Fast Algorithms for Energy Games in Special Cases

In this paper, we study algorithms for special cases of energy games, a class of turn-based games on graphs that show up in the quantitative analysis of reactive systems. In an energy game, the vertices of a weighted directed graph belong either to Alice or to Bob. A token is moved to a next vertex by the player controlling its current location, and its energy is changed by the weight of the edge. Given a fixed starting vertex and initial energy, Alice wins the game if the energy of the token remains nonnegative at every moment. If the energy goes below zero at some point, then Bob wins. The problem of determining the winner in an energy game lies in $\mathsf{NP} \cap \mathsf{coNP}$. It is a long standing open problem whether a polynomial time algorithm for this problem exists. We devise new algorithms for three special cases of the problem. The first two results focus on the single-player version, where either Alice or Bob controls the whole game graph. We develop an $\tilde{O}(n^\omega W^\omega)$ time algorithm for a game graph controlled by Alice, by providing a reduction to the All-Pairs Nonnegative Prefix Paths problem (APNP), where $W$ is the maximum weight and $\omega$ is the best exponent for matrix multiplication. Thus we study the APNP problem separately, for which we develop an $\tilde{O}(n^\omega W^\omega)$ time algorithm. For both problems, we improve over the state of the art of $\tilde O(mn)$ for small $W$. For the APNP problem, we also provide a conditional lower bound, which states that there is no $O(n^{3-\epsilon})$ time algorithm for any $\epsilon > 0$, unless the APSP Hypothesis fails. For a game graph controlled by Bob, we obtain a near-linear time algorithm. Regarding our third result, we present a variant of the value iteration algorithm, and we prove that it gives an $O(mn)$ time algorithm for game graphs without negative cycles

Computing Smallest Convex Intersecting Polygons

Funding Information: Funding Mark de Berg is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003. Antonis Skarlatos: Part of the work was done during an internship at the Max Planck Institute for Informatics in Saarbrücken, Germany. Publisher Copyright: © 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.A polygon C is an intersecting polygon for a set O of objects in R2 if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the minimum-area convex intersecting polygon for a given set O of objects. We present an FPTAS for both problems for the case where O is a set of possibly intersecting convex polygons in the plane of total complexity n. Furthermore, we present an exact polynomial-time algorithm for the minimum-perimeter intersecting polygon for the case where O is a set of n possibly intersecting segments in the plane. So far, polynomial-time exact algorithms were only known for the minimum perimeter intersecting polygon of lines or of disjoint segments.Peer reviewe

Bootstrapping Dynamic Distance Oracles

Designing approximate all-pairs distance oracles in the fully dynamic setting is one of the central problems in dynamic graph algorithms. Despite extensive research on this topic, the first result breaking the O(√n) barrier on the update time for any non-trivial approximation was introduced only recently by Forster, Goranci and Henzinger [SODA'21] who achieved m^{1/ρ+o(1)} amortized update time with a O(log n)^{3ρ-2} factor in the approximation ratio, for any parameter ρ ≥ 1. In this paper, we give the first constant-stretch fully dynamic distance oracle with small polynomial update and query time. Prior work required either at least a poly-logarithmic approximation or much larger update time. Our result gives a more fine-grained trade-off between stretch and update time, for instance we can achieve constant stretch of O(1/(ρ²))^{4/ρ} in amortized update time Õ(n^{ρ}), and query time Õ(n^{ρ/8}) for any constant parameter 0 < ρ < 1. Our algorithm is randomized and assumes an oblivious adversary. A core technical idea underlying our construction is to design a black-box reduction from decremental approximate hub-labeling schemes to fully dynamic distance oracles, which may be of independent interest. We then apply this reduction repeatedly to an existing decremental algorithm to bootstrap our fully dynamic solution

Dynamic algorithms for k-center on graphs

In this paper we give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into $k$ sets by choosing $k$ centers such that the maximum distance from any data point to the closest center is minimized. It is known that it is NP-hard to get a better than $2$ approximation for this problem. While in many applications the input may naturally be modeled as a graph, all prior works on $k$-center problem in dynamic settings are on metrics. In this paper, we give a deterministic decremental $(2+\epsilon)$-approximation algorithm and a randomized incremental $(4+\epsilon)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+\epsilon)$-approximation algorithm for the $k$-center problem, with worst-case update time that is within a factor $k$ of the state-of-the-art upper bound for maintaining $(1+\epsilon)$-approximate single-source distances in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a $(2+\epsilon)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances

