Improved online algorithms for the batch scheduling of equal-length jobs with incompatible families to maximize the weighted number of early jobs

We consider the online scheduling of equal-length jobs with incompatible families on \(m\) identical batch machines. Each job has a release time, a deadline and a weight. Each batch machine can process up to \(b\) jobs (which come from the same family) simultaneously as a batch, where \(b\) is called the capacity of the machines. Our goal is to determine a preemption-restart schedule which maximizes the weighted number of early jobs. For this problem, Li et al. (Inf Process Lett 112:503–508, 2012) provided an online algorithm of competitive ratio \(3+2\sqrt{2}\) for both \(b=\infty \) and \(b<\infty \) . In this paper, we study two special cases of this problem. For the case that \(m=2\) , we first present a lower bound 2, and then provide an online algorithm with a competitive ratio of 3 for both \(b=\infty \) and \(b<\infty \) . For the case in which \(m=3\) , \(b=\infty \) and all jobs come from a common family, we present an online algorithm with a competitive ratio of \((8+2\sqrt{15})/3\approx 5.249\) .     

A constant-factor approximation algorithm for multi-vehicle collection for processing problem

We define the multiple-vehicle collection for processing problem (mCfPP) as a vehicle routing and scheduling problem in which items that accumulate at customer sites over time should be transferred by a series of tours to a processing facility. We show that this problem with the makespan objective (mCfPP( $C_{\max }$ )) is NP-hard using an approximation preserving reduction from a two-stage, hybrid flowshop scheduling problem. We develop a polynomial-time, constant-factor approximation algorithm to solve mCfPP( $C_{\max }$ ). The problem with a single site is analyzed as a special case with two purposes. First, we identify the minimum number of vehicles required to achieve a lower bound on the makespan, and second, we characterize the optimal makespan when a single vehicle is utilized.     

Preemptive and non-preemptive generalized min sum set cover

In the (non-preemptive) Generalized Min Sum Set Cover Problem, we are given $n$ ground elements and a collection of sets $\mathcal{S }= \{S_1, S_2, \ldots , S_m\}$ where each set $S_i \in 2^{[n]}$ has a positive requirement $\kappa (S_i)$ that has to be fulfilled. We would like to order all elements to minimize the total (weighted) cover time of all sets. The cover time of a set $S_i$ is defined as the first index $j$ in the ordering such that the first $j$ elements in the ordering contain $\kappa (S_i)$ elements in $S_i$ . This problem was introduced by Azar et al. (2009) with interesting motivations in web page ranking and broadcast scheduling. For this problem, constant approximations are known by Bansal et al. (2010) and Skutella and Williamson (Oper Res Lett 39(6):433–436, 2011). We study the version where preemption is allowed. The difference is that elements can be fractionally scheduled and a set $S$ is covered in the moment when $\kappa (S)$ amount of elements in $S$ are scheduled. We give a 2-approximation for this preemptive problem. Our linear programming relaxation and analysis are completely different from the aforementioned previous works. We also show that any preemptive solution can be transformed into a non-preemptive one by losing a factor of 6.2 in the objective function. As a byproduct, we obtain an improved $12.4$ -approximation for the non-preemptive problem.     

Supply chain scheduling to minimize holding costs with outsourcing

This paper addresses a scheduling problem in a flexible supply chain, in which the jobs can be either processed in house, or outsourced to a third-party supplier. The goal is to minimize the sum of holding and delivery costs. This problem is proved to be strongly \(\mathcal{NP}\) -hard. Consider two special cases, in which the jobs have identical processing times. For the problem with limited outsourcing budgets, a \(\mathcal{NP}\) -hardness proof, a pseudo-polynomial algorithm and a fully polynomial time approximation scheme are presented. For the problem with unlimited outsourcing budgets, the problem is shown to be equivalent to the shortest path problem, and therefore it is in class  \(\mathcal{P}\) . This shortest-path-problem solution approach is further shown to be applicable to a similar but more applicable problem, in which the number of deliveries is upper bounded.     

