An exact algorithm for scheduling identical coupled tasks

The coupled task problem is to schedule n jobs on one machine where each job consists of two subtasks with required delay time between them. The objective is to minimize the makespan. This problem was analyzed in depth by Orman and Potts [3]. They investigated the complexity of different cases depending on the lengths a_i and b_i of the two subtasks and the delay time L_i. -hardness proofs or polynomial algorithms were given for all cases except for the one where a_i=a, b_i=b and L_i=L. In this paper we present an exact algorithm for this problem with time complexity O(nr^2L) where holds. Therefore the algorithm is linear in the number of jobs for fixed L.Acknowledgements. The authors are grateful to Hans Kellerer who called their attention to this problem.Research was supported by DAAD exchange program 324 PPP-Ungarn.     

Construction of a Jacobi matrix from spectral data

A proof is given for the existence and uniqueness of a correspondence between two pairs of sequences {a},{b} and {ω},{μ}, satisfying b_i>0 for i=1,…,n?1 and ω₁<μ₁<?<μ_n?1<ω_n, under which the symmetric Jacobi matrices J(n,a,b) and J(n?1,a,b) have eigenvalues {ω} and {μ} respectively. Here J(m,a,b) is symmetric and tridiagonal with diagonal elements a_i (i=1,…,m) and off diagonal elements b_i (i=1,…,m?1). A new concise proof is given for the known uniqueness result. The existence result is new.     

Quasi sure quadratic variation of local times of smooth semimartingales

In this paper, we prove that the process of the quadratic variation of local times of smooth semimartingales can be constructed as the quasi sure limit of the form ∑_Δn(L_t^a_i+1n−L_t^a_in)², where Δ_n=(a_iⁿ,a_i+1ⁿ) is a sequence of subdivisions of [a,b], a_iⁿ=i(b−a)/2ⁿ+a, i=0,1,…,2ⁿ.     

Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u₁, u₂, . . . , u_k} together with a "size" v_i ≡ v(u_i) ∈ Z⁺, such that v_i ≠ v_j for i ≠ j, a "frequency" a_i ≡ a(u_i) ∈ Z⁺, and a positive integer (shelf length) L ∈ Z⁺ with the following conditions: (i) L = ∏ⁿ_j=1p_j(p_j ∈ Z⁺ ∀j, p_j ≠ p_l for j ≠ l) and v_i = ∏ _j∈Aip_j, A_i ⊆ {l, 2, . . . , n} for i = 1, . . . , n; (ii) (A_i\{⋂^k_j=1A_j}) ∩ (A_l\{⋂^k_j=1A_j}) = ⊘∀i ≠ l. Note that v_i|L (divides L) for each i. If for a given m ∈ Z⁺, ∑ⁿ_i=1a_iv_i = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b₁₁, b₁₂, . . . , b_1m, b₂₁, . . . , b_n1, . . . , b_nm}⊆ N such that ∑^m_j=1b_ij = a_i, i = 1, . . . , k, and ∑^k_i=1b_ijv_i = L, j =1, . . . , m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.     

Approximation of functions of finite variation by superpositions of a Sigmoidal function

The aim of this note is to generalize a result of Barron [1] concerning the approximation of functions, which can be expressed in terms of the Fourier transform, by superpositions of a fixed sigmoidal function. In particular, we consider functions of the type h(x) = ∫ℝ^d ƒ (〈t, x〉)dμ(t), where μ is a finite Radon measure on ℝ^d and ƒ : ℝ → ℂ is a continuous function with bounded variation in ℝ We show (Theorem 2.6) that these functions can be approximated in L₂-norm by elements of the set G_n = {Σ_i=0^staggeredn c_ig(〈a_i, x〉 + b_i) : a_iℝ^d, b_i, c_iℝ}, where g is a fixed sigmoidal function, with the error estimated by C/n^1/2, where C is a positive constant depending only on f. The same result holds true (Theorem 2.9) for f : ℝ → ℂ satisfying the Lipschitz condition under an additional assumption that ∫ℝ^d6t6e^d|u(t)| > ∞     

Approximation schemes for parallel machine scheduling with availability constraints

