Class Of Decision Problems Research Articles

On the parameterized complexity of freezing dynamics

In this paper we establish how alphabet size, treewidth and maximum degree of the underlying graph are key parameters which influence the overall computational complexity of finite freezing automata networks. First, we define a general decision problem, called Specification Checking Problem, that captures many classical decision problems such as prediction, nilpotency, predecessor, asynchronous reachability.Then, we present a fast-parallel algorithm that solves the general model checking problem when the three parameters are bounded, hence showing that the problem is in NC. Moreover, we show that the problem is in XP on the parameters tree-width and maximum degree.Finally, we show that these problems are hard from two different perspectives. First, the general problem is W[2]-hard when taking either treewidth or alphabet as single parameter and fixing the others. Second, the classical problems are hard in their respective classes when restricted to families of graph with sufficiently large treewidth.

Forced Exploration in Bandit Problems

The multi-armed bandit(MAB) is a classical sequential decision problem. Most work requires assumptions about the reward distribution (e.g., bounded), while practitioners may have difficulty obtaining information about these distributions to design models for their problems, especially in non-stationary MAB problems. This paper aims to design a multi-armed bandit algorithm that can be implemented without using information about the reward distribution while still achieving substantial regret upper bounds. To this end, we propose a novel algorithm alternating between greedy rule and forced exploration. Our method can be applied to Gaussian, Bernoulli and other subgaussian distributions, and its implementation does not require additional information. We employ a unified analysis method for different forced exploration strategies and provide problem-dependent regret upper bounds for stationary and piecewise-stationary settings. Furthermore, we compare our algorithm with popular bandit algorithms on different reward distributions.

Solving the SAT problem with the string multiset rewriting calculus

In this paper, we develop computing machinery within the framework of the String Multiset Rewriting calculus (SMSR), as defined by Barbuti et al. [4], to solve the SAT problem in linear time regarding the number of variables of a given conjunctive normal form. This shows that SMSR can be considered a computational model capable of significantly reducing the time requirement of classical decision problems.

Prophet Inequalities with Linear Correlations and Augmentations

In a classical online decision problem, a decision-maker who is trying to maximize her value inspects a sequence of arriving items to learn their values (drawn from known distributions) and decides when to stop the process by taking the current item. The goal is to prove a “prophet inequality”: that she can do approximately as well as a prophet with foreknowledge of all the values. In this work, we investigate this problem when the values are allowed to be correlated. Since nontrivial guarantees are impossible for arbitrary correlations, we consider a natural “linear” correlation structure introduced by Bateni et al. as a generalization of the common-base value model of Chawla et al. A key challenge is that threshold-based algorithms, which are commonly used for prophet inequalities, no longer guarantee good performance for linear correlations. We relate this roadblock to another “augmentations” challenge that might be of independent interest: many existing prophet inequality algorithms are not robust to slight increases in the values of the arriving items. We leverage this intuition to prove bounds (matching up to constant factors) that decay gracefully with the amount of correlation of the arriving items. We extend these results to the case of selecting multiple items by designing a new (1+ o (1))-approximation ratio algorithm that is robust to augmentations.

Regret theory, Allais’ paradox, and Savage’s omelet

We study a sufficiently general regret criterion for choosing between two probabilistic lotteries. For independent lotteries, the criterion is consistent with stochastic dominance and can be made transitive by a unique choice of the regret function. Together with additional (and intuitively meaningful) super-additivity property, the regret criterion resolves the Allais’ paradox including the cases were the paradox disappears, and the choices agree with the expected utility. This super-additivity property is also employed for establishing consistency between regret and stochastic dominance for dependent lotteries. Furthermore, we demonstrate how the regret criterion can be used in Savage’s omelet, a classical decision problem in which the lottery outcomes are not fully resolved. The expected utility cannot be used in such situations, as it discards important aspects of lotteries.

