Greedy multiprocessor server scheduling

We show that the greedy highest density first (HDF) algorithm is (1+ε)-speed O(1)-competitive for the problem of minimizing the ?_p norms of weighted flow time on m identical machines. Similar results for minimizing unweighted flow provide insight into the power of migration.     

Competitive Algorithms for Relaxed List Update and Multilevel Caching

We study relaxed list update problem (RLUP), in which access requests are made to items stored in a list. The cost to access the jth item x_j is c_j, where c_i ≤ c_{i + 1} for all i. After the access, x_j can be repeatedly swapped, at no cost, with any item that precedes it in the list. This problem was introduced by Aggarwal et al. (1987, “Proc. 19th Symp. Theory of Computing,” pp. 305–313) as a model for the management of hierarchical memory that consists of a number of caches of increasing size and access time. They also proved that a version of LRU is C-competitive, for some C, for a restricted class of cost functions. We give an efficient offline algorithm that computes the optimal strategy for RLUP. We also show an elegant characterization of work functions for RLUP. We prove that move-to-front (MTF) is optimally competitive for RLUP with any cost function. An interesting feature of the proof is that it does not involve any estimates on the competitive ratio. Finally, we give a lower bound on the competitive ratio of online algorithms for RLUP.     

Online competitive algorithms for maximizing weighted throughput of unit jobs

We study an online unit-job scheduling problem arising in buffer management. Each job is specified by its release time, deadline, and a nonnegative weight. Due to overloading conditions, some jobs have to be dropped. The goal is to maximize the total weight of scheduled jobs. We present several competitive online algorithms for various versions of unit-job scheduling, as well as some lower bounds on the competitive ratios.We first give a randomized algorithm RMix with competitive ratio of e/(e−1)≈1.582. This is the first algorithm for this problem with competitive ratio smaller than 2.Then we consider s-bounded instances, where the span of each job (deadline minus release time) is at most s. We give a 1.25-competitive randomized algorithm for 2-bounded instances, matching the known lower bound. We also give a deterministic algorithm Edf_α, whose competitive ratio on s-bounded instances is 2−2/s+o(1/s). For 3-bounded instances its ratio is ≈1.618, matching the known lower bound.In s-uniform instances, the span of each job is exactly s. We show that no randomized algorithm can be better than 1.25-competitive on s-uniform instances, if the span s is unbounded. For s=2, our proof gives a lower bound of . Also, in the 2-uniform case, we prove a lower bound of for deterministic memoryless algorithms, matching a known upper bound.Finally, we investigate the multiprocessor case and give a -competitive algorithm for m processors. We also show improved lower bounds for the general and s-uniform cases.     

Batching identical jobs

We study the problems of scheduling jobs, with different release dates and equal processing times, on two types of batching machines. All jobs of the same batch start and are completed simultaneously. On a serial batching machine, the length of a batch equals the sum of the processing times of its jobs and, when a new batch starts, a constant setup time s occurs. On a parallel batching machine, there are at most b jobs per batch and the length of a batch is the largest processing time of its jobs. We show that in both environments, for a large class of so called “ordered” objective functions, the problems are polynomially solvable by dynamic programming. This allows us to derive that the problems where the objective is to minimize the weighted number of late jobs, or the weighted flow time, or the total tardiness, or the maximal tardiness are polynomial. In other words, we show that 1|p-batch,b<n, r _i, p _i=p|F and 1|s-batch, r _i, p _i=p|F, are polynomial for F∈{∑w _i U _i,∑w _i C _i,∑T _i, T _max}. The complexity status of these problems was unknown before.     

Online C-benevolent job scheduling on multiple machines

We consider scheduling a sequence of C-benevolent jobs on multiple homogeneous machines. For two machines, we propose a 2-competitive Cooperative Greedy algorithm and provide a lower bound of 2 for the competitive ratio of any deterministic online scheduling algorithms on two machines. For multiple machines, we propose a Pairing-m Greedy algorithm, which is deterministic 2-competitive for even number of machines and randomized \((2+2/{\hbox {m}})\)-competitive for odd number of machines. We provide a lower bound of 1.436 for the competitive ratio of any deterministic online scheduling algorithms on three machines, which is the best known lower bound for competitive ratios of deterministic scheduling algorithms on three machines.     