We investigate the problems of scheduling n weighted jobs to m parallel machines with availability constraints. We consider two different models of availability constraints: the preventive model, in which the unavailability is due to preventive machine maintenance, and the fixed job model, in which the unavailability is due to a priori assignment of some of the n jobs to certain machines at certain times. Both models have applications such as turnaround scheduling or overlay computing. In both models, the objective is to minimize the total weighted completion time. We assume that m is a constant, and that the jobs are non-resumable.For the preventive model, it has been shown that there is no approximation algorithm if all machines have unavailable intervals even if w_i=p_i for all jobs. In this paper, we assume that there is one machine that is permanently available and that the processing time of each job is equal to its weight for all jobs. We develop the first polynomial-time approximation scheme (PTAS) when there is a constant number of unavailable intervals. One main feature of our algorithm is that the classification of large and small jobs is with respect to each individual interval, and thus not fixed. This classification allows us (1) to enumerate the assignments of large jobs efficiently; and (2) to move small jobs around without increasing the objective value too much, and thus derive our PTAS. Next, we show that there is no fully polynomial-time approximation scheme (FPTAS) in this case unless P=NP.For the fixed job model, it has been shown that if job weights are arbitrary then there is no constant approximation for a single machine with 2 fixed jobs or for two machines with one fixed job on each machine, unless P=NP. In this paper, we assume that the weight of a job is the same as its processing time for all jobs. We show that the PTAS for the preventive model can be extended to solve this problem when the number of fixed jobs and the number of machines are both constants.     

A class of Fibonacci-type sequences

Let {L_n:n ? 1} be a sequence of the form

where {a_j} and {b_j} are positive integers, and e=max_i,j{a_i, b_j}. A necessary and sufficient condition on the integers {a_j} and {b_j} is given so that, for all choices of positive initial values L₁, L₂,…,L_e,L_n=Σ^p_j=1L_n?aj for all large enough n.     

The spectrum of a finite pseudocomplemented lattice

Let L be a finite pseudocomplemented lattice. Every interval [0, a] in L is pseudocomplemented, so by Glivenko’s theorem, the set S(a) of all pseudocomplements in [0, a] forms a boolean lattice. Let B _i denote the finite boolean lattice with i atoms. We describe all sequences (s ₀, s ₁, . . . , s _n ) of integers for which there exists a finite pseudocomplemented lattice L with s _i = |{ a ∈ L | S(a) ? B _i }|, for all i, and there is no a ∈ L with S(a) ? B _n+1. This result settles a problem raised by the first author in 1971.     

Indecomposable modules of the intermediate series over W(a, b)

For any complex parameters a and b,W(a,b)is the Lie algebra with basis{Li,Wi|i∈Z}and relations[Li,Lj]=(j i)Li+j,[Li,Wj]=(a+j+bi)Wi+j,[Wi,Wj]=0.In this paper,indecomposable modules of the intermediate series over W(a,b)are classified.It is also proved that an irreducible Harish-Chandra W(a,b)-module is either a highest/lowest weight module or a uniformly bounded module.Furthermore,if a∈/Q,an irreducible weight W(a,b)-module is simply a Vir-module with trivial actions of Wk’s.     

On constant discrete programming problems

Let H be a subset of the set S_n of all permutations

$\text{1}\text{2}\text{???}\text{n}\text{s(1)}\text{s(2)}\text{???}\text{s(n)}$

C=6c_ij6 a real n?n matrix L_c(s)=c_1s(1)+c_2s(2)+???+c_ns(n) for s ? H. A pair (H, C) is the existencee of reals a₁,b₁,a₂,b₂,…a_n,b_n, for which c_ij=a₁+b_j if (i,j)?D(H), where D(H)={(i,j):(?h?H)(j=h(i))}.For a pair (H,C) the specifity of it is proved in the case, when H is either a special cyclic class of permutations or a special union of cyclic classes. Specific pairs with minimal sets H are in some sense described.     

Polynomial time approximation schemes for general multiprocessor job shop scheduling

We study preemptive and non-preemptive versions of the general multiprocessor job shop scheduling problem: Given a set of n tasks each consisting of at most μ ordered operations that can be processed on different (possibly all) subsets of m machines with different processing times, compute a schedule (preemptive or non-preemptive, depending on the model) with minimum makespan where operations belonging to the same task have to be scheduled according to the specified order. We propose algorithms for both preemptive and non-preemptive variants of this problem that compute approximate solutions of any positive ε accuracy and run in O(n) time for any fixed values of m, μ, and ε. These results include (as special cases) many recent developments on polynomial time approximation schemes for scheduling jobs on unrelated machines, multiprocessor tasks, and classical open, flow and job shops.     

Probabilities of dominant candidates based on first-place votes

Suppose each of an odd number n of voters has a strict preference order on the three ‘candidates’ in {1,2,3} and votes for his most preferred candidate on a plurality ballot. Assume that a voter who votes for i is equally likely to have ijk and ikj as his preference order when {i,j,k} = {1,2,3}.Fix an integer m between $\text{1}\text{2}$(n + 1) and n inclusive. Then, given that n_i of the n voters vote for i, let f_m(n₁,n₂,n₃) be the probability that one of the three candidates is preferred by m or more voters to each of the other two.This paper examines the behavior of f_m over the lattice points in L_n, the set of triples of non-negative integers that sum to n. It identifies the regions in L_n where f_m is 1 and where f_m is 0, then shows that f_m(a,b + 1, c)>f_m(a + 1,b,c) whenever a + b + c + 1 = n, a≤c≤b, a<c<m and c≤n ? m. These results are used to partially identify the points in L_n where f_m is minimized subject to f_m>0. It is shown that at least two of the n_i are equal at minimizing points.     

