations in a stage-by-stage fashion. By exploiting an interesting connection to the classical $K$-server problem, we present a constant-competitive algorithm for $K = 2$ facilities

Online Multistage Subset Maximization Problems

Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground set N={1,...,n}, a collection F subseteq 2^N of subsets thereof such that the empty set is in F, and an objective (profit) function p: F -> R_+. The task is to choose a set S in F that maximizes p(S). We consider the multistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function p_t (and possibly the set of feasible solutions F_t) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors. We study multistage subset maximization problems in the online setting, that is, p_t (along with possibly F_t) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future. We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one

Clustering Dynamique par Rayon

National audienceComprendre les dynamiques d'évolution de réseaux sociaux et d'infrastructuresest un problème crucial dans les domaines comme l'épidémiologie, l'urbanismeou la marketing viral. Une quantité croissante de données dynamiquessur des réseaux divers est disponible depuis plusieurs années maisles outils pour analyser ces données ne sont pas toujours adaptés.Nous proposons d'utiliser ces données dynamique pour faire des groupesd'individus de comportement similaire restant stables avec le temps.Pour cela nous introduisons une variante dynamique du problème Sum-Radii Clustering, en utilisant le formalisme du problème DynamicFacility Location, avec la distinction que nous cherchons à minimiser le diamètre des groupes aulieu de la somme des distances au centre. Nous étudions deux adaptations naturelles d'algorithmes probabilistes utilisés pour Dynamic Facility Location (marchant respectivement dans le cas général et quand on se restreint à des espaces métriques). Dans le premier cas, l'algorithme atteint la même borne d'approximation et nous proposons une amélioration, aussi valable pour l'algorithme original (faisant passer le facteur d'approximation de $O(\log nT)$ à $O(\log n)$, où $n$ est le nombre de clients et $T$ la durée en nombre de pas de temps).Enfin, nous montrons que dans le cas métrique, les outils actuels ne permettent pas encorede donner un meilleur résultat, et exhibons un contre-exemple pour le deuxième algorithme, prouvant qu'il ne peut pas atteindre une approximation constante

Multistage Matchings

We consider a multistage version of the Perfect Matching problem which models the scenario where the costs of edges change over time and we seek to obtain a solution that achieves low total cost, while minimizing the number of changes from one instance to the next. Formally, we are given a sequence of edge-weighted graphs on the same set of vertices V, and are asked to produce a perfect matching in each instance so that the total edge cost plus the transition cost (the cost of exchanging edges), is minimized. This model was introduced by Gupta et al. (ICALP 2014), who posed as an open problem its approximability for bipartite instances. We completely resolve this question by showing that Minimum Multistage Perfect Matching (Min-MPM) does not admit an n^{1-epsilon}-approximation, even on bipartite instances with only two time steps. Motivated by this negative result, we go on to consider two variations of the problem. In Metric Minimum Multistage Perfect Matching problem (Metric-Min-MPM) we are promised that edge weights in each time step satisfy the triangle inequality. We show that this problem admits a 3-approximation when the number of time steps is 2 or 3. On the other hand, we show that even the metric case is APX-hard already for 2 time steps. We then consider the complementary maximization version of the problem, Maximum Multistage Perfect Matching problem (Max-MPM), where we seek to maximize the total profit of all selected edges plus the total number of non-exchanged edges. We show that Max-MPM is also APX-hard, but admits a constant factor approximation algorithm for any number of time steps

A note on the inflating enclosing ball problem

International audienc

Dynamic Sum-Radii Clustering

International audienceReal networks have in common that they evolve over time and their dynamics have a huge impact on their structure. Clustering is an efficient tool to reduce the complexity to allow representation of the data. In 2014, Eisenstat et al. introduced a dynamic version of this classic problem where the distances evolve with time and where coherence over time is enforced by introducing a cost for clients to change their assigned facility. They designed a Θ(ln n)-approximation. An O(1)-approximation for the metric case was proposed later on by An et al. (2015). Both articles aimed at minimizing the sum of all client-facility distances; however, other metrics may be more relevant. In this article we aim to minimize the sum of the radii of the clusters instead. We obtain an asymptotically optimal Θ(ln n)-approximation algorithm where n is the number of clients and show that existing algorithms from An et al. (2015) do not achieve a constant approximation in the metric variant of this setting

Dynamic Sum-Radii Clustering

International audienceReal networks have in common that they evolve over time and their dynamics have a huge impact on their structure. Clustering is an efficient tool to reduce the complexity to allow representation of the data. In 2014, Eisenstat et al. introduced a dynamic version of this classic problem where the distances evolve with time and where coherence over time is enforced by introducing a cost for clients to change their assigned facility. They designed a Θ(ln n)-approximation. An O(1)-approximation for the metric case was proposed later on by An et al. (2015). Both articles aimed at minimizing the sum of all client-facility distances; however, other metrics may be more relevant. In this article we aim to minimize the sum of the radii of the clusters instead. We obtain an asymptotically optimal Θ(ln n)-approximation algorithm where n is the number of clients and show that existing algorithms from An et al. (2015) do not achieve a constant approximation in the metric variant of this setting