Relaxing the Irrevocability Requirement for Online Graph Algorithms

Online graph problems are considered in models where the irrevocability requirement is relaxed. Motivated by practical examples where, for example, there is a cost associated with building a facility and no extra cost associated with doing it later, we consider the Late Accept model, where a request can be accepted at a later point, but any acceptance is irrevocable. Similarly, we also consider a Late Reject model, where an accepted request can later be rejected, but any rejection is irrevocable (this is sometimes called preemption). Finally, we consider the Late Accept/Reject model, where late accepts and rejects are both allowed, but any late reject is irrevocable. For Independent Set, the Late Accept/Reject model is necessary to obtain a constant competitive ratio, but for Vertex Cover the Late Accept model is sufficient and for Minimum Spanning Forest the Late Reject model is sufficient. The Matching problem has a competitive ratio of 2, but in the Late Accept/Reject model, its competitive ratio is 3/2

Approximation Algorithms for Stochastic Boolean Function Evaluation and Stochastic Submodular Set Cover

Stochastic Boolean Function Evaluation is the problem of determining the value of a given Boolean function f on an unknown input x, when each bit of x_i of x can only be determined by paying an associated cost c_i. The assumption is that x is drawn from a given product distribution, and the goal is to minimize the expected cost. This problem has been studied in Operations Research, where it is known as "sequential testing" of Boolean functions. It has also been studied in learning theory in the context of learning with attribute costs. We consider the general problem of developing approximation algorithms for Stochastic Boolean Function Evaluation. We give a 3-approximation algorithm for evaluating Boolean linear threshold formulas. We also present an approximation algorithm for evaluating CDNF formulas (and decision trees) achieving a factor of O(log kd), where k is the number of terms in the DNF formula, and d is the number of clauses in the CNF formula. In addition, we present approximation algorithms for simultaneous evaluation of linear threshold functions, and for ranking of linear functions. Our function evaluation algorithms are based on reductions to the Stochastic Submodular Set Cover (SSSC) problem. This problem was introduced by Golovin and Krause. They presented an approximation algorithm for the problem, called Adaptive Greedy. Our main technical contribution is a new approximation algorithm for the SSSC problem, which we call Adaptive Dual Greedy. It is an extension of the Dual Greedy algorithm for Submodular Set Cover due to Fujito, which is a generalization of Hochbaum's algorithm for the classical Set Cover Problem. We also give a new bound on the approximation achieved by the Adaptive Greedy algorithm of Golovin and Krause

Online Maximum Matching with Recourse

We study the online maximum matching problem in a model in which the edges are associated with a known recourse parameter k. An online algorithm for this problem has to maintain a valid matching while edges of the underlying graph are presented one after the other. At any moment the algorithm can decide to include an edge into the matching or to exclude it, under the restriction that at most k such actions per edge take place, where k is typically a small constant. This problem was introduced and studied in the context of general online packing problems with recourse by Avitabile et al. [Avitabile et al., 2013], whereas the special case k=2 was studied by Boyar et al. [Boyar et al., 2017]. In the first part of this paper, we consider the edge arrival model, in which an arriving edge never disappears from the graph. Here, we first show an improved analysis on the performance of the algorithm AMP given in [Avitabile et al., 2013], by exploiting the structure of the matching problem. In addition, we extend the result of [Boyar et al., 2017] and show that the greedy algorithm has competitive ratio 3/2 for every even k and ratio 2 for every odd k. Moreover, we present and analyze an improvement of the greedy algorithm which we call L-Greedy, and we show that for small values of k it outperforms the algorithm of [Avitabile et al., 2013]. In terms of lower bounds, we show that no deterministic algorithm better than 1+1/(k-1) exists, improving upon the lower bound of 1+1/k shown in [Avitabile et al., 2013]. The second part of the paper is devoted to the edge arrival/departure model, which is the fully dynamic variant of online matching with recourse. The analysis of L-Greedy and AMP carry through in this model; moreover we show a lower bound of (k^2-3k+6)/(k^2-4k+7) for all even k >= 4. For k in {2,3}, the competitive ratio is 3/2

An FPTAS for Stochastic Unbounded Min-Knapsack Problem

In this paper, we study the stochastic unbounded min-knapsack problem ($\textbf{Min-SUKP}$). The ordinary unbounded min-knapsack problem states that: There are $n$ types of items, and there is an infinite number of items of each type. The items of the same type have the same cost and weight. We want to choose a set of items such that the total weight is at least $W$ and the total cost is minimized. The \prob~generalizes the ordinary unbounded min-knapsack problem to the stochastic setting, where the weight of each item is a random variable following a known distribution and the items of the same type follow the same weight distribution. In \prob, different types of items may have different cost and weight distributions. In this paper, we provide an FPTAS for $\textbf{Min-SUKP}$, i.e., the approximate value our algorithm computes is at most $(1+\epsilon)$ times the optimum, and our algorithm runs in $poly(1/\epsilon,n,\log W)$ time.Comment: 24 page

オンラインナップサックと関連する諸問題に対するアルゴリズム論的研究

学位の種別:課程博士University of Tokyo(東京大学

Effective task allocation frameworks for large-scale multiple agent systems.

