9 research outputs found

    Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

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    The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph G=(V,E)G=(V,E) and a set DV×V\mathcal{D}\subseteq V\times V of kk demand pairs. The aim is to compute the cheapest network NGN\subseteq G for which there is an sts\to t path for each (s,t)D(s,t)\in\mathcal{D}. It is known that this problem is notoriously hard as there is no k1/4o(1)k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parametrizing the runtime by kk [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter kk. For the bi-DSNPlanar_\text{Planar} problem, the aim is to compute a planar optimum solution NGN\subseteq G in a bidirected graph GG, i.e., for every edge uvuv of GG the reverse edge vuvu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for kk. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSNPlanar_\text{Planar}, unless FPT=W[1]. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network NGN\subseteq G needs to strongly connect a given set of kk terminals. It has been observed before that for SCSS a parameterized 22-approximation exists when parameterized by kk [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for kk no parameterized (2ε)(2-\varepsilon)-approximation algorithm exists under Gap-ETH. Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for kk

    Exploiting Hopsets: Improved Distance Oracles for Graphs of Constant Highway Dimension and Beyond

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    For fixed h >= 2, we consider the task of adding to a graph G a set of weighted shortcut edges on the same vertex set, such that the length of a shortest h-hop path between any pair of vertices in the augmented graph is exactly the same as the original distance between these vertices in G. A set of shortcut edges with this property is called an exact h-hopset and may be applied in processing distance queries on graph G. In particular, a 2-hopset directly corresponds to a distributed distance oracle known as a hub labeling. In this work, we explore centralized distance oracles based on 3-hopsets and display their advantages in several practical scenarios. In particular, for graphs of constant highway dimension, and more generally for graphs of constant skeleton dimension, we show that 3-hopsets require exponentially fewer shortcuts per node than any previously described distance oracle, and also offer a speedup in query time when compared to simple oracles based on a direct application of 2-hopsets. Finally, we consider the problem of computing minimum-size h-hopset (for any h >= 2) for a given graph G, showing a polylogarithmic-factor approximation for the case of unique shortest path graphs. When h=3, for a given bound on the space used by the distance oracle, we provide a construction of hopset achieving polylog approximation both for space and query time compared to the optimal 3-hopset oracle given the space bound

    JFPC 2019 - Actes des 15es Journées Francophones de Programmation par Contraintes

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    National audienceLes JFPC (Journées Francophones de Programmation par Contraintes) sont le principal congrès de la communauté francophone travaillant sur les problèmes de satisfaction de contraintes (CSP), le problème de la satisfiabilité d'une formule logique propositionnelle (SAT) et/ou la programmation logique avec contraintes (CLP). La communauté de programmation par contraintes entretient également des liens avec la recherche opérationnelle (RO), l'analyse par intervalles et différents domaines de l'intelligence artificielle.L'efficacité des méthodes de résolution et l'extension des modèles permettent à la programmation par contraintes de s'attaquer à des applications nombreuses et variées comme la logistique, l'ordonnancement de tâches, la conception d'emplois du temps, la conception en robotique, l'étude du génôme en bio-informatique, l'optimisation de pratiques agricoles, etc.Les JFPC se veulent un lieu convivial de rencontres, de discussions et d'échanges pour la communauté francophone, en particulier entre doctorants, chercheurs confirmés et industriels. L'importance des JFPC est reflétée par la part considérable (environ un tiers) de la communauté francophone dans la recherche mondiale dans ce domaine.Patronnées par l'AFPC (Association Française pour la Programmation par Contraintes), les JFPC 2019 ont lieu du 12 au 14 Juin 2019 à l'IMT Mines Albi et sont organisées par Xavier Lorca (président du comité scientifique) et par Élise Vareilles (présidente du comité d'organisation)

    Approximation Schemes for Machine Scheduling

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    In the classical problem of makespan minimization on identical parallel machines, or machine scheduling for short, a set of jobs has to be assigned to a set of machines. The jobs have a processing time and the goal is to minimize the latest finishing time of the jobs. Machine scheduling is well known to be NP-hard and thus there is no polynomial time algorithm for this problem that is guaranteed to find an optimal solution unless P=NP. There is, however, a polynomial time approximation scheme (PTAS) for machine scheduling, that is, a family of approximation algorithms with ratios arbitrarily close to one. Whether a problem admits an approximation scheme or not is a fundamental question in approximation theory. In the present work, we consider this question for several variants of machine scheduling. We study the problem where the machines are partitioned into a constant number of types and the processing time of the jobs is also dependent on the machine type. We present so called efficient PTAS (EPTAS) results for this problem and variants thereof. We show that certain cases of machine scheduling with assignment restrictions do not admit a PTAS unless P=NP. Moreover, we introduce a graph framework based on the restrictions of the jobs and use it in the design of approximation schemes for other variants. We introduce an enhanced integer programming formulation for assignment problems, show that it can be efficiently solved, and use it in the EPTAS design for variants of machine scheduling with setup times. For one of the problems, we show that there is also a PTAS in the case with uniform machines, where machines have speeds influencing the processing times of the jobs. We consider cases in which each job requires a certain amount of a shared renewable resource and the processing time is depended on the amount of resource it receives or not. We present so called asymptotic fully polynomial time approximation schemes (AFPTAS) for the problems

    Parameterized approximation algorithms for bidirected steiner network problems

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    The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph G=(V,E) and a set {D}subseteq V x V of k demand pairs. The aim is to compute the cheapest network N subseteq G for which there is an s -> t path for each (s,t)in {D}. It is known that this problem is notoriously hard as there is no k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parameterizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k. For the bi-DSN_Planar problem, the aim is to compute a planar optimum solution N subseteq G in a bidirected graph G, i.e. for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSN_Planar, unless FPT=W[1]. Additionally we study several generalizations of bi-DSN_Planar and obtain upper and lower bounds on obtainable runtimes parameterized by k. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network N subseteq G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists when parameterized by k [Chitnis et al., IPEC 2013]. We show a tight inapproximability result: under Gap-ETH there is no (2-{epsilon})-approximation algorithm parameterized by k (for any epsilon>0). To the best of our knowledge, this is the first example of a W[1]-hard problem admitting a non-trivial parameterized approximation factor which is also known to be tight! Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k

    An Indexing Framework for Queries on Probabilistic Graphs

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    Structural Parameters, Tight Bounds, and Approximation for (k,r)-Center

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    In (k,r)-Center we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically: - For any r>=1, we show an algorithm that solves the problem in O*((3r+1)^cw) time, where cw is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm\u27s performance. As a corollary, for r=1, this closes the gap that previously existed on the complexity of Dominating Set parameterized by cw. - We strengthen previously known FPT lower bounds, by showing that (k,r)-Center is W[1]-hard parameterized by the input graph\u27s vertex cover (if edge weights are allowed), or feedback vertex set, even if k is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs. - We show that the complexity of the problem parameterized by tree-depth is 2^Theta(td^2) by showing an algorithm of this complexity and a tight ETH-based lower bound. We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth which work efficiently independently of the values of k,r. In particular, we give algorithms which, for any epsilon>0, run in time O*((tw/epsilon)^O(tw)), O*((cw/epsilon)^O(cw)) and return a (k,(1+epsilon)r)-center, if a (k,r)-center exists, thus circumventing the problem\u27s W-hardness

    A Polyhedral Study of Mixed 0-1 Set

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    We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set
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