Extensions to online delay management on a single train line: new bounds for delay minimization and profit maximization

We present extensions to the Online Delay Management Problem on a Single Train Line. While a train travels along the line, it learns at each station how many of the passengers wanting to board the train have a delay of δ. If the train does not wait for them, they get delayed even more since they have to wait for the next train. Otherwise, the train waits and those passengers who were on time are delayed by δ. The problem consists in deciding when to wait in order to minimize the total delay of all passengers on the train line. We provide an improved lower bound on the competitive ratio of any deterministic online algorithm solving the problem using game tree evaluation. For the extension of the original model to two possible passenger delays δ ₁ and δ ₂, we present a 3-competitive deterministic online algorithm. Moreover, we study an objective function modeling the refund system of the German national railway company, which pays passengers with a delay of at least Δ a part of their ticket price back. In this setting, the aim is to maximize the profit. We show that there cannot be a deterministic competitive online algorithm for this problem and present a 2-competitive randomized algorithm.     

Hamilton cycles,avoiding prescribed arcs,in close-to-regular tournaments

The local irregularity of a digraph D is defined as i_l(D) = max {|d⁺ (x) − d⁻ (x)| : x ϵ V(D)}. Let T be a tournament, let Γ = {V₁, V₂, …, V_c} be a partition of V(T) such that |V₁| ≥ |V₂| ≥ … ≥ |V_c|, and let D be the multipartite tournament obtained by deleting all the arcs with both end points in the same set in Γ. We prove that, if |V(T)| ≥ max{2i_l(T) + 2|V₁| + 2|V₂| − 2, i_l(T) + 3|V₁| − 1}, then D is Hamiltonian. Furthermore, if T is regular (i.e., i_l(T) = 0), then we state slightly better lower bounds for |V(T)| such that we still can guarantee that D is Hamiltonian. Finally, we show that our results are best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 123–136, 1999     

New dominance rules and exploration strategies for?the?1|ri|∑Ui scheduling problem

The paper proposes a new exact approach, based on a Branch, Bound, and Remember (BB&R) algorithm that uses the Cyclic Best First Search (CBFS) strategy, for the 1|r _i |∑U _i scheduling problem, a single machine scheduling problem, where the objective is to find a schedule with the minimum number of tardy jobs. The search space is reduced using new and improved dominance properties and tighter upper bounds, based on a new dynamic programming algorithm. Computational results establish the effectiveness of the BB&R algorithm with CBFS for a broad spectrum of problem instances. In particular, this algorithm was able to solve all problems instances, up to 300 jobs, while existing best known algorithms only solve problems instances up to 200 jobs. Furthermore, the BB&R algorithm with CBFS runs one to two orders of magnitude faster than the current best known algorithm on comparable instances.     

Scheduling uniform machines on-line requires nondecreasing speed ratios

We consider the following on-line scheduling problem. We have to schedulen independent jobs, wheren is unknown, onm uniform parallel machines so as to minimize the makespan; preemption is allowed. Each job becomes available at its release date, and this release date is not known beforehand; its processing requirement becomes known at its arrival. We show that if only a finite number of preemptions is allowed, there exists an algorithm that solves the problem if and only ifs _i–1/s_i s_i/s_i+1 for alli = 2,,m – 1, wheres _i denotes theith largest machine speed. We also show that if this condition is satisfied, then O(mn) preemptions are necessary, and we provide an example to show that this bound is tight. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

An approximation algorithm for the generalized assignment problem

The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing jobj on machinei requires timep _ij and incurs a cost ofc _ij ; each machinei is available forT _i time units, and the objective is to minimize the total cost incurred. Our main result is as follows. There is a polynomial-time algorithm that, given a valueC, either proves that no feasible schedule of costC exists, or else finds a schedule of cost at mostC where each machinei is used for at most 2T _i time units.We also extend this result to a variant of the problem where, instead of a fixed processing timep _ij , there is a range of possible processing times for each machine—job pair, and the cost linearly increases as the processing time decreases. We show that these results imply a polynomial-time 2-approximation algorithm to minimize a weighted sum of the cost and the makespan, i.e., the maximum job completion time. We also consider the objective of minimizing the mean job completion time. We show that there is a polynomial-time algorithm that, given valuesM andT, either proves that no schedule of mean job completion timeM and makespanT exists, or else finds a schedule of mean job completion time at mostM and makespan at most 2T. Research partially supported by an NSF PYI award CCR-89-96272 with matching support from UPS, and Sun Microsystems, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.Research supported in part by a Packard Fellowship, a Sloan Fellowship, an NSF PYI award, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

