Marc Uetz

An algorithm-based topographical biomaterials library to instruct cell fate

It is increasingly recognized that material surface topography is able to evoke specific cellular responses, endowing materials with instructive properties that were formerly reserved for growth factors. This opens the window to improve upon, in a cost-effective manner, biological performance of any surface used in the human body. Unfortunately, the interplay between surface topographies and cell behavior is complex and still incompletely understood. Rational approaches to search for bioactive surfaces will therefore omit previously unperceived interactions. Hence, in the present study, we use mathematical algorithms to design nonbiased, random surface features and produce chips of poly(lactic acid) with 2,176 different topographies. With human mesenchymal stromal cells (hMSCs) grown on the chips and using high-content imaging, we reveal unique, formerly unknown, surface topographies that are able to induce MSC proliferation or osteogenic differentiation. Moreover, we correlate parameters of the mathematical algorithms to cellular responses, which yield novel design criteria for these particular parameters. In conclusion, we demonstrate that randomized libraries of surface topographies can be broadly applied to unravel the interplay between cells and surface topography and to find improved material surfaces.

Solving Project Scheduling Problems by Minimum Cut Computations

In project scheduling, a set of precedence-constrained jobs has to be scheduled so as to minimize a given objective. In resource-constrained project scheduling, the jobs additionally compete for scarce resources. Due to its universality, the latter problem has a variety of applications in manufacturing, production planning, project management, and elsewhere. It is one of the most intractable problems in operations research, and has therefore become a popular playground for the latest optimization techniques, including virtually all local search paradigms. We show that a somewhat more classical mathematical programming approach leads to both competitive feasible solutions and strong lower bounds, within reasonable computation times. The basic ingredients of our approach are the Lagrangian relaxation of a time-indexed integer programming formulation and relaxation-based list scheduling, enriched with a useful idea from recent approximation algorithms for machine scheduling problems. The efficiency of the algorithm results from the insight that the relaxed problem can be solved by computing a minimum cut in an appropriately defined directed graph. Our computational study covers different types of resource-constrained project scheduling problems, based on several notoriously hard test sets, including practical problem instances from chemical production planning.

Approximation in stochastic scheduling: the power of LP-based priority policies

We consider the problem to minimize the total weighted completion time of a set of jobs with individual release dates which have to be scheduled on identical parallel machines. Job processing times are not known in advance, they are realized on-line according to given probability distributions. The aim is to find a scheduling policy that minimizes the objective in expectation. Motivated by the success of LP-based approaches to deterministic scheduling, we present a polyhedral relaxation of the performance space of stochastic parallel machine scheduling. This relaxation extends earlier relaxations that have been used, among others, by Hall et al. [1997] in the deterministic setting. We then derive constant performance guarantees for priority policies which are guided by optimum LP solutions, and thereby generalize previous results from deterministic scheduling. In the absence of release dates, the LP-based analysis also yields an additive performance guarantee for the WSEPT rule which implies both a worst-case performance ratio and a result on its asymptotic optimality, thus complementing previous work by Weiss [1990]. The corresponding LP lower bound generalizes a previous lower bound from deterministic scheduling due to Eastman et al. [1964], and exhibits a relation between parallel machine problems and corresponding problems with only one fast single machine. Finally, we show that all employed LPs can be solved in polynomial time by purely combinatorial algorithms.

Stochastic Machine Scheduling with Precedence Constraints

We consider parallel, identical machine scheduling problems, where the jobs are subject to precedence constraints and release dates, and where the processing times of jobs are governed by independent probability distributions. Our objective is to minimize the expected value of the total weighted completion time. Building upon a linear programming relaxation by Mohring, Schulz, and Uetz [J. ACM, 46 (1999), pp. 924--942] and a delayed list scheduling algorithm by Chekuri et al. [SIAM J. Comput., 31 (2001), pp. 146--166], we derive the first constant-factor approximation algorithms for this model.