An efficient polynomial time approximation scheme for load balancing on uniformly related machines

We consider basic problems of non-preemptive scheduling on uniformly related machines. For a given schedule, defined by a partition of the jobs into m subsets corresponding to the m machines, \(C_i\) denotes the completion time of machine i. Our goal is to find a schedule that minimizes or maximizes \(\sum _{i=1}^m C_i^p\) for a fixed value of p such that \(0 . For \(p>1\) the minimization problem is equivalent to the well-known problem of minimizing the \(\ell _p\) norm of the vector of the completion times of the machines, and for \(0 , the maximization problem is of interest. Our main result is an efficient polynomial time approximation scheme (EPTAS) for each one of these problems. Our schemes use a non-standard application of the so-called shifting technique. We focus on the work (total size of jobs) assigned to each machine and introduce intervals of work that are forbidden. These intervals are defined so that the resulting effect on the goal function is sufficiently small. This allows the partition of the problem into sub-problems (with subsets of machines and jobs) whose solutions are combined into the final solution using dynamic programming. Our results are the first EPTAS’s for this natural class of load balancing problems.     

Complexity of unconstrained L_2-L_p minimization

We consider the unconstrained $L_q$ - $L_p$ minimization: find a minimizer of $\Vert Ax-b\Vert ^q_q+\lambda \Vert x\Vert ^p_p$ for given $A \in R^{m\times n}$ , $b\in R^m$ and parameters $\lambda >0$ , $p\in [0, 1)$ and $q\ge 1$ . This problem has been studied extensively in many areas. Especially, for the case when $q=2$ , this problem is known as the $L_2-L_p$ minimization problem and has found its applications in variable selection problems and sparse least squares fitting for high dimensional data. Theoretical results show that the minimizers of the $L_q$ - $L_p$ problem have various attractive features due to the concavity and non-Lipschitzian property of the regularization function $\Vert \cdot \Vert ^p_p$ . In this paper, we show that the $L_q$ - $L_p$ minimization problem is strongly NP-hard for any $p\in [0,1)$ and $q\ge 1$ , including its smoothed version. On the other hand, we show that, by choosing parameters $(p,\lambda )$ carefully, a minimizer, global or local, will have certain desired sparsity. We believe that these results provide new theoretical insights to the studies and applications of the concave regularized optimization problems.     

Coset intersection of irreducible plane cubics

In a projective plane $PG(2,\mathbb K )$ over an algebraically closed field $\mathbb K $ of characteristic $p\ge 0$ , let $\Omega $ be a pointset of size $n$ with $5\le n \le 9$ . The coset intersection problem relative to $\Omega $ is to determine the family $\mathbf F$ of irreducible cubics in $PG(2,\mathbb K )$ for which $\Omega $ is a common coset of a subgroup of the additive group $(\mathcal F ,+)$ for every $\mathcal F \in \mathbf F$ . In this paper, a complete solution of this problem is given.     

Euclidean Steiner trees optimal with respect to swapping 4-point subtrees

The Steiner tree problem in Euclidean space $E^3$ asks for a minimum length network $T$ , called a Euclidean Steiner Minimum Tree (ESMT), spanning a given set of points. This problem is NP-hard and the hardness is inherently due to the number of feasible topologies (underlying graph structure of $T$ ) which increases exponentially as the number of given points increases. Planarity is a very strong condition that gives a big difference between the ESMT problem in the Euclidean plane $E^2$ and in Euclidean $d$ -space $E^d (d\ge 3)$ : the ESMT problem in the plane is practically solvable whereas the ESMT problem in $d$ -space is really intractable. The simplest tree rearrangement technique is to repeatedly replace a subtree spanning four nodes in $T$ with another subtree spanning the same four nodes. This technique is referred to as the Swapping 4-point Topology/ Tree technique in this paper. An indicator (or quasi-indicator) of $T$ plays a similar role in the optimization of the length $L(T)$ of $T$ in the discrete topology space (the underlying graph structure of $T$ ) to the derivative of a differentiable function which indicates a fastest direction of descent. $T$ will be called S4pT-optimal if it is optimal with respect to swapping 4-point subtrees. In this paper we first make a complete analysis of 4-point trees in Euclidean space exploring all possible types of 4-point trees and their properties. We review some known indicators of 4-point ESMTs in $E^2$ , and give some simple geometric proofs of these indicators. Then, we translate these indicators to $E^3$ , producing eight quasi-indicators in $E^3$ using computational experiments, the best quasi-indicator $\rho _\mathrm{osr}$ is sifted with an effectiveness for randomly generated 4-point sets as high as 98.62 %. Finally we show how $\rho _\mathrm{osr}$ is used to find an S4pT-optimal ESMT on 14 probability vectors in $4d$ -space with a detailed example.     

