Class Of Decision Problems Research Articles

The Price of Connectivity for Vertex Cover

Graph Theory The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the \em Price of Connectivity, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number t. We obtain forbidden induced subgraph characterizations for every real value t ≤q 3/2. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most t. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NP-hard. In fact, we show that it is even complete for the class Θ₂^P = P^NP[\log], the class of decision problems that can be solved in polynomial time, provided we can make O(\log n) queries to an NP-oracle. This paves the way for a thorough investigation of the complexity of problems involving ratios of graph invariants.

Towards a complexity theory for local distributed computing

A central theme in distributed network algorithms concerns understanding and coping with the issue of locality . Yet despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems . In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard LOCAL model of computation and define LD( t ) (for local decision ) as the class of decision problems that can be solved in t communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD( t , p , q ), containing all languages for which there exists a randomized algorithm that runs in t rounds, accepts correct instances with probability at least p , and rejects incorrect ones with probability at least q . We show that p 2 + q = 1 is a threshold for the containment of LD( t ) in BPLD( t , p , q ). More precisely, we show that there exists a language that does not belong to LD( t ) for any t = o ( n ) but does belong to BPLD( 0 , p , q ) for any p , q ∈ (0,1) such that p 2 + q ≤ 1. On the other hand, we show that, restricted to hereditary languages, BPLD( t , p , q )=LD( O ( t )), for any function t , and any p , q ∈ (0,1) such that p 2 + q > 1. In addition, we investigate the impact of nondeterminism on local decision, and establish several structural results inspired by classical computational complexity theory. Specifically, we show that nondeterminism does help, but that this help is limited, as there exist languages that cannot be decided locally nondeterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with nondeterminism that enables to decide all languages in constant time . Finally, we introduce the notion of local reduction, and establish a couple of completeness results.

Treatment decisions under ambiguity

Many health risks are ambiguous in the sense that reliable and credible information about these risks is unavailable. In health economics, ambiguity is usually handled through sensitivity analysis, which implicitly assumes that people are neutral towards ambiguity. However, empirical evidence suggests that people are averse to ambiguity and react strongly to it. This paper studies the effects of ambiguity aversion on two classical medical decision problems. If there is ambiguity regarding the diagnosis of a patient, ambiguity aversion increases the decision maker's propensity to opt for treatment. On the other hand, in the case of ambiguity regarding the effects of treatment, ambiguity aversion leads to a reduction in the propensity to choose treatment.

THE FRONTIER OF DECIDABILITY IN PARTIALLY OBSERVABLE RECURSIVE GAMES

The classical decision problem associated with a game is whether a given player has a winning strategy, i.e. some strategy that leads almost surely to a victory, regardless of the other players' strategies. While this problem is relevant for deterministic fully observable games, for a partially observable game the requirement of winning with probability 1 is too strong. In fact, as shown in this paper, a game might be decidable for the simple criterion of almost sure victory, whereas optimal play (even in an approximate sense) is not computable. We therefore propose another criterion, the decidability of which is equivalent to the computability of approximately optimal play. Then, we show that (i) this criterion is undecidable in the general case, even with deterministic games (no random part in the game), (ii) that it is in the jump 0', and that, even in the stochastic case, (iii) it becomes decidable if we add the requirement that the game halts almost surely whatever maybe the strategies of the players.

Estimating Optimal Decision Rules in the Presence of Model Parameter Uncertainty

This article proposes a bootstrap procedure for estimating the risk minimizing decision-rule from within a parameterized family of rules. The procedure is conceptually simple and applicable to a broad class of decision problems involving parameter uncertainty. Moreover, when applied to Markowitz's (1952) model of portfolio selection--a leading example in which parameter uncertainty arises--it is shown to be capable of generating decision rules with lower risk than estimators derived from conventional methods. Theoretical results establishing the asymptotic optimality of the procedure are also presented. Copyright The Author, 2012. Published by Oxford University Press. All rights reserved., Oxford University Press.

Estimating Optimal Decision Rules in the Presence of Model Parameter Uncertainty

This paper proposes a bootstrap procedure for estimating the risk minimizing decision rule from within a parameterized family of rules. The procedure is conceptually simple and applicable to a broad class of decision problems involving parameter uncertainty. Moreover, when applied to Markowitz’s (1952) model of portfolio selection — a leading example in which parameter uncertainty arises — it is shown to be capable of generating decision rules with lower risk than estimators derived from conventional methods. Theoretical results establishing the asymptotic optimality of the procedure are also presented.

Hardness results on the gapped consecutive-ones property problem

Motivated by problems in comparative genomics and paleogenomics, we study the computational complexity of the Gapped Consecutive-Ones Property ((k,δ)-C1P) Problem: given a binary matrix M and two integers k and δ, decide if the columns of M can be permuted such that each row contains at most k blocks of ones and no two neighboring blocks of ones are separated by a gap of more than δ zeros. The classical C1P decision problem, which is known to be polynomial-time solvable is equivalent to the (1,0)-C1P problem. We extend our earlier results on this problem [C. Chauve, J. Mauch, M. Patterson, On the gapped consecutive-ones property, in: Proceedings of the European Conference on Combinatorics, Graphs Theory and Applications (EuroComb), in: Electronic Notes in Discrete Mathematics, vol. 34, 2009, pp. 121–125] to show that for every k≥2,δ≥1,(k,δ)≠(2,1), the (k,δ)-C1P Problem is NP-complete, and that for every δ≥1, the (∞,δ)-C1P Problem is NP-complete. On the positive side, we also show that if k,δ and the maximum degree of M are constant, the problem is related to the classical Graph Bandwidth Problem and can be solved in polynomial time using a variant of an algorithm of Saxe [J.B. Saxe, Dynamic-programming algorithms for recognizing small-bandwidth graphs in polynomial time, SIAM Journal on Algebraic and Discrete Methods 1 (4) (1980) 363–369].

