Parameterized (in)approximability of subset problems

We discuss approximability and inapproximability in FPT-time for a large class of subset problems where a feasible solution $S$ is a subset of the input data and the value of $S$ is $|S|$. The class handled encompasses many well-known graph, set, or satisfiability problems such as Dominating Set, Vertex Cover, Set Cover, Independent Set, Feedback Vertex Set, etc. In a first time, we introduce the notion of intersective approximability that generalizes the one of safe approximability and show strong parameterized inapproximability results for many of the subset problems handled. Then, we study approximability of these problems with respect to the dual parameter $n-k$ where $n$ is the size of the instance and $k$ the standard parameter. More precisely, we show that under such a parameterization, many of these problems, while W[$\cdot$]-hard, admit parameterized approximation schemata.Comment: 7 page

A Computable Economist’s Perspective on Computational Complexity

A computable economist's view of the world of computational complexity theory is described. This means the model of computation underpinning theories of computational complexity plays a central role. The emergence of computational complexity theories from diverse traditions is emphasised. The unifications that emerged in the modern era was codified by means of the notions of efficiency of computations, non-deterministic computations, completeness, reducibility and verifiability - all three of the latter concepts had their origins on what may be called 'Post's Program of Research for Higher Recursion Theory'. Approximations, computations and constructions are also emphasised. The recent real model of computation as a basis for studying computational complexity in the domain of the reals is also presented and discussed, albeit critically. A brief sceptical section on algorithmic complexity theory is included in an appendix

Bridging gap between standard and differential polynomial approximation: The case of bin-packing

AbstractThe purpose of this paper is mainly to prove the following theorem: for every polynomial time algorithm running in time T(n) and guaranteeing standard-approximation ratio ϱ for bin-packing, there exists an algorithm running in time O(nT(n)) and achieving differential-approximation ratio 2 − ϱ for BP. This theorem has two main impacts. The first one is “operational”, deriving a polynomial time differential-approximation schema for bin-packing. The second one is structural, establishing a kind of reduction (to our knowledge not existing until now) between standard approximation and differential one

Dynamic O-D demand estimation: Application of SPSA AD-PI method in conjunction with different assignment strategies

This paper examines the impact of applying dynamic traffic assignment (DTA) and quasi-dynamic traffic assignment (QDTA) models, which apply different route choice approaches (shortest paths based on current travel times, User Equilibrium: UE, and system optimum: SO), on the accuracy of the solution of the offline dynamic demand estimation problem. The evaluation scheme is based on the adoption of a bilevel approach, where the upper level consists of the adjustment of a starting demand using traffic measures and the lower level of the solution of the traffic network assignment problem. The SPSA AD-PI (Simultaneous Perturbation Stochastic Approximation Asymmetric Design Polynomial Interpolation) is adopted as a solution algorithm. A comparative analysis is conducted on a test network and the results highlight the importance of route choicemodel and information for the stability and the quality of the offline dynamic demand estimations

Mathematical models for planning support

In this paper we describe how computer systems can provide planners with active planning support, when these planners are carrying out their daily planning activities. This means that computer systems actively participate in the planning process by automatically generating plans or partial plans. Active planning support by computer systems requires the application of mathematical models and solution techniques. In this paper we describe the modeling process in general terms, as well as several modeling and solution techniques. We also present some background information on computational complexity theory, since most practical planning problems are hard to solve. We also describe how several objective functions can be handled, since it is rare that solutions can be evaluated by just one single objective. Furthermore, we give an introduction into the use of mathematical modeling systems, which are useful tools in a modeling context, especially during the development phases of a mathematical model. We finish the paper with a real life example related to the planning process of the rolling stock circulation of a railway operator.optimization;mathematical models;modeling process;planning support;Planning

Dynamic O-D demand estimation: Application of SPSA AD-PI method in conjunction with different assignment strategies

This paper examines the impact of applying dynamic traffic assignment (DTA) and quasi-dynamic traffic assignment (QDTA) models, which apply different route choice approaches (shortest paths based on current travel times, User Equilibrium: UE, and system optimum: SO), on the accuracy of the solution of the offline dynamic demand estimation problem. The evaluation scheme is based on the adoption of a bilevel approach, where the upper level consists of the adjustment of a starting demand using traffic measures and the lower level of the solution of the traffic network assignment problem. The SPSA AD-PI (Simultaneous Perturbation Stochastic Approximation Asymmetric Design Polynomial Interpolation) is adopted as a solution algorithm. A comparative analysis is conducted on a test network and the results highlight the importance of route choice model and information for the stability and the quality of the offline dynamic demand estimations

Approximation algorithms for the traveling salesman problem

We first prove that the minimum and maximum traveling salesman problems, their metric versions as well as some versions defined on parameterized triangle inequalities (called sharpened and relaxed metric traveling salesman) are all equi-approximable under an approximation measure, called differential-approximation ratio, that measures how the value of an approximate solution is placed in the interval between the worst- and the best-value solutions of an instance. We next show that the 2-OPT, one of the most-known traveling salesman algorithms, approximately solves all these problems within differential-approximation ratio bounded above by 1/2. We analyze the approximation behavior of 2-OPT when used to approximately solve traveling salesman problem in bipartite graphs and prove that it achieves differential-approximation ratio bounded above by 1/2 also in this case. We also prove that, it is NP-hard to differentially approximate metric traveling salesman within better than 649/650 and traveling salesman with distances 1 and 2 within better than 741/742. Finally, we study the standard approximation of the maximum sharpened and relaxed metric traveling salesman problems. These are versions of maximum metric traveling salesman defined on parameterized triangle inequalities and, to our knowledge, they have not been studied until now

A Computable Economist’s Perspective on Computational Complexity

A computable economist.s view of the world of computational complexity theory is described. This means the model of computation underpinning theories of computational complexity plays a central role. The emergence of computational complexity theories from diverse traditions is emphasised. The unifications that emerged in the modern era was codified by means of the notions of efficiency of computations, non-deterministic computations, completeness, reducibility and verifiability - all three of the latter concepts had their origins on what may be called "Post's Program of Research for Higher Recursion Theory". Approximations, computations and constructions are also emphasised. The recent real model of computation as a basis for studying computational complexity in the domain of the reals is also presented and discussed, albeit critically. A brief sceptical section on algorithmic complexity theory is included in an appendix.