Post's Correspondence Problem for hyperbolic and virtually nilpotent groups

AbstractPost's Correspondence Problem (the PCP) is a classical decision problem in theoretical computer science that asks whether for pairs of free monoid morphisms there exists any non‐trivial such that . PCP for a group takes pairs of group homomorphisms instead, and similarly asks whether there exists an such that holds for non‐elementary reasons. The restrictions imposed on in order to get non‐elementary solutions lead to several interpretations of the problem; we mainly consider the natural restriction asking that and prove that the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic , but decidable when is virtually nilpotent, en route also studying this problem for finite extensions. We also consider a different interpretation of the PCP due to Myasnikov, Nikolaev and Ushakov, proving decidability for torsion‐free nilpotent groups.

Augmented intuition: a bridge between theory and practice

Motivated by the celebrated paper of Hooker (J Heuristics 1(1): 33–42, 1995) published in the first issue of this journal, and by the relative lack of progress of both approximation algorithms and fixed-parameter algorithms for the classical decision and optimization problems related to covering edges by vertices, we aimed at developing an approach centered in augmenting our intuition about what is indeed needed. We present a case study of a novel design methodology by which algorithm weaknesses will be identified by computer-based and fixed-parameter tractable algorithmic challenges on their performance. Comprehensive benchmarkings on all instances of small size then become an integral part of the design process. Subsequent analyses of cases where human intuition “fails”, supported by computational testing, will then lead to the development of new methods by avoiding the traps of relying only on human perspicacity and ultimately will improve the quality of the results. Consequently, the computer-aided design process is seen as a tool to augment human intuition. It aims at accelerating and foster theory development in areas such as graph theory and combinatorial optimization since some safe reduction rules for pre-processing can be mathematically proved via theorems. This approach can also lead to the generation of new interesting heuristics. We test our ideas with a fundamental problem in graph theory that has attracted the attention of many researchers over decades, but for which seems it seems to be that a certain stagnation has occurred. The lessons learned are certainly beneficial, suggesting that we can bridge the increasing gap between theory and practice by a more concerted approach that would fuel human imagination from a data-driven discovery perspective.

Computational complexity of flat and generic Assumption-Based Argumentation, with and without probabilities

Reasoning with probabilistic information has recently attracted considerable attention in argumentation, and formalisms of Probabilistic Abstract Argumentation (PAA), Probabilistic Bipolar Argumentation (PBA) and Probabilistic Structured Argumentation (PSA) have been proposed. These foundational advances have been complemented with investigations on the complexity of some approaches to PAA and PBA, but not to PSA. We study the complexity of an existing form of PSA, namely Probabilistic Assumption-Based Argumentation (PABA), a powerful, implemented formalism which subsumes several forms of PAA and other forms of PSA. Specifically, we establish membership (general upper bounds) and completeness (instantiated lower bounds) of reasoning in PABA for the class FP#P (of functions with a #P-oracle for counting the solutions of an NP problem) with respect to newly introduced probabilistic verification, credulous and sceptical acceptance function problems under several ABA semantics. As a by-product necessary to establish PABA complexity results, we provide a comprehensive picture of the ABA complexity landscape (for both flat and generic, possibly non-flat ABA) for the classical decision problems of verification, existence, credulous and sceptical acceptance under those ABA semantics.

The ecological rationality of decision criteria

Standard evolutionary game theory investigates the evolutionary fitness of alternative behaviors in a fixed and single decision problem. This paper instead focuses on decision criteria, rather than on simple behaviors, as the general behavioral rules under selection in the population: the evolutionary fitness of classic decision criteria for rational choice is analyzed through Monte Carlo simulations over various classes of decision problems. Overall, quantifying the uncertainty in a probabilistic way and maximizing expected utility turns out to be evolutionarily beneficial in general. Minimizing regret also finds some evolutionary justifications in our results, while maximin seems to be always disadvantaged by differential selection.

Optimality, Equilibrium, and Curb Sets in Decision Problems Without Commitment

The paper considers a class of decision problems with an infinite time horizon that contains Markov decision problems as an important special case. Our interest concerns the case where the decision maker cannot commit himself to his future action choices. We model the decision maker as consisting of multiple selves, where each history of the decision problem corresponds to one self. Each self is assumed to have the same utility function as the decision maker. Our results are twofold: Firstly, we demonstrate that the set of subgame optimal policies coincides with the set of subgame perfect equilibria of the decision problem. Furthermore, the set of subgame optimal policies is contained in the set of optimal policies and the set of optimal policies is contained in the set of Nash equilibria. Secondly, we show that the set of pure subgame optimal policies is the unique minimal curb set of the decision problem. The concept of a subgame optimal policy is therefore robust to the absence of commitment technologies.