A note on lattice chains and Delannoy numbers

Fix nonnegative integers n₁,…,n_d and let L denote the lattice of integer points (a₁,…,a_d)∈Z^d satisfying 0?a_i?n_i for 1?i?d. Let L be partially ordered by the usual dominance ordering. In this paper we offer combinatorial derivations of a number of results concerning chains in L. In particular, the results obtained are established without recourse to generating functions or recurrence relations. We begin with an elementary derivation of the number of chains in L of a given size, from which one can deduce the classical expression for the total number of chains in L. Then we derive a second, alternative, expression for the total number of chains in L when d=2. Setting n₁=n₂ in this expression yields a new proof of a result of Stanley [Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999] relating the total number of chains to the central Delannoy numbers. We also conjecture a generalization of Stanley's result to higher dimensions.     

Representations of lattices via neutral elements

For any neutral element a in a bounded latticeL, the mapping x→(x∧,x∨a) representsL as a subdirect product of [0, a]×[a, 1]. It is first shown that for certain neutral elements, the image ofL under this mapping is completely determined by a homomorphism of [0, a] into [a, 1]. Iterating this process, a representation ofL can be obtained as a subdirect product of the intervals [a_i, a_i+1] for any chain 0=a₀1... nn+1=1 where each a_i is such a neutral element relative to [0, a_i+1]. The image in this case is completely determined by a family of homomorphisms π_i,j:A_i →A_j(ii=[a_i, a_i+1].     

The inverse M-matrix problem

One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=(a_ij) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ? a_ij ? x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A^?1=B=(b_ij), then b_ii> 0 and b_ij ? 0 for i≠_j. If n=2 or x=y no further conditions are needed, but if n ? 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x²=sy+(1?s)y²; then B is an M-matrix if s^?1 ? n?2. Moreover, if all off-diagonal elements of A have the value y except for a_ij=a_jj=x when i=n?1, n and 1 ? _j ? n?2, then the condition on both necessary and sufficient for B to be an M-matrix.     

Strip generation algorithms for constrained two-dimensional two-staged cutting problems

The constrained two-dimensional cutting (C_TDC) problem consists of determining a cutting pattern of a set of n small rectangular piece types on a rectangular stock plate of length L and width W, as to maximize the sum of the profits of the pieces to be cut. Each piece type i, i = 1, …, n, is characterized by a length l_i, a width w_i, a profit (or weight) c_i and an upper demand value b_i. The upper demand value is the maximum number of pieces of type i which can be cut on rectangle (L, W). In this paper, we study the two-staged fixed orientation C_TDC, noted FC_2TDC. It is a classical variant of the C_TDC where each piece is produced, in the final cutting pattern, by at most two guillotine cuts, and each piece has a fixed orientation. We solve the FC_2TDC problem using several approximate algorithms, that are mainly based upon a strip generation procedure. We evaluate the performance of these algorithms on instances extracted from the literature.     

On the mean square of the error term for the asymmetric two-dimensional divisor problem (I)

Suppose a and b are two fixed positive integers such that (a, b) = 1. In this paper we shall establish an asymptotic formula for the mean square of the error term Δ _a,b (x) of the general two-dimensional divisor problem.     

On a partition problem of Frobenius

We consider the following problem, which was raised by Frobenius: Given n relatively prime positive integers, what is the largest integer M(a₁, a₂, …, a_n) omitted by the linear form Σ_i=1ⁿa_ix_i, where the x_i are variable nonnegative integers. We give the solution for certain special cases when n = 3.     

Optimal schedulings with gaps for independent jobs in a service system withN servers

One considers the problem of forming the optimal schedulings with gaps for a service system withN identical parallel processors. In the service one performsK jobs, each of which consists of V_i homogeneous independent operations and has lower and upper directive times d_i and D_i. For the operations which constitute the jobs, one considers linear penalty functions outside the interval [d_i,D_i]. One solves the problem of finding the schedulings with a minimal total penalty and having the origin in a given interval [t₁,t₂]. It is proved that for an arbitrary set Z of jobs, the penalty function F_Z(t), where t is the origin of the scheduling, has a unique minimum for t∈(?∞,+∞). We present an algorithm for the construction of the optimal scheduling requiring \(C \cdot K\left( {\mathop {\max }\limits_i \left\{ {D_i } \right\} - \mathop {\min }\limits_i \left\{ {d_i } \right\} + \sum\limits_1^\kappa {V_i } } \right)\) operations on an electronic computer.