Approximation Algorithms for Modern Multi-Processor Scheduling Problems

This thesis is devoted to the design and analysis of algorithms for scheduling problems. These problems are ubiquitous in the modern world. Examples include the optimization of local transportation, managing access to concurrent resources like runways at airports and efficient execution of computing tasks on server systems. Problem instances that appear in the real world often are so large and complex that it is not possible to solve them “by hand”. This rises the need for strong algorithmic approaches, which motivates our focus of study. In this work we consider two types of scheduling problems which gained in importance due to recent technological advances. The first problem comes from the avionics industry and deals with scheduling periodically recurring tasks in a parallel computer network on a plane: Each task comes with a period p and execution time c, and needs to use a processor exclusively for c time units every p time units. The scheduling problem is to assign starting offsets for the first execution of the tasks so that no collision occurs. The second problem is a scheduling problem that arises in highly parallelized processing environments with a shared common resource, e.g., modern multi-core computer architectures. In addition to classical makespan minimization problems such as scheduling on identical machines, each job has an additional resource constraint. The scheduler must ensure that at no time, the accumulated requirement of all active jobs at that time exceeds a given limit. For both types of problems we study their algorithmic complexity in a mathematical, rigorous way by designing approximation algorithms and establishing inapproximability results. We thereby give a characterization of the approximation landscape of these problems. We also consider a more practical perspective: For an engineer from the industry, a rigorous proof that an algorithm finds a solution of certain guaranteed quality for all possible kinds of problem instances is usually not that relevant. It is rather of interest to find “good enough” or even optimal solutions for particular instances that actually appear in the real world in “reasonable” time. We show that structural insights gained in the more theoretical process of designing approximation algorithms can be highly beneficial also for obtaining practical results. In particular, we develop integer programming formulations for the avionics problem based on structural properties revealed in the design of approximation algorithms. These formulations lead to strong tools that, for the first time, enable to algorithmically solve real-world instances from our industrial partner.

New Graph Algorithms via Polyhedral Techniques

Jakub Tarnawski

In this thesis we give new algorithms for two fundamental graph problems. We develop novel ways of using linear programming formulations, even exponential-sized ones, to extract structure from problem instances and to guide algorithms in making progress. Somewhat surprisingly, similar polyhedral techniques can be harnessed in the two seemingly disparate settings. In the first part of the thesis we address a benchmark problem in combinatorial optimization: the asymmetric traveling salesman problem (ATSP). It consists in finding the shortest tour that visits all vertices of a given directed graph with weights on edges. Due to its NP-hardness, the theoretical study of algorithms for ATSP has focused on approximation algorithms: ones that are provably both efficient and give solutions competitive with the optimum. Specifically, a rho-approximation algorithm for ATSP is one that runs in polynomial time and always outputs a tour that is at most rho times longer than the shortest tour. Finding such an approximation algorithm with rho bounded (i.e., a constant factor) had been a long-standing open problem. In this thesis, we give such an algorithm. Our approximation guarantee is analyzed with respect to the standard linear programming relaxation, and thus our result also confirms the conjectured constant integrality gap of that relaxation. Our techniques build upon the constant-factor approximation algorithm for the special case of node-weighted metrics due to Svensson. In particular, we give a generic reduction to structured instances that resemble but are more general than those arising from node-weighted metrics. This reduction takes advantage of a laminar family of vertex sets that arises from the linear programming relaxation. In the second part of the thesis we address the perfect matching problem. The first polynomial-time algorithm for it, given by Edmonds in 1965, is historically associated with the introduction of the class P and our notion that polynomial-time'' means efficient''. That algorithm is sequential and deterministic. We have also known since the 1980s that the matching problem has efficient parallel algorithms if the use of randomness is allowed. Formally, it is in the class RNC, i.e., it has randomized algorithms that use polynomially many processors and run in polylogarithmic time. However, we do not know if randomness is necessary - that is, whether the matching problem is in the class NC. In this thesis we show that the matching problem is in quasi-NC. That is, we give a deterministic parallel algorithm that runs in O(log^3 n) time on n^{O(log^2 n)} processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani to obtain an RNC algorithm. Our proof extends the framework of Fenner, Gurjar and Thierauf, who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.

