Minimizing the Maximum Flow Time in the Online Food Delivery Problem

We study a common delivery problem encountered in nowadays online food-ordering platforms: Customers order dishes online, and the restaurant delivers the food after receiving the order. Specifically, we study a problem where k vehicles of capacity c are serving a set of requests ordering food from one restaurant. After a request arrives, it can be served by a vehicle moving from the restaurant to its delivery location. We are interested in serving all requests while minimizing the maximum flow-time, i.e., the maximum time length a customer waits to receive his/her food after submitting the order. We show that the problem is hard in both offline and online settings even when k = 1 and c = ?: There is a hardness of approximation of ?(n) for the offline problem, and a lower bound of ?(n) on the competitive ratio of any online algorithm, where n is number of points in the metric. We circumvent the strong negative results in two directions. Our main result is an O(1)-competitive online algorithm for the uncapacitated (i.e, c = ?) food delivery problem on tree metrics; we also have negative result showing that the condition c = ? is needed. Then we explore the speed-augmentation model where our online algorithm is allowed to use vehicles with faster speed. We show that a moderate speeding factor leads to a constant competitive ratio, and we prove a tight trade-off between the speeding factor and the competitive ratio

Experimental Evaluation of Approximation and Heuristic Algorithms for Maximum Distance-Bounded Subgraph Problems

In this paper, we consider two distance-based relaxed variants of the maximum clique problem (Max Clique), named Maxd-Clique and Maxd-Club for positive integers d. Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of n1−ε for any real ε>0 unless P=NP , since they are identical to Max Clique (Håstad in Acta Math 182(1):105–142, 1999; Zuckerman in Theory Comput 3:103–128, 2007). In addition, it is NP -hard to approximate Maxd-Clique and Maxd-Club to within a factor of n1/2−ε for any fixed integer d≥2 and any real ε>0 (Asahiro et al. in Approximating maximum diameter-bounded subgraphs. In: Proc of LATIN 2010, Springer, pp 615–626, 2010; Asahiro et al. in Optimal approximation algorithms for maximum distance-bounded subgraph problems. In: Proc of COCOA, Springer, pp 586–600, 2015). As for approximability of Maxd-Clique and Maxd-Club, a polynomial-time algorithm, called ReFindStar d, that achieves an optimal approximation ratio of O(n1/2) for Maxd-Clique and Maxd-Club was designed for any integer d≥2 in Asahiro et al. (2015, Algorithmica 80(6):1834–1856, 2018). Moreover, a simpler algorithm, called ByFindStar d, was proposed and it was shown in Asahiro et al. (2010, 2018) that although the approximation ratio of ByFindStar d is much worse for any odd d≥3, its time complexity is better than ReFindStar d. In this paper, we implement those approximation algorithms and evaluate their quality empirically for random graphs. The experimental results show that (1) ReFindStar d can find larger d-clubs (d-cliques) than ByFindStar d for odd d, (2) the size of d-clubs (d-cliques) output by ByFindStar d is the same as ones by ReFindStar d for even d, and (3) ByFindStar d can find the same size of d-clubs (d-cliques) much faster than ReFindStar d. Furthermore, we propose and implement two new heuristics, Hclub d for Maxd-Club and Hclique d for Maxd-Clique. Then, we present the experimental evaluation of the solution size of ReFindStar d, Hclub d, Hclique d and previously known heuristic algorithms for random graphs and Erdős collaboration graphs

Hysteretic optimization for the Sherrington-Kirkpatrick spin glass

Hysteretic optimization is a heuristic optimization method based on the observation that magnetic samples are driven into a low energy state when demagnetized by an oscillating magnetic field of decreasing amplitude. We show that hysteretic optimization is very good for finding ground states of Sherrington-Kirkpatrick spin glass systems. With this method it is possible to get good statistics for ground state energies for large samples of systems consisting of up to about 2000 spins. The way we estimate error rates may be useful for some other optimization methods as well. Our results show that both the average and the width of the ground state energy distribution converges faster with increasing size than expected from earlier studies.Comment: Physica A, accepte