A note on an optimal choice problem

An item has probabilityp ₀ of not being workable. Before, going into use it has to be controlled by some deliberately chosen checksC _i which costsc _i 0,1in. Total checking costs must be not larger thanc ₀>0. It may happen that a check breaks a workable item, and failures may be overlooked. The problem is to determine an optimal sequence of checks subject to the cost constraints, such that the probability is maximized that an item leaving the checks is workable. In [1] this problem is solved by W. Stadje, but numerically the solution method only is applicable to problems of modest size.In this paper a simple reformulation of the original problem is presented. This first allows a simpler derivation of some of the results in [1]. Further, a dynamic programming algorithm is presented, which is pseudopolynomial, if costsc _i are integers. It then requiresO(n·c ₀) time.     

Energetic reasoning and bin-packing problem, for bounding a parallel machine scheduling problem

This paper deals with an extension of energetic reasoning, using some efficient lower bounds of the bin-packing problem, to get tight lower bounds for the P|r _i , q _i |C _max. The link between P||C _max and bin-packing problem is well-known. Our purpose is to extend the use of efficient lower bounds of the bin-packing problem to P|r _i , q _i |C _max. We focus on some time-intervals, to compute the mandatory parts of activities within this time-interval and then to deduce an associated bin-packing instance. Thus, lower bounds of the bin-packing problem are used to get new satisfiability tests for the parallel machine problem. We also propose to extend the classical time-bound adjustments of release dates and deadlines to efficiently use bin-packing lower bounds. Experimental results that prove the efficiency of our approach on several kind of instances are reported.     

h‐monotonically computable real numbers

Let h : ? → ? be a computable function. A real number x is called h‐monotonically computable (h‐mc, for short) if there is a computable sequence (x_s) of rational numbers which converges to x h‐monotonically in the sense that h(n)|x – x_n| ≥ |x – x_m| for all n andm > n. In this paper we investigate classes h‐ MC of h‐mc real numbers for different computable functions h. Especially, for computable functions h : ? → (0, 1)_?, we show that the class h‐ MC coincides with the classes of computable and semi‐computable real numbers if and only if Σ_i∈?(1 – h(i)) = ∞and the sum Σ_i∈?(1 – h(i)) is a computable real number, respectively. On the other hand, if h(n) ≥ 1 and h converges to 1, then h‐ MC = SC (the class of semi‐computable reals) no matter how fast h converges to 1. Furthermore, for any constant c > 1, if h is increasing and converges to c, then h‐ MC = c‐ MC . Finally, if h is monotone and unbounded, then h‐ MC contains all ω‐mc real numbers which are g‐mc for some computable function g. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Divisibility and structure constraints on automorphism groups of almost perfect 1-factorizations of

Let G be a permutation group acting on a set with N elements such that every permutation with more than m fixed points is the identity. It is easy to verify that |G| divides N(N − 1) ··· (N − m). We show that if gcd(|G|, m!) = 1, then |G| divides (N − i)(N − j) for some i and j satisfying 0 ≤ i < j ≤ m. We use this to show that any almost perfect 1-factorization of K_2n has an automorphism group whose cardinality divides (2n − i)(2n − j) for some i and j with 0 ≤ i < j ≤ 2 as long as n is odd. An almost perfect 1-factorization (or APOF) is a 1-factorization in which the union of any three distinct 1-factors is connected. This result contrasts with an example of an APOF on K₁₂ given by Cameron which has PSL(2, ℤ₁₁) as its automorphism group [with cardinality 12(11)(5)]. When n is even and the automorphism group is solvable, we show that either G acts vertex transitively and n is a power of two, or |G| divides 2n − 2^a for some integer a with 2^a dividing 2n, or else |G| divides (2n − i)(2n − j) for some i and j with 0 ≤ i < j ≤ 2. We also give a number of structure results concerning these automorphism groups. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 355–380, 1998     

How Close to Regular Must a Semicomplete Multipartite Digraph Be to Secure Hamiltonicity?