EPFL2019

Strengths and Limitations of Linear Programming Relaxations

Abbas Bazzi

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than

2

and

\omega(1)

, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than

1.5

for the Vertex Cover Problem, and better than

2

for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of

2-\varepsilon

for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

EPFL2017

On the computational complexity of periodic scheduling

Thomas Rothvoss

In periodic scheduling jobs arrive regularly and have to be executed on one or several machines or processors. An enormous amount of literature has been published on this topic, e.g. the founding work of Liu & Layland [LL73] is among the top cited computer science papers, according to Citeseer database. The majority of these papers is mainly concerned about practical applications and several important theoretical questions remained unsolved. Let S be a set of periodic tasks, where each task τ ∈ S releases a job of running time c(τ) and relative deadline d(τ) at each integer multiple of its period p(τ). In the single-processor scheduling one considers rules like Earliest-Deadline-First (EDF) or Rate-monotonic (RM) scheduling, that determine the order, in which the jobs have to be processed. The principal question here is to predict, whether such a schedule is feasible, i.e. whether all jobs are finished before their individual deadlines. It was well-known that to validate the feasibility of an EDF-schedule, one has to evaluate the condition But the complexity status of this test remained unknown, despite of many heuristic approaches in the literature. We prove that testing this condition is coNP-hard even in special cases, answering an open question of Baruah & Pruhs [BP09]. For a static-priority schedule of implicit-deadline tasks, i.e. d(τ) = p(τ), it is known that the jobs of a task τ will meet their deadlines if and only if there exists a fix-point r ∈ [0,p(τ)] of the equation where the sum ranges over all tasks with priority higher than τ . We settle the complexity status of this problem by proving its NP-hardness, even if one asks for modest approximations. Both results are achieved by bridging the more practically oriented area of Real-time scheduling and the field of algorithmic number theory. In fact, the intractability follows by a chain of reductions to simultaneous Diophantine approximation (SDA), which is a classical problem in the geometry of numbers and deals with finding a small denominator Q ∈ {1,…,N}, that yields a good approximation to a given vector of rational numbers α ∈ Qn, i.e. maxi=1,…,n |Qαi - ⎡Qαi⎦| ≤ ε. We strengthen existing hardness results for SDA such that they admit intractability results also for a directed version, where the goal is to find a Q ∈ {1,…,N} with maxi=1,…,n(⎡Qαi⎤ - Qαi) ≤ ε. We show that this problem remains NP-hard if one asks for an approximation that may exceed the bound N by a factor of nO(1/loglogn) and the error bound ε by 2nO(1). As an application we are able to answer an open question of Conforti et al. [CSW08] negatively by obtaining NP-hardness for the problem of optimizing a linear function over a mixing set {(s,y ∈ R≥0 × Zn | s + aiyi ≥ bi ∀i = 1,…, n}. Furthermore we obtain that, regardless to the used ‖ · ‖p-norm, the problem of finding a shortest positive vector in a lattice is not approximable within a factor of nO(1/loglogn), unless NP = P. For scheduling constrained-deadline tasks (i.e. d(τ) ≤ p(τ)) with a static-priority policy one possibly needs machines, which are a factor f > 1 faster than it is needed for the EDF policy. Baruah & Burns [BB08] sandwiched this factor between 1.5 and 2. We pinpoint this value to f = 1/Ω ≈ 1.76 by characterizing the properties of task systems, which attain this value. Here, Ω ≈ 0.56 is a well known constant from complex calculus, defined as real root of x · ex = 1. We further consider a multiprocessor setting, where a given set of tasks has to be assigned to machines, such that each partition is RM-schedulable and the number of occupied processors is minimized. For the popular case that the tasks are equipped with implicit-deadlines, i.e. d(τ) = p(τ), we develop new algorithms as well as intractability results: We state an asymptotic PTAS under resource augmentation. This is achieved by two concepts: First we derive that using a merging and clustering procedure, the instance can be modified, such that the utilization of tasks is lowerbounded by a constant, without excluding near-optimal solutions. Secondly, we introduce a relaxed notion of feasibility, which yields a drastic reduction of complexity and admits to compute solutions via dynamic programming. We introduce a simple First-Fit-like heuristic and obtain that this algorithm behaves nearly optimal on average, by proving that the expected waste is upperbounded by O(n3/4(log n)3/8), if the instance of n tasks is generated randomly. Here, we assume that the utilization values c(τ)/p(τ) of the tasks are drawn uniformly at random from [0, 1]. We state a new approximation algorithm with a running time of O(n3), which can be sketched as follows: Create a graph with the tasks being the nodes and insert edges, if both incident tasks can be scheduled together on a processor. Then define suitable vertex costs and compute a min-cost matching, where the costs are the sum of edge costs plus the vertex costs of not covered nodes. From such a matching solution we can then extract a feasible multiprocessor schedule. The approximation ratio of this algorithm can be proven to tend to 3/2, improving over the previously best known ratio of 7/4 [BLOS95]. We consider a column-based linear programming relaxation for this multiprocessor scheduling problem and derive that the asymptotic integrality gap is located between 4/3 ≈ 1.33 and 1 + ln(2) ≈ 1.69. We disprove the existence of an asymptotic FPTAS, which yields that the problem is strictly harder to approximate than its special case of Bin Packing.

EPFL2009