Hairong Zhao

Fault-tolerant geometric spanners

We present two new results about vertex and edge fault-tolerant spanners in Euclidean spaces.We describe the first construction of vertex and edge fault-tolerant spanners having optimal bounds for maximum degree and total cost. We present a greedy algorithm that for any t > 1 and any non-negative integer k, constructs a k-fault-tolerant t-spanner in which every vertex is of degree O(k) and whose total cost is O(k2) times the cost of minimum spanning tree; these bounds are asymptotically optimal.Our next contribution is an efficient algorithm for constructing good fault-tolerant spanners. We present a new, sufficient condition for a graph to be a k-fault-tolerant spanner. Using this condition, we design an efficient algorithm that finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bounds for the total cost.

Bi-criteria scheduling problems: Number of tardy jobs and maximum weighted tardiness

Abstract Consider a single machine and a set of n jobs that are available for processing at time 0. Job j has a processing time p j , a due date d j and a weight w j . We consider bi-criteria scheduling problems involving the maximum weighted tardiness and the number of tardy jobs. We give NP-hardness proofs for the scheduling problems when either one of the two criteria is the primary criterion and the other one is the secondary criterion. These results answer two open questions posed by Lee and Vairaktarakis in 1993. We consider complexity relationships between the various problems, give polynomial-time algorithms for some special cases, and propose fast heuristics for the general case. The effectiveness of the heuristics is measured by empirical study. Our results show that one heuristic performs extremely well compared to optimal solutions.

Complexity of two dual criteria scheduling problems

In this article we answer the complexity question of two dual criteria scheduling problems which had been open for a long time. We show that both problems are binary NP-hard.

Exponential inapproximability and FPTAS for scheduling with availability constraints

We investigate the problems of scheduling n weighted jobs to one or more identical machines with the constraint that the machines may be unavailable in some specified time intervals. The objective is to find a schedule that minimizes the total weighted completion time. We consider both non-resumable and resumable schedules. Our first contributions concern approximability. For both resumable problem and non-resumable problem, we show that they cannot be approximated within an exponential factor by any polynomial time algorithm for multiple machines where each of them has an unavailable interval, even if the weight of each job equals to its processing time. Additionally, the non-resumable problem is also exponentially inapproximable for a single machine with two or more unavailable intervals. Then we develop the first FPTASs for the problems with a single unavailable interval among all machines. The running time is O(cnlog^dn(1@elogw)^d) for the non-resumable problem, and O(cnlog^dn(1@elogw)^d^+^1) for the resumable problem, where w is the product of the total weight and the total processing time of all jobs, c is the number of machines that are always available and d=6c+12. Thus our results give a clear boundary delineating the inapproximable cases and approximable cases. When there is a single machine and w=O(n^l^o^g^n^^^O^^^(^^^1^^^)), our algorithms greatly improve the current results. Note that instead of conventional ways of sequentially processing the jobs, our fast schemes process jobs in a divide-and-conquer fashion, which greatly reduces the running time. This may give some insight for some other related problems.

Approximation schemes for minimum 2-connected spanning subgraphs in weighted planar graphs

We present new approximation schemes for various classical problems of finding the minimum-weight spanning subgraph in edge-weighted undirected planar graphs that are resistant to edge or vertex removal. We first give a PTAS for the problem of finding minimum-weight 2-edge-connected spanning subgraphs where duplicate edges are allowed. Then we present a new greedy spanner construction for edge-weighted planar graphs, which augments any connected subgraph A of a weighted planar graph G to a (1+e)-spanner of G with total weight bounded by weight(A)/e. From this we derive quasi-polynomial time approximation schemes for the problems of finding the minimum-weight 2-edge-connected or biconnected spanning subgraph in planar graphs. We also design approximation schemes for the minimum-weight 1-2-connectivity problem, which is the variant of the survivable network design problem where vertices have 1 or 2 connectivity constraints. Prior to our work, for all these problems no polynomial or quasi-polynomial time algorithms were known to achieve an approximation ratio better than 2.

Polynomial-Time Approximation Schemes for the Euclidean Survivable Network Design Problem

The survivable network design problem is a classical problem in combinatorial optimization of constructing a minimum-cost subgraph satisfying predetermined connectivity requirements. In this paper we consider its geometric version in which the input is a complete Euclidean graph. We assume that each vertex v has been assigned a connectivity requirement rv. The output subgraph is supposed to have the vertex- (or edge-, respectively) connectivity of at least min{rv, ru} for any pair of vertices v, u.We present the first polynomial-time approximation schemes (PTAS) for basic variants of the survivable network design problem in Euclidean graphs. We first show a PTAS for the Steiner tree problem, which is the survivable network design problem with rv ? {0, 1} for any vertex v. Then, we extend it to include the most widely applied case where rv ? {0, 1, 2} for any vertex v. Our polynomial-time approximation schemeswork for both vertex- and edge-connectivity requirements in time O(n log n), where the constants depend on the dimension and the accuracy of approximation. Finally, we observe that our techniques yield also a PTAS for the multigraph variant of the problem where the edge-connectivity requirements satisfy rv ? {0, 1, . . . , k} and k = O(1).

Minimizing sum of completion times and makespan in master-slave systems

We consider scheduling problems in the master-slave model. In this model, each job has to be processed sequentially in three stages. In the first stage, a preprocessing task runs on a master machine, in the second stage, a slave task runs on a dedicated slave machine, and, in the last stage, a postprocessing task again runs on a master machine, possibly different from the master machine in the first stage. It has been shown that the problem of minimizing the makespan or the sum of completion times is NP-hard in the strong sense even if preemption is allowed. In this paper, we design efficient approximation algorithms to minimize the sum of completion times in various settings. These are the first general results for the minsum problem in the master-slave model. We also show that these algorithms generate schedules with small makespan as well

Minimizing mean flowtime and makespan on master--slave systems

The master-slave scheduling model is a new model recently introduced by Sahni. It has many important applications in parallel computer scheduling and industrial settings such as semiconductor testing, machine scheduling, etc. In this model each job is associated with a preprocessing task, a slave task and a postprocessing task that must be executed in this order. While the preprocessing and postprocessing tasks are scheduled on the master machine, the slave tasks are scheduled on the slave machines. In this paper, we consider scheduling problems on single-master master-slave systems. We first strengthen some previously known complexity results for makespan problems, by showing them to be strongly NP-hard. We then show that the problem of minimizing the mean flowtime is strongly NP-hard even under severe constraints. Finally, we propose some heuristics for the mean flowtime and makespan problems subject to some constraints, and we analyze the worst-case performance of these heuristics.

Total completion time minimization on multiple machines subject to machine availability and makespan constraints

This paper studies preemptive bi-criteria scheduling on m parallel machines with machine unavailable intervals. The goal is to minimize the total completion time subject to the constraint that the makespan is at most a constant T. We study the unavailability model such that the number of available machines cannot go down by 2 within any period of pmax where pmax is the maximum processing time among all jobs. We show that there is an optimal polynomial time algorithm.

Minimizing total completion time in two-machine flow shops with exact delays

We study the problem of minimizing total completion time in the two-machine flow shop with exact delay model. This problem is a generalization of the no-wait flow shop problem which is known to be strongly NP-hard. Our problem has many applications but little results are given in the literature so far. We focus on permutation schedules. We first prove that some simple algorithms can be used to find the optimal schedules for some special cases. Then for the general case, we design some heuristics as well as metaheuristics whose performance are shown to be effective by computational experiments.