Polynomial-time approximation scheme

In computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an instance of an optimization problem and a parameter ε > 0 and produces a solution that is within a factor 1 + ε of being optimal (or 1 – ε for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most (1 + ε)L, with L being the length of the shortest tour. The running time of a PTAS is required to be polynomial in the problem size for every fixed ε, but can be different for different ε. Thus an algorithm running in time O(n^1/ε) or even O(n^exp(1/ε)) counts as a PTAS. A practical problem with PTAS algorithms is that the exponent of the polynomial could increase dramatically as ε shrinks, for example if the runtime is O(n^(1/ε)!). One way of addressing this is to define the efficient polynomial-time approximation scheme or EPTAS, in which the running time is required to be O(n^c) for a constant c independent of ε. This ensures that an increase in problem size has the same relative effect on runtime regardless of what ε is being used; however, the constant under the big-O can still depend on ε arbitrarily. In other words, an EPTAS runs in FPT time where the parameter is ε. Even more restrictive, and useful in practice, is the fully polynomial-time approximation scheme or FPTAS, which requires the algorithm to be polynomial in both the problem size n and 1/ε. Unless P = NP, it holds that FPTAS ⊊ PTAS ⊊ APX. Consequently, under this assumption, APX-hard problems do not have PTASs. Another deterministic variant of the PTAS is the quasi-polynomial-time approximation scheme or QPTAS. A QPTAS has time complexity n^polylog(n) for each fixed ε > 0. Furthermore, a PTAS can run in FPT time for some parameterization of the problem, which leads to a parameterized approximation scheme.

From Massive Parallelization to Quantum Computing: Seven Novel Approaches to Query Optimization

Immanuel Trummer

The goal of query optimization is to map a declarative query (describing data to generate) to a query plan (describing how to generate the data) with optimal execution cost. Query optimization is required to support declarative query interfaces. It is a core problem in the area of database systems and has received tremendous attention in the research community, starting with an initial publication in 1979. In this thesis, we revisit the query optimization problem. This visit is motivated by several developments that change the context of query optimization. That change is not reflected in prior literature. First, advances in query execution platforms and processing techniques have changed the context of query optimization. Novel provisioning models and processing techniques such as Cloud computing, crowdsourcing, or approximate processing allow to trade between different execution cost metrics (e.g., execution time versus monetary execution fees in case of Cloud computing). This makes it necessary to compare alternative execution plans according to multiple cost metrics in query optimization. While this is a common scenario nowadays, the literature on query optimization with multiple cost metrics (a generalization of the classical problem variant with one execution cost metric) is surprisingly sparse. While prior methods take hours to optimize even moderately sized queries when considering multiple cost metrics, we propose a multitude of approaches to make query optimization in such scenarios practical. A second development that we address in this thesis is the availability of novel software and hardware platforms that can be exploited for optimization. We will show that integer programming solvers, massively parallel clusters (which nowadays are commonly used for query execution), and adiabatic quantum annealers enable us to solve query optimization problem instances that are far beyond the capabilities of prior approaches. In summary, we propose seven novel approaches to query optimization that significantly increase the size of the problem instances that can be addressed (measured by the query size and by the number of considered execution cost metrics). Those novel approaches can be classified into three broad categories: moving query optimization before run time to relax constraints on optimization time, trading optimization time for relaxed optimality guarantees (leading to approximation schemes, incremental algorithms, and randomized algorithms for query optimization with multiple cost metrics), and reducing optimization time by leveraging novel software and hardware platforms (integer programming solvers, massively parallel clusters, and adiabatic quantum annealers). Those approaches are novel since they address novel problem variants of query optimization, introduced in this thesis, since they are novel for their respective problem variant (e.g., we propose the first randomized algorithm for query optimization with multiple cost metrics), or because they have never been used for optimization problems in the database domain (e.g., this is the first time that quantum computing is used to solve a database-specific optimization problem).

EPFL2016

From Massive Parallelization to Quantum Computing: Seven Novel Approaches to Query Optimization

Immanuel Trummer

EPFL2016

Polynomial-time approximation scheme

From Massive Parallelization to Quantum Computing: Seven Novel Approaches to Query Optimization

An algorithm for source location in directed graphs

An algorithm for source location in directed graphs

From Massive Parallelization to Quantum Computing: Seven Novel Approaches to Query Optimization