Smoothed performance guarantees for local search

We study popular local search and greedy algorithms for standard machine scheduling problems. The performance guarantee of these algorithms is well understood, but the worst-case lower bounds seem somewhat contrived and it is questionable whether they arise in practical applications. To find out how robust these bounds are, we study the algorithms in the framework of smoothed analysis, in which instances are subject to some degree of random noise. While the lower bounds for all scheduling variants with restricted machines are rather robust, we find out that the bounds are fragile for unrestricted machines. In particular, we show that the smoothed performance guarantee of the jump and the lex-jump algorithm are (in contrast to the worst case) independent of the number of machines. They are  \(\Theta (\phi )\) and  \(\Theta (\log \phi )\) , respectively, where  \(1/\phi \) is a parameter measuring the magnitude of the perturbation. The latter immediately implies that also the smoothed price of anarchy is  \(\Theta (\log \phi )\) for routing games on parallel links. Additionally, we show that for unrestricted machines also the greedy list scheduling algorithm has an approximation guarantee of   \(\Theta (\log \phi )\) .     

Optimal Reconstruction Might be Hard

Given as input a point set $\mathcal S $ that samples a shape $\mathcal A $ , the condition required for inferring Betti numbers of $\mathcal A $ from $\mathcal S $ in polynomial time is much weaker than the conditions required by any known polynomial time algorithm for producing a topologically correct approximation of $\mathcal A $ from $\mathcal S $ . Under the former condition which we call the weak precondition, we investigate the question whether a polynomial time algorithm for reconstruction exists. As a first step, we provide an algorithm which outputs an approximation of the shape with the correct Betti numbers under a slightly stronger condition than the weak precondition. Unfortunately, even though our algorithm terminates, its time complexity is unbounded. We then identify at the heart of our algorithm a test which requires answering the following question: given 2 two-dimensional simplicial complexes $L \subset K$ , does there exist a simplicial complex containing $L$ and contained in $K$ which realizes the persistent homology of $L$ into $K$ ? We call this problem the homological simplification of the pair $(K,L)$ and prove that this problem is NP-complete, using a reduction from 3SAT.     

The generalization of Minkowski problems for polytopes

The generalization of Minkowski problems, such as the $L_p$ and Orlicz Minkowski problems, have caused wide concern recently. In this paper, we will establish the existence of the Orlicz Minkowski problem for polytopes. In particular, a solution to the $L_p$ Minkowski problem for polytopes with $p>1$ is given. By the uniqueness of this solution, we present a new proof of the $L_p$ Minkowski inequality that demonstrates the relationship between these two fundamental theorems of the $L_p$ Brunn–Minkowski theory.     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.     

Algorithms on Minimizing the Maximum Sensor Movement for Barrier Coverage of a Linear Domain

In this paper, we study the problem of moving $n$ n sensors on a line to form a barrier coverage of a specified segment of the line such that the maximum moving distance of the sensors is minimized. Previously, it was an open question whether this problem on sensors with arbitrary sensing ranges is solvable in polynomial time. We settle this open question positively by giving an $O(n^2\log n)$ O ( n 2 log n ) time algorithm. For the special case when all sensors have the same-size sensing range, the previously best solution takes $O(n^2)$ O ( n 2 ) time. We present an $O(n\log n)$ O ( n log n ) time algorithm for this case; further, if all sensors are initially located on the coverage segment, our algorithm takes $O(n)$ O ( n ) time. Also, we extend our techniques to the cycle version of the problem where the barrier coverage is for a simple cycle and the sensors are allowed to move only along the cycle. For sensors with the same-size sensing range, we solve the cycle version in $O(n)$ O ( n ) time, improving the previously best $O(n^2)$ O ( n 2 ) time solution.     

Weak Fenchel and weak Fenchel-Lagrange conjugate duality for nonconvex scalar optimization problems

In this work, by using weak conjugate maps given in (Azimov and Gasimov, in Int J Appl Math 1:171–192, 1999), weak Fenchel conjugate dual problem, ${(D_F^w)}$ , and weak Fenchel Lagrange conjugate dual problem ${(D_{FL}^w)}$ are constructed. Necessary and sufficient conditions for strong duality for the ${(D_F^w)}$ , ${(D_{FL}^w)}$ and primal problem are given. Furthermore, relations among the optimal objective values of dual problem constructed by using Augmented Lagrangian in (Azimov and Gasimov, in Int J Appl Math 1:171–192, 1999), ${(D_F^w)}$ , ${(D_{FL}^w)}$ dual problems and primal problem are examined. Lastly, necessary and sufficient optimality conditions for the primal and the dual problems ${(D_F^w)}$ and ${(D_{FL}^w)}$ are established.     