The curve number concept as a driver for delineating Hydrological Response Units

In this paper, a new methodology for delineating Hydrological Response Units (HRUs), based on the Curve Number (CN) concept, is presented. Initially, a semi-automatic procedure in a GIS environment is used to produce basin maps of distributed CN values as the product of the three classified layers, soil permeability, land use/land cover characteristics and drainage capacity. The map of CN values is used in the context of model parameterization, in order to identify the essential number and spatial extent of HRUs and, consequently, the number of control variables of the calibration problem. The new approach aims at reducing the subjectivity introduced by the definition of HRUs and providing parsimonious modelling schemes. In particular, the CN-based parameterization (1) allows the user to assign as many parameters as can be supported by the available hydrological information, (2) associates the model parameters with anticipated basin responses, as quantified in terms of CN classes across HRUs, and (3) reduces the effort for model calibration, simultaneously ensuring good predictive capacity. The advantages of the proposed approach are demonstrated in the hydrological simulation of the Nedontas River Basin, Greece, where parameterizations of different complexities are employed in a recently improved version of the HYDROGEIOS model. A modelling experiment with a varying number of HRUs, where the parameter estimation problem was handled through automatic optimization, showed that the parameterization with three HRUs, i.e., equal to the number of flow records, ensured the optimal performance. Similarly, tests with alternative HRU configurations confirmed that the optimal scores, both in calibration and validation, were achieved by the CN-based approach, also resulting in parameters values across the HRUs that were in agreement with their physical interpretation

Computing Smallest Convex Intersecting Polygons

A polygon C is an intersecting polygon for a set O of objects in the plane if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the minimum-area convex intersecting polygon for a given set O of objects. We present an FPTAS for both problems for the case where O is a set of possibly intersecting convex polygons in the plane of total complexity n. Furthermore, we present an exact polynomial-time algorithm for the minimum-perimeter intersecting polygon for the case where O is a set of n possibly intersecting segments in the plane. So far, polynomial-time exact algorithms were only known for the minimum perimeter intersecting polygon of lines or of disjoint segments

Dynamic algorithms for k-center on graphs

In this paper we give the first efficient algorithms for the k-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into k sets by choosing k centers such that the maximum distance from any data point to its closest center is minimized. It is known that it is NP-hard to get a better than 2 approximation for this problem. While in many applications the input may naturally be modeled as a graph, all prior works on k-center problem in dynamic settings are on point sets in arbitrary metric spaces. In this paper, we give a deterministic decremental (2+ϵ)-approximation algorithm and a randomized incremental (4+ϵ)-approximation algorithm, both with amortized update time kno(1) for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic (2+ϵ)-approximation algorithm for the k-center problem, with worst-case update time that is within a factor k of the state-of-the-art fully dynamic (1+ϵ)-approximation single-source shortest paths algorithm in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a (2+ϵ)-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances

The Curve Number Concept as a Driver for Delineating Hydrological Response Units

In this paper, a new methodology for delineating Hydrological Response Units (HRUs), based on the Curve Number (CN) concept, is presented. Initially, a semi-automatic procedure in a GIS environment is used to produce basin maps of distributed CN values as the product of the three classified layers, soil permeability, land use/land cover characteristics and drainage capacity. The map of CN values is used in the context of model parameterization, in order to identify the essential number and spatial extent of HRUs and, consequently, the number of control variables of the calibration problem. The new approach aims at reducing the subjectivity introduced by the definition of HRUs and providing parsimonious modelling schemes. In particular, the CN-based parameterization (1) allows the user to assign as many parameters as can be supported by the available hydrological information, (2) associates the model parameters with anticipated basin responses, as quantified in terms of CN classes across HRUs, and (3) reduces the effort for model calibration, simultaneously ensuring good predictive capacity. The advantages of the proposed approach are demonstrated in the hydrological simulation of the Nedontas River Basin, Greece, where parameterizations of different complexities are employed in a recently improved version of the HYDROGEIOS model. A modelling experiment with a varying number of HRUs, where the parameter estimation problem was handled through automatic optimization, showed that the parameterization with three HRUs, i.e., equal to the number of flow records, ensured the optimal performance. Similarly, tests with alternative HRU configurations confirmed that the optimal scores, both in calibration and validation, were achieved by the CN-based approach, also resulting in parameters values across the HRUs that were in agreement with their physical interpretation