Complex Theoretical Analysis Research Articles

Missio Dei through complexity theory: Complexity to total salvation

Complexity theory continues to fascinate the academics owing to its capability to describe complex systems holistically. This theory makes it possible to understand a complex system by studying its constituent subsystems. Understanding the missio Dei has become somewhat complex, particularly from the perspective of the 1952 Willingen Conference. This conference emphasised mission as a divine activity embodied in the core missions of the Son (missio Christos) and the Holy Spirit (missio Spiritus). It highlighted that the triune God, and not the church, is the source and centre of the missio Dei. Conversely, missio hominum and missio ecclesiae denote the commissions of human participation in the mission of the triune God. Observably, the complex interaction of these core missions and commissions aimed at attaining salvation as the main objective of the missio Dei. This arguably warrants innovative research to explore the missio Dei as a rather complex phenomenon. Correspondingly, as there seems to be no direct complexity-theoretic study of the missio Dei, this article analyses the literature to propose a complexity-theoretic analysis of the missio Dei. Further research is therefore warranted – first to assess the feasibility of this proposal, and then to further use complexity theory to deepen the understanding of the missio Dei.Contribution: Although the study does not purport to provide a comprehensive understanding of the missio Dei, it will hopefully increase understanding and generate interest in using complexity theory to analyse the missio Dei.

A Complexity-theoretic Analysis of Green Pickup-and-Delivery Problems

In a Green Pickup-and-Delivery problem (GPD), vehicles traveling in a transport network achieving pickup-and-delivery tasks are in particular subject to the two \textit{green} constraints: limited vehicle fuel capacity thus short vehicle traveling range, and limited availability of refueling infrastructure for the vehicles. GPD adds additional but probably insignificant computational complexity to the classic and already NP-hard Pickup-and-Delivery problem and Vehicle Routing Problem. Nevertheless, we demonstrate in this paper an inherent intractability of these green components themselves. More precisely, we show that GPD problems whose total constraints are reduced to almost the green ones only, remain to be NP-complete in the strong sense. We figure out a specifically constrained variant of GPD that, however, is weakly NP-complete -- a practical pseudo-polynomial time algorithm solving the variant problem is identified. Insight obtained from this complexity-theoretic analysis would shed light for a deeper understanding of GPDs, and on better development of heuristics for solving these problems, leading to promisingly many real-world applications.

Acceptance in incomplete argumentation frameworks

Abstract argumentation frameworks (AFs), originally proposed by Dung, constitute a central formal model for the study of computational aspects of argumentation in AI. Credulous and skeptical acceptance of arguments in a given AF are well-studied problems both in terms of theoretical analysis—especially computational complexity—and the development of practical decision procedures for the problems. However, AFs make the assumption that all attacks between arguments are certain (i.e., present attacks are known to exist, and missing attacks are known to not exist), which can in various settings be a restrictive assumption. A generalization of AFs to incomplete AFs was recently proposed as a formalism that allows the representation of both uncertain attacks and uncertain arguments in AFs. In this article, we explore the impact of allowing for modeling such uncertainties in AFs on the computational complexity of natural generalizations of acceptance problems to incomplete AFs under various central AF semantics. Complementing the complexity-theoretic analysis, we also develop the first practical decision procedures for all of the NP-hard variants of acceptance in incomplete AFs. In terms of complexity analysis, we establish a full complexity landscape, showing that depending on the variant of acceptance and property/semantics, the complexity of acceptance in incomplete AFs ranges from polynomial-time decidable to completeness for Σ3p. In terms of algorithms, we show through an extensive empirical evaluation that an implementation of the proposed decision procedures, based on boolean satisfiability (SAT) solving, is effective in deciding variants of acceptance under uncertainties. We also establish conditions for what type of atomic changes are guaranteed to be redundant from the perspective of preserving extensions of completions of incomplete AFs, and show that the results allow for considerably improving the empirical efficiency of the proposed SAT-based counterexample-guided abstraction refinement algorithms for acceptance in incomplete AFs for problem variants with complexity beyond NP.

Global optimality in k-means clustering

We study the problem of finding an optimum clustering, a problem known to be NP-hard. Existing literature contains algorithms running in time proportional to the number of points raised to a power that depends on the dimensionality and on the number of clusters. Published validations of some of these algorithms are unfortunately incomplete; besides, the constant factors (with respect to the number of points) in their running time bounds have seen several published important improvements but are still huge, exponential on the dimension and on the number of clusters, making the corresponding algorithms fully impractical. We provide a new algorithm, with its corresponding complexity-theoretic analysis. It reduces both the exponent and the constant factor, to the extent that it becomes feasible for relevant particular cases. Additionally, it parallelizes extremely well, so that its implementation on current high-performance hardware is quite straightforward. Our proposal opens the door to potential improvements along a research line that had no practical significance so far; besides, a long but single-shot run of our algorithm allows one to identify absolutely optimum solutions for benchmark problems, whereby alternative heuristic proposals can evaluate the goodness of their solutions and the precise price paid for their faster running times.

Two optimization problems in linear regression with interval data

We consider a linear regression model where neither regressors nor the dependent variable is observable; only intervals are available which are assumed to cover the unobservable data points. Our task is to compute tight bounds for the residual errors of minimum-norm estimators of regression parameters with various norms (corresponding to least absolute deviations (LAD), ordinary least squares (OLS), generalized least squares (GLS) and Chebyshev approximation). The computation of the error bounds can be formulated as a pair of max–min and min–min box-constrained optimization problems. We give a detailed complexity-theoretic analysis of them. First, we prove that they are NP-hard in general. Then, further analysis explains the sources of NP-hardness. We investigate three restrictions when the problem is solvable in polynomial time: the case when the parameter space is known apriori to be restricted into a particular orthant, the case when the regression model has a fixed number of regression parameters, and the case when only the dependent variable is observed with errors. We propose a method, called orthant decomposition of the parameter space, which is the main tool for obtaining polynomial-time computability results.