Semidefinite relaxations of ordering problems

Ordering problems assign weights to each ordering and ask to find an ordering of maximum weight. We consider problems where the cost function is either linear or quadratic. In the first case, there is a given profit if the element $u$ is before $v$ in the ordering. In the second case, the profit depends on whether $u$ is before $v$ and $r$ is before $s$ . The linear ordering problem is well studied, with exact solution methods based on polyhedral relaxations. The quadratic ordering problem does not seem to have attracted similar attention. We present a systematic investigation of semidefinite optimization based relaxations for the quadratic ordering problem, extending and improving existing approaches. We show the efficiency of our relaxations by providing computational experience on a variety of problem classes.     

The Arboreal Jump Number of an Order

Let ${\mathcal P}$ be a partial order and ${\mathcal A}$ an arboreal extension of it (i.e. the Hasse diagram of ${\mathcal A}$ is a rooted tree with a unique minimal element). A jump of ${\mathcal A}$ is a relation contained in the Hasse diagram of ${\mathcal A}$ , but not in the order ${\mathcal P}$ . The arboreal jump number of ${\mathcal A}$ is the number of jumps contained in it. We study the problem of finding the arboreal extension of ${\mathcal P}$ having minimum arboreal jump number—a problem related to the well-known (linear) jump number problem. We describe several results for this problem, including NP-completeness, polynomial time solvable cases and bounds. We also discuss the concept of a minimal arboreal extension, namely an arboreal extension whose removal of one jump makes it no longer arboreal.     

On Sets Defining Few Ordinary Lines

Let $P$ P be a set of $n$ n points in the plane, not all on a line. We show that if $n$ n is large then there are at least $n/2$ n / 2 ordinary lines, that is to say lines passing through exactly two points of $P$ P . This confirms, for large $n$ n , a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than $n-C$ n - C ordinary lines for some absolute constant $C$ C . We also solve, for large $n$ n , the “orchard-planting problem”, which asks for the maximum number of lines through exactly 3 points of $P$ P . Underlying these results is a structure theorem which states that if $P$ P has at most $Kn$ K n ordinary lines then all but O(K) points of $P$ P lie on a cubic curve, if $n$ n is sufficiently large depending on $K$ K .     

ParNes: a rapidly convergent algorithm for accurate recovery of sparse and approximately sparse signals

In this article, we propose an algorithm, nesta-lasso, for the lasso problem, i.e., an underdetermined linear least-squares problem with a 1-norm constraint on the solution. We prove under the assumption of the restricted isometry property (rip) and a sparsity condition on the solution, that nesta-lasso is guaranteed to be almost always locally linearly convergent. As in the case of the algorithm nesta, proposed by Becker, Bobin, and Candès, we rely on Nesterov’s accelerated proximal gradient method, which takes $O(\sqrt {1/\varepsilon })$ iterations to come within $\varepsilon > 0$ of the optimal value. We introduce a modification to Nesterov’s method that regularly updates the prox-center in a provably optimal manner. The aforementioned linear convergence is in part due to this modification. In the second part of this article, we attempt to solve the basis pursuit denoising (bpdn) problem (i.e., approximating the minimum 1-norm solution to an underdetermined least squares problem) by using nesta-lasso in conjunction with the Pareto root-finding method employed by van den Berg and Friedlander in their spgl1 solver. The resulting algorithm is called parnes. We provide numerical evidence to show that it is comparable to currently available solvers.     

Multi-bump positive solutions for a nonlinear elliptic problem in expanding tubular domains

In this paper we study the existence of multi-bump positive solutions of the following nonlinear elliptic problem: $$\begin{aligned} -\Delta u=u^p \quad \text{ in }\; \Omega _t,\quad u=0 \quad \text{ on }\; \partial \Omega _t. \end{aligned}$$ Here $1<p<\frac{N+2}{N-2}$ when $N\ge 3,\,1<p<\infty $ when $N=2$ and $\Omega _t$ is a tubular domain which expands as $t\rightarrow \infty $ . See (1.6) below for a precise definition of expanding tubular domain. When the section $D$ of $\Omega _t$ is a ball, the existence of multi-bump positive solutions is shown by Dancer and Yan (Commun Partial Differ Equ, 27(1–2), 23–55, 2002) and by Ackermann et al. (Milan J Math, 79(1), 221–232, 2011) under the assumption of a non-degeneracy of a solution of a limit problem. In this paper we introduce a new local variational method which enables us to show the existence of multi-bump positive solutions without the non-degeneracy condition for the limit problem. In particular, we can show the existence for all $N\ge 2$ without the non-degeneracy condition. Moreover we can deal with more general domains, for example, a domain whose section is an annulus, for which least energy solutions of the limit problem are really degenerate.