Online scheduling with partial job values: Does timesharing or randomization help?

We study the following online preemptive scheduling problem: given a set of jobs with release times, deadlines, processing times and weights, schedule them so as to maximize the total value obtained. Unlike traditional scheduling problems, partially completed jobs can get partial values proportional to their amounts processed. Recently Chrobak et al. gave improved lower and upper bounds [1.236, 1.8] on the competitive ratio for this problem, the upper bound being achieved by using timesharing to simulate two equal-speed processors. In this paper we (1) give a new algorithm MIXED-κ with competitive ratio 1/(1 - (κ/(κ + 1))κ) which approaches e/(e-1) ≈ 1.582 when κ → ∞, by using timesharing to simulate κ equal-speed processors; (2) give an equivalent but much more practical algorithm MIX, which is e/(e - 1)-competitive (independent of κ), by timesharing the processor with different speeds (depending on the job weights), and use its interesting properties to devise an efficient implementation; (3) improve the lower bound to 1.25 by showing an identical lower bound for randomized algorithms; and (4) prove a lower bound of 1.618 on the competitive ratio when timesharing is not allowed, thus answering an open problem raised by Chang and Yap, showing that timesharing provably helps in giving better algorithms for this problem.postprin

Approximation for minimum triangulation of convex polyhedra

The minimum triangulation of a convex polyhedron is a triangulation that contains the minimum number of tetrahedra over all its possible triangulations. Since finding the minimum triangulation of convex polyhedra was recently shown to be NP-hard, it becomes significant to find algorithms that give good approximation. In this paper, we give a new triangulation algorithm with an improved approximation ratio 2 - &OHgr;(l/√). We also show that this is best possible for algorithms that only consider the combinatorial structure of the polyhedra. Copyright © 2009 ACM, Inc.published_or_final_versio

Linear-time haplotype inference on pedigrees without recombinations and mating loops

In this paper, an optimal linear-time algorithm is presented to solve the haplotype inference problem for pedigree data when there are no recombinations and the pedigree has no mating loops. The approach is based on the use of graphs to capture SNP, Mendelian, and parity constraints of the given pedigree. This representation allows us to capture the constraints as the edges in a graph, rather than as a system of linear equations as in previous approaches. Graph traversals are then used to resolve the parity of these edges, resulting in an optimal running time. © 2009 Society for Industrial and Applied Mathematics.published_or_final_versio

Improved competitive algorithms for online scheduling with partial job values

This paper considers an online scheduling problem arising from Quality-of-Service (QoS) applications. We are required to schedule a set of jobs, each with release time, deadline, processing time and weight. The objective is to maximize the total value obtained for scheduling the jobs. Unlike the traditional model of this scheduling problem, in our model unfinished jobs also get partial values proportional to their amounts processed. No non-timesharing algorithm for this problem with competitive ratio better than 2 is known. We give a new non-timesharing algorithm GAP that improves this ratio for bounded values of m, where m can be the number of concurrent jobs or the number of weight classes. The competitive ratio is improved from 2 to 1.618 (golden ratio) which is optimal for m = 2, and when applied to cases with m > 2 it still gives a competitive ratio better than 2, e.g. 1.755 when m = 3. We also give a new study of the problem in the multiprocessor setting, giving an upper bound of 2 and a lower bound of 1.25 for the competitiveness. Finally, we consider resource augmentation and show that O(log α) speedup or extra processors is sufficient to achieve optimality, where x is the importance ratio. We also give a tradeoff result, showing that in fact a small amount of extra resources is sufficient for achieving close-to-optimal competitiveness. © 2004 Elsevier B.V. All rights reserved.link_to_subscribed_fulltex

Approximation of minimum triangulation for polyhedron with bounded degrees

Finding minimum triangulations of convex polyhedra is NPhard. The best approximation algorithms only give a ratio 2 for this problem, and for combinatorial algorithms it is shown to be the best possible asymptotically. In this paper we improve the approximation ratio of finding minimum triangulations for some special classes of 3-dimensional convex polyhedra. (1) For polyhedra without 3-cycles and degree-4 vertices we achieve a tight approximation ratio 3/2. (2) For polyhedra with vertices of degree-5 or above, we achieve an upper bound 2 - 1/12 on the approximation ratio. (3) For polyhedra with n vertices and vertex degrees bounded by a constant Δ we achieve an asymptotic tight ratio 2 - ω (1/Δ) - ω (1/n). © 2001 Springer Berlin Heidelberg.link_to_subscribed_fulltex

Approximating the minimum triangulation of convex 3-polytopes with bounded degrees

Finding minimum triangulations of convex 3-polytopes is NP-hard. The best approximation algorithms only give an approximation ratio of 2 for this problem, which is the best possible asymptotically when only combinatorial structures of the polytopes are considered. In this paper we improve the approximation ratio of finding minimum triangulations for some special classes of 3-dimensional convex polytopes. (1) For polytopes without 3-cycles and degree-4 vertices we achieve a tight approximation ratio of 3/2. (2) For polytopes where all vertices have degrees at least 5, we achieve an upper bound of 2-112 on the approximation ratio. (3) For polytopes with n vertices and vertex degrees bounded above by Δ we achieve an asymptotic tight ratio of 2-Ω(1/Δ)-Ω(Δ/n). When Δ is constant the ratio can be shown to be at most 2-2/(Δ+1). © 2005 Elsevier B.V. All rights reserved.link_to_subscribed_fulltex

Laxity helps in broadcast scheduling

We study the effect of laxity, or slack time, on the online scheduling of broadcasts with deadlines. The laxity of a request is defined to be the ratio between its span (difference between release time and deadline) and its processing time. All requests have a minimum guaranteed laxity. We give different algorithms and lower bounds on the competitive ratio for different ranges of values of laxity, which not only represents a tradeoff between the laxity and the competitive ratio of the system, but also bridges between interval scheduling and job scheduling techniques and results. We also give an improved algorithm for general instances in the case when requests can have different processing times. © Springer-Verlag Berlin Heidelberg 2005.link_to_subscribed_fulltex

Approximation for minimum triangulations of simplicial convex 3-polytopes

A minimum triangulation of a convex 3-polytope is a triangulation that contains the minimum number of tetrahedra over all its possible triangulations. Since finding minimum triangulations of convex 3-polytopes was recently shown to be NP-hard, it becomes significant to find algorithms that give good approximation. In this paper we give a new triangulation algorithm with an improved approximation ratio 2 - Ω(1/√n), where n is the number of vertices of the polytope. We further show that this is the best possible for algorithms that only consider the combinatorial structure of the polytopes.link_to_subscribed_fulltex