This research aims to develop innovative and transformative decision-making frameworks that enable a large-scale multi-robot system, called robotic swarm, to autonomously address multi-robot task allocation problem: given a set of complicated tasks, requiring cooperation, how to partition themselves into subgroups (or called coalitions) and assign the subgroups to each task while maximising the system performance. The frameworks should be executable based on local information in a decentralised manner, operable for a wide range of the system size (i.e., scalable), predictable in terms of collective behaviours, adaptable to dynamic environments, operable asynchronously, and preferably able to accommodate heterogeneous agents. Firstly, for homogeneous robots, this thesis proposes two frameworks based on biological inspiration and game theories, respectively. The former, called LICA-MC (Markov-Chan-based approach under Local Information Consistency Assumption), is inspired by fish in nature: despite insufficient awareness of the entire group, they are well-coordinated by sensing social distances from neighbours. Analogously, each agent in the framework relies only on local information and requires its local consistency over neighbouring agents to adaptively generate the stochastic policy. This feature offers various advantages such as less inter-agent communication, a shorter timescale for using new information, and the potential to accommodate asynchronous behaviours of agents. We prove that the agents can converge to a desired collective status without resorting to any global information, while maintaining scalability, flexibility, and long-term system efficiency. Numerical experiments show that the framework is robust in a realistic environment where information sharing over agents is partially and temporarily disconnected. Furthermore, we explicitly present the design requirements to have all these advantages, and implementation examples concerning travelling costs minimisation, over-congestion avoidance, and quorum models, respectively. The game-theoretical framework, called GRAPE (GRoup Agent Partitioning and placing Event), regards each robot as a self-interested player attempting to join the most preferred coalition according to its individual preferences regarding the size of each coalition. We prove that selfish agents who have social inhibition can always converge to a Nash stable partition (i.e., a social agreement) within polynomial time under the proposed framework. The framework is executable based on local interactions with neighbour agents under a strongly-connected communication network and even in asynchronous environments. This study analyses an outcome’s minimum-guaranteed suboptimality, and additionally shows that at least 50% is guaranteed if social utilities are non-decreasing functions with respect to the number of co-working agents. Numerical experiments confirm that the framework is scalable, fast adaptable against dynamical environments, and robust even in a realistic situation where some of the agents temporarily halt operation during a mission. The two proposed frameworks are compared in the domain of division of labour. Empirical results show that LICA-MC provides excellent scalability with respect to the number of agents, whereas GRAPE has polynomial complexity but is more efficient in terms of convergence time (especially when accommodating a moderate number of robots) and total travelling costs. It also turns out that GRAPE is sensitive to traffic congestion, meanwhile LICA-MC suffers from slower robot speed. We discuss other implicit advantages of the frameworks such as mission suitability and additionally-builtin decision-making functions. Importantly, it is found that GRAPE has the potential to accommodate heterogeneous agents to some extent, which is not the case for LICA-MC. Accordingly, this study attempts to extend GRAPE to incorporate the heterogeneity of agents. Particularly, we consider the case where each task has its minimum workload requirement to be fulfilled by multiple agents and the agents have different work capacities and costs depending on the tasks. The objective is to find an assignment that minimises the total cost of assigned agents while satisfying the requirements. GRAPE cannot be directly used because of the heterogeneity, so we adopt tabu-learning heuristics where an agent penalises its previously chosen coalition whenever it changes decision: this variant is called T-GRAPE. We prove that, by doing so, a Nash stable partition is always guaranteed to be determined in a decentralised manner. Experi-mental results present the properties of the proposed approach regarding suboptimality and algorithmic complexity. Finally, the thesis addresses a more complex decision-making problem involving team formation, team-to-task assignment, agent-to-working-position selection, fair resource allocation concerning tasks’ minimum requirements for completion, and trajectory optimisation with collision avoidance. We propose an integrated framework that decouples the original problem into three subproblems (i.e., coalition formation, position allocation, and path planning) and deals with them sequentially by three respective modules. The coalition formation module based on T-GRAPE deals with a max-min problem, balancing the work resources of agents in proportion to the task’s requirements. We show that, given reasonable assumptions, the position allocation subproblem can be solved efficiently in terms of computational complexity. For the path planning, we utilise an MPC-SCP (Model Predictive Control and Sequential Convex Programming) approach that enables the agents to produce collision-free trajectories. As a proof of concept, we implement the framework into a cooperative stand-in jamming mission scenario using multiple UAVs. Numerical experiments suggest that the framework could be computationally feasible, fault-tolerant, and near-optimal. Comparison of the proposed frameworks for multi-robot task allocation is discussed in the last chapter regarding the desired features described at first (i.e., decentralisation, scalability, predictability, flexibility, asynchronisation, heterogeneity), along with future work and possible applications in other domains.PhD in Aerospac