This work examines a strongly NP-hard routing problem on trees, in which multiple servers need to serve a given set of requests (on vertices), where the routes of the servers start from a common source and end at their respective terminals. Each server can travel free of cost on its source-to-terminal path but has to pay for travel on other edges. The objective is to minimize the maximum cost over all servers. As the servers may pay different costs for traveling through a common edge, balancing the loads of the servers can be difficult. We propose a polynomial-time 4-approximation algorithm that applies the parametric pruning framework but consists of two phases. The first phase of the algorithm partitions the requests into packets, and the second phase of the algorithm assigns the packets to the servers. Unlike the standard parametric pruning techniques, the challenge of our algorithm design and analysis is to harmoniously relate the quality of the partition in the first phase, the balances of the servers' loads in the second phase, and the hypothetical optimal values of the framework. For the problem in general graphs, we show that there is no algorithm better than 2-approximate unless P = NP. The problem is a generalization of unrelated machine scheduling and other classic scheduling problems. It also models scheduling problems where the job processing times depend on the machine serving the job and the other jobs served by that machine. This modeling provides a framework that physicalizes scheduling problems through the graph’s point of view.

We consider the online traveling salesman problem on the real line (OLTSPL) in which a salesman begins at the origin, traveling at no faster than unit speed along the real line, and wants to serve a sequence of requests, arriving online over time on the real line and return to the origin as quickly as possible. The problem has been widely investigated for more than two decades, but was just optimally solved by a deterministic algorithm with a competitive ratio of $(9+\sqrt{17})/8$, reported in~[Bjelde A. et al., in Proc. SODA 2017, pp.994--1005]. In this study we present lower bounds and upper bounds for randomized algorithms in the OLTSPL. Precisely, we show, for the first time, that a simple randomized \emph{zealous} algorithm can improve the optimal deterministic algorithm. Here an algorithm is called zealous if waiting strategies are not allowed to use for the salesman as long as there are unserved requests. Moreover, we incorporate a natural waiting scheme into the randomized algorithm, which can even achieve the lower bound we propose for any randomized algorithms, and thus it is optimal. We also consider randomized algorithms against a \emph{fair} adversary, i.e. an adversary with restricted power that requires the salesman to move within the convex hull of the origin and the requests released so far. The randomized non-zealous algorithm can outperform the optimal deterministic algorithm against the fair adversary as well.

In the online variant of the traveling repairperson problem (TRP), requests arrive in time at points of a metric space X and must be eventually visited by a server. The server starts at a designated point of X and travels at most at unit speed. Each request has a given weight and once the server visits its position, the request is considered serviced; we call such time completion time of the request. The goal is to minimize the weighted sum of completion times of all requests. In this paper, we give a 5.429-competitive deterministic algorithm for line metrics improving over 5.829-competitive solution by Krumke et al. (TCS 2003). Our result is obtained by modifying the schedule by serving requests that are close to the origin first. To compute the competitive ratio of our approach, we use a charging scheme, and later evaluate its properties using a factor-revealing linear program which upper-bounds the competitive ratio.

We study a scheduling problem arising in demand response management in smart grid. Consumers send in power requests with a flexible feasible time interval during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective is to schedule all requests with the minimum total electricity cost. Previous work has studied cases where jobs have unit power requirement and unit duration. We extend the study to arbitrary power requirement and duration, which has been shown to be NP-hard. We give the first online algorithm for the general problem, and prove that the worst case competitive ratio is asymptotically optimal. We also prove that the problem is fixed parameter tractable. Due to space limit, the missing proofs are presented in the full paper.