Optimality, Equilibrium, and Curb Sets in Decision Problems Without Commitment

The paper considers a class of decision problems with in_nite time horizon that contains Markov decision problems as an important special case. Our interest concerns the case where the decision maker cannot commit himself to his future action choices. We model the decision maker as consisting of multiple selves, where each history of the decision problem corresponds to one self. Each self is assumed to have the same utility function as the decision maker. We introduce the notions of Nash equilibrium, subgame perfect equilibrium, and curb sets for decision problems. An optimal policy at the initial history is a Nash equilibrium but not vice versa. Both subgame perfect equilibria and curb sets are equivalent to subgame optimal policies. The concept of a subgame optimal policy is therefore robust to the absence of commitment technologies.

∃R-Completeness for Decision Versions of Multi-Player (Symmetric) Nash Equilibria

As a result of a series of important works [7--9, 15, 23], the complexity of two-player Nash equilibrium is by now well understood, even when equilibria with special properties are desired and when the game is symmetric. However, for multi-player games, when equilibria with special properties are desired, the only result known is due to Schaefer and Štefankovič [28]: that checking whether a three-player Nash Equilibrium (3-Nash) instance has an equilibrium in a ball of radius half in l ∞ -norm is ∃R-complete, where ∃R is the class of decision problems that can be reduced in polynomial time to Existential Theory of the Reals. Building on their work, we show that the following decision versions of 3-Nash are also ∃R-complete: checking whether (i) there are two or more equilibria, (ii) there exists an equilibrium in which each player gets at least h payoff, where h is a rational number, (iii) a given set of strategies are played with non-zero probability, and (iv) all the played strategies belong to a given set. Next, we give a reduction from 3-Nash to symmetric 3-Nash, hence resolving an open problem of Papadimitriou [25]. This yields ∃R-completeness for symmetric 3-Nash for the last two problems stated above as well as completeness for the class FIXP a , a variant of FIXP for strong approximation. All our results extend to k -Nash for any constant k ≥ 3.

Decision making under measure-based granular uncertainty

In situations, where an uncertain variable takes its value in a large space, we can use granulation of the space to simplify the knowledge acquisition process. We observe, however, that this use of granulation can come at the price of introducing imprecision into the related decision process. We note that answers to questions about values related to the original space can often only be answered using interval values. We look at two classes of decision problems where uncertain information about relevant variables is expressed indirectly via information about the uncertainty on related granular objects. To make a precise decision, we require the decision maker to provide some subjective preference information to resolve the interval uncertainty induced by the granulation. In this paper, our major contribution is that we provide an approach to decision making in the face of uncertainty where the uncertain information is expressed on a space of granular objects and the underlying uncertainty is most generally represented by a measure.

Halting Operations for Algorithmic Alignment

Departing from the discourse on whether a specific (social, ethical) responsibility is attached to the creation and manipulation of algorithms, this article questions the prerequisite of having an identity of algorithms to which that responsibility could be attached. After showing that such identity is partly fictional due to the fact that algorithms are connected to other algorithms and their identity is always a selective reading of a series of transitions through which algorithms come into existence, the perspective is shifted to the algorithmic as the medium of algorithms and as the actual agential domain. This shift translates responsibility into the ability to respond to otherness and non-identity through sensitive forms of alignment. Comparing the algorithmic with the desiring-machines of Deleuze and Guattari, this article proposes that its dynamics of flows and interruptions could be artistically reflected as halting operations that controvert the superficial evaluation of algorithms, for example under the classical decision problem or halting problem. A possible strategy for making the inner dynamics perceivable is proposed through a balancing act between the credible and the incredible, the plausible and the implausible.