 Let D be a semicomplete multipartite digraph, with partite sets V ₁, V ₂,…, V _c, such that |V ₁|≤|V ₂|≤…≤|V _c|. Define f(D)=|V(D)|−3|V _c|+1 and . We define the irregularity i(D) of D to be max|d ⁺(x)−d ⁻(y)| over all vertices x and y of D (possibly x=y). We define the local irregularity i _l(D) of D to be max|d ⁺(x)−d ⁻(x)| over all vertices x of D and we define the global irregularity of D to be i _g(D)=max{d ⁺(x),d ⁻(x) : x∈V(D)}−min{d ⁺(y),d ⁻(y) : y∈V(D)}. In this paper we show that if i _g(D)≤g(D) or if i _l(D)≤min{f(D), g(D)} then D is Hamiltonian. We furthermore show how this implies a theorem which generalizes two results by Volkmann and solves a stated problem and a conjecture from [6]. Our result also gives support to the conjecture from [6] that all diregular c-partite tournaments (c≥4) are pancyclic, and it is used in [9], which proves this conjecture for all c≥5. Finally we show that our result in some sense is best possible, by giving an infinite class of non-Hamiltonian semicomplete multipartite digraphs, D, with i _g(D)=i(D)=i _l(D)=g(D)+?≤f(D)+1. Revised: September 17, 1998     

Pareto optima for total weighted completion time and maximum lateness on a single machine

We consider the single-machine bicriterion scheduling problem of enumerating the Pareto-optimal sequences with respect to the total weighted completion time and the maximum lateness objectives. We show that the master sequence concept originally introduced for 1|r_j|∑w_jU_j by Dauzère-Pérès and Sevaux is also applicable to our problem and a large number of other sequencing problems. Our unified development is based on exploiting common order-theoretic structures present in all these problems. We also show that the master sequence implies the existence of global dominance orders for these scheduling problems. These dominance results were incorporated into a new branch and bound algorithm, which was able to enumerate all the Pareto optima for over 90% of the 1440 randomly generated problems with up to n=50 jobs. The identification of each Pareto optimum implicitly requires the optimal solution of a strongly NP-hard problem. The instances solved had hundreds of these Pareto solutions and to the best of our knowledge, this is the first algorithm capable of completely enumerating all Pareto sequences within reasonable time and space for a scheduling problem with such a large number of Pareto optima.     

Batch sizing and job sequencing on a single machine

We study a single-machine scheduling problem in which the items to be processed have to be batched as well as sequenced. Since processed items become available in batches, flow times are defined to be the same for all items in the same batch. A constant set-up delay is incurred between consecutive batches. For any fixed, but arbitrary item sequence, we present an algorithm that finds a sequence of batches such that the total flow time of the items is minimized; we prove that for a set ofn items, the algorithm runs inO(n) time. We show that, among all sequences, the one leading to the minimum flow time has the items in non-decreasing order of running times. Thus, the optimal algorithm for the combined problem, called thebatch-sizing problem, runs inO(n logn) time. We also prove that this algorithm yields an improved solution to a scheduling problem recently studied by Baker [1].     

Two algorithms for weighted matroid intersection

Consider a finite setE, a weight functionw:E→R, and two matroidsM ₁ andM ₂ defined onE. The weighted matroid intersection problem consists of finding a setI⊆E, independent in both matroids, that maximizes Σ{w(e):e inI}. We present an algorithm of complexity O(nr(r+c+logn)) for this problem, wheren=|E|,r=min(rank(M ₁), rank (M ₂)),c=max (c ₁,c ₂) and, fori=1,2,c _i is the complexity of finding the circuit ofI∪{e} inM _i (or show that none exists) wheree is inE andI⊆E is independent inM ₁ andM ₂. A related problem is to find a maximum weight set, independent in both matroids, and of given cardinalityk (if one exists). Our algorithm also solves this problem. In addition, we present a second algorithm that, given a feasible solution of cardinalityk, finds an optimal one of the same cardinality. A sensitivity analysis on the weights is easy to perform using this approach. Our two algorithms are related to existing algorithms. In fact, our framework provides new simple proofs of their validity. Other contributions of this paper are the existence of nonnegative reduced weights (Theorem 6), allowing the improved complexity bound, and the introduction of artificial elements, allowing an improved start and flexibility in the implementation of the algorithms. This research was supported in part by NSF grant ECS 8503192 to Carnegie-Mellon University.     

On number system constructions

We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ and expansive matrix M: Λ → Λ if ρ(M ⁻¹) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M ⁻¹. We shall prove further that if the polynomial f(x) = c ₀ + c ₁ x + ··· + c _k x ^k ∈ Z[x], c _k = 1 satisfies the condition |c ₀| > 2 Σ _i=1 ^k |c _i | then there is a suitable digit set D for which (Z ^k , M, D) is a number system, where M is the companion matrix of f(x). The research was supported by OTKA-T043657 and Bolyai Fellowship Committee.