Characterization of Revenue Equivalence

A saw blade guard for a sawing power tool, particularly a mitre saw, comprises a protective element having two side walls each formed as a part of a circle, and a peripheral wall connecting the side walls with one another, the side walls having a center and an edge spaced from the center, the protective element being formed as an integral element and provided with a member in the region of the edge for turning the protective element as a whole.

On project scheduling with irregular starting time costs

Maniezzo and Mingozzi (Oper. Res. Lett. 25 (1999) 175-182) study a project scheduling problem with irregular starting time costs. Starting from the assumption that its computational complexity status is open, they develop a branch-and-bound procedure and they identify special cases that are solvable in polynomial time. In this note, we present a collection of previously established results which show that the general problem is solvable in polynomial time. This collection may serve as a useful guide to the literature, since this polynomial-time solvability has been rediscovered in different contexts or using different methods. In addition, we briefly review some related results for specializations and generalizations of the problem.

Machine scheduling with resource dependent processing times

We consider machine scheduling on unrelated parallel machines with the objective to minimize the schedule makespan. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of an integer linear programming formulation for a relaxation of the problem, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham’s list scheduling, we show how to derive a 4-approximation algorithm. We also show how to tune our approach to yield a 3.75-approximation algorithm. This is achieved by applying the same rounding technique to a slightly modified linear programming relaxation, and by using a more sophisticated scheduling algorithm that is inspired by the harmonic algorithm for bin packing. We finally derive inapproximability results for two special cases, and discuss tightness of the integer linear programming relaxations.

On the generation of circuits and minimal forbidden sets

Abstract.We present several complexity results related to generation and counting of all circuits of an independence system. Our motivation to study these problems is their relevance in the solution of resource constrained scheduling problems, where an independence system arises as the subsets of jobs that may be scheduled simultaneously. We are interested in the circuits of this system, the so-called minimal forbidden sets, which are minimal subsets of jobs that must not be scheduled simultaneously. As a consequence of the complexity results for general independence systems, we obtain several complexity results in the context of resource constrained scheduling. On that account, we propose and analyze a simple backtracking algorithm that generates all minimal forbidden sets for such problems. The performance of this algorithm, in comparison to a previously suggested divide-and-conquer approach, is evaluated empirically using instances from the project scheduling library PSPLIB.

Resource-Constrained Project Scheduling: Computing Lower Bounds by Solving Minimum Cut Problems

We present a novel approach to compute Lagrangian lower bounds on the objective function value of a wide class of resource-constrained project scheduling problems. The basis is a polynomial-time algorithm to solve the following scheduling problem: Given a set of activities with start-time dependent costs and temporal constraints in the form of time windows, find a feasible schedule of minimum total cost. In fact, we show that any instance of this problem can be solved by a minimum cut computation in a certain directed graph.We then discuss the performance of the proposed Lagrangian approach when applied to various types of resource-constrained project scheduling problems. An extensive computational study based on different established test beds in project scheduling shows that it can significantly improve upon the quality of other comparably fast computable lower bounds.

How to sell a graph: guidelines for graph retailers

We consider a profit maximization problem where we are asked to price a set of m items that are to be assigned to a set of n customers. The items can be represented as the edges of an undirected (multi)graph G, where an edge multiplicity larger than one corresponds to multiple copies of the same item. Each customer is interested in purchasing a bundle of edges of G, and we assume that each bundle forms a simple path in G. Each customer has a known budget for her respective bundle, and is interested only in that particular bundle. The goal is to determine item prices and a feasible assignment of items to customers in order to maximize the total profit. When the underlying graph G is a path, we derive a fully polynomial time approximation scheme, complementing a recent NP-hardness result. If the underlying graph is a tree, and edge multiplicities are one, we show that the problem is polynomially solvable, contrasting its APX-hardness for the case of unlimited availability of items. However, if the underlying graph is a grid, and edge multiplicities are one, we show that it is even NP-complete to approximate the maximum profit to within a factor n1−−e.