The Logical View on Continuous Petri Nets

Continuous Petri nets are a relaxation of classical discrete Petri nets in which transitions can be fired a fractional number of times, and consequently places may contain a fractional number of tokens. Such continuous Petri nets are an appealing object to study, since they over-approximate the set of reachable configurations of their discrete counterparts, and their reachability problem is known to be decidable in polynomial time. The starting point of this article is to show that the reachability relation for continuous Petri nets is definable by a sentence of linear size in the existential theory of the rationals with addition and order. Using this characterization, we obtain decidability and complexity results for a number of classical decision problems for continuous Petri nets. In particular, we settle the open problem about the precise complexity of reachability set inclusion. Finally, we show how continuous Petri nets can be incorporated inside the classical backward coverability algorithm for discrete Petri nets as a pruning heuristic to tackle the symbolic state explosion problem. The cornerstone of the approach we present is that our logical characterization enables us to leverage the power of modern SMT-solvers to yield a highly performant and robust decision procedure for coverability in Petri nets. We demonstrate the applicability of our approach on a set of standard benchmarks from the literature.

N-Queens Problem Resolution Using the Quantum Computing Model

This paper aims to propose solution for a classical and difficult decision problem using quantum computing. At first, it provides experimental samples taking well-known search problems, where it is possible to establish a plausible comparison between classical search algorithms and a similar quantum algorithm. Moreover, a quantum algorithm is construct in order to deliver solution for a sophisticated mathematical problem called N-Queens. The algorithm was implemented in a quantum computing simulator obtaining solutions for problems with up to 128 queens at a reasonable cost.

Problems on Finite Automata and the Exponential Time Hypothesis

We study several classical decision problems on finite automata under the (Strong) Exponential Time Hypothesis. We focus on three types of problems: universality, equivalence, and emptiness of intersection. All these problems are known to be CoNP-hard for nondeterministic finite automata, even when restricted to unary input alphabets. A different type of problems on finite automata relates to aperiodicity and to synchronizing words. We also consider finite automata that work on commutative alphabets and those working on two-dimensional words.

Analytical solution for the size of the minimum dominating set in complex networks

Domination is the fastest-growing field within graph theory with a profound diversity and impact in real-world applications, such as the recent breakthrough approach that identifies optimized subsets of proteins enriched with cancer-related genes. Despite its conceptual simplicity, domination is a classical NP-complete decision problem which makes analytical solutions elusive and poses difficulties to design optimization algorithms for finding a dominating set of minimum cardinality in a large network. Here, we derive for the first time an approximate analytical solution for the density of the minimum dominating set (MDS) by using a combination of cavity method and ultra-discretization (UD) procedure. The derived equation allows us to compute the size of MDS by only using as an input the information of the degree distribution of a given network.

Decision-making without a brain: how an amoeboid organism solves the two-armed bandit.

Several recent studies hint at shared patterns in decision-making between taxonomically distant organisms, yet few studies demonstrate and dissect mechanisms of decision-making in simpler organisms. We examine decision-making in the unicellular slime mould Physarum polycephalum using a classical decision problem adapted from human and animal decision-making studies: the two-armed bandit problem. This problem has previously only been used to study organisms with brains, yet here we demonstrate that a brainless unicellular organism compares the relative qualities of multiple options, integrates over repeated samplings to perform well in random environments, and combines information on reward frequency and magnitude in order to make correct and adaptive decisions. We extend our inquiry by using Bayesian model selection to determine the most likely algorithm used by the cell when making decisions. We deduce that this algorithm centres around a tendency to exploit environments in proportion to their reward experienced through past sampling. The algorithm is intermediate in computational complexity between simple, reactionary heuristics and calculation-intensive optimal performance algorithms, yet it has very good relative performance. Our study provides insight into ancestral mechanisms of decision-making and suggests that fundamental principles of decision-making, information processing and even cognition are shared among diverse biological systems.

Embedding Ordered Sets into Distributive Lattices

This paper investigates the class of ordered sets that are embeddable into a distributive lattice in such a way that all existing finite meets and joins are preserved. The main result is that the following decision problem is NP-complete: Given a finite ordered set, is it embeddable into a distributive lattice with preservation of existing meets and joins? The NP-hardness of the problem is proved by polynomial reduction of the classical 3SAT decision problem into it, and the NP-completeness by presenting a suitable NP-algorithm.