Generalized constraint diagrams and the classical decision problem

Constraint diagrams were proposed as a means of modelling software systems, with generalized constraint diagrams being a recent refinement. It is known that generalized constraint diagrams are more expressive than constraint diagrams and can express any first-order logic statement that uses monadic or dyadic predicates. Thus, the generalized constraint diagram logic is undecidable. In this article, we develop a decision procedure for the so-called existential fragment of generalized constraint diagrams, building on previous work for a simpler, less expressive, fragment of the logic.

H.264/SVC Mode Decision Based on Optimal Stopping Theory

Fast mode decision algorithms have been widely used in the video encoder implementation to reduce encoding complexity yet without much sacrifice in the coding performance. Optimal stopping theory, which addresses early termination for a generic class of decision problems, is adopted in this paper to achieve fast mode decision for the H.264/Scalable Video Coding standard. A constrained model is developed with optimal stopping, and the solutions to this model are employed to initialize the candidate mode list and predict the early termination. Comprehensive simulation results are conducted to demonstrate that the proposed method strikes a good balance between low encoding complexity and high coding efficiency.

Quantitative languages

Quantitative generalizations of classical languages, which assign to each word a real number instead of a Boolean value, have applications in modeling resource-constrained computation. We use weighted automata (finite automata with transition weights) to define several natural classes of quantitative languages over finite and infinite words; in particular, the real value of an infinite run is computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We define the classical decision problems of automata theory (emptiness, universality, language inclusion, and language equivalence) in the quantitative setting and study their computational complexity. As the decidability of the language-inclusion problem remains open for some classes of weighted automata, we introduce a notion of quantitative simulation that is decidable and implies language inclusion. We also give a complete characterization of the expressive power of the various classes of weighted automata. In particular, we show that most classes of weighted automata cannot be determinized.

Investment-Based Cost Accounting as a Fundamental Basis of Decision-Oriented Management Accounting

Decision making concerns over cost allocations, especially common cost allocations, have a long history. They are well canvassed in Thomas (1969) and Wells (1978). This article revisits the cost allocation debate, albeit in a new setting, and rehearses arguments relevant to long- and short-term decision contexts. Here a means is proposed to address those problems, namely to adopt the investment-based approach to cost accounting. This approach draws on ideas of Hotelling (1925), Preinreich (1938) and Schneider (1961), and applies the notion of net present value in another setting, namely to cost accounting theory. Research has revealed no discussion of this in the Anglo-American literature. This article shows analytically that the investment-based approach offers a general basis for decision-oriented cost accounting, as it combines investment theory with cost accounting and thereby connects long-term with short-term decisions. While reviewing primarily European literature, it also examines several Anglo-American works. The analysis reveals how for three classical decision problems—production program planning, purchase order lot sizes, and break-even price limits—two different types of costs, namely depreciation and material costs, have to be based on cash flows and net present value. The proposed investment-based approach permits an examination of the extent to which cost accounting concepts and cost information are relevant to those decisions. This theoretical concept is used to derive pertinent cost dimensions and to solve traditional problems of cost allocation. A caution is that the investment approach is limited to decision facilitating cost accounting. Whether it may be possible to couple it with agency theory and its focus on decision influencing has not been explored and is an issue for further research.

On problems without polynomial kernels

Kernelization is a strong and widely-applied technique in parameterized complexity. A kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomially-bounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world ( i.e. non-parametric complexity), and revolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which is of independent interest. Using the notion of distillation algorithms, we develop a generic lower-bound engine that allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include k- Path, k- Cycle, k- Exact Cycle, k- Short Cheap Tour, k- Graph Minor Order Test, k- Cutwidth, k- Search Number, k- Pathwidth, k- Treewidth, k- Branchwidth, and several optimization problems parameterized by treewidth and other structural parameters.

Highly Undecidable Problems For Infinite Computations

We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and ``highly In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $\Pi_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.

A New Model for Team Optimization: The Effects of Uncertainty on Interaction

The objective of this paper is twofold. First, a new model of team optimization is formulated. Second, this model is used to investigate the effects of uncertainty on interaction. A model of team optimization that encompasses the classical team decision problem is introduced. This model is suitable for problems where agents' posterior information is not shared and is possibly inconsistent with the mutual prior information. For a broad class of problems, every agent's dominant beliefs about the posterior information of the other agents are derived. Then, the level of interaction and the level of uncertainty are defined, and the relationship between these two levels is studied. It is shown that the optimal level of interaction decreases as the level of uncertainty increases, and in some cases, the optimal level of interaction tends to zero, suggesting that the optimization problem may be decomposed. The theoretical results are demonstrated on sensor network examples.

Two dimensional cross-layer optimization for packet transmission over fading channel

In this paper a single-input-single-output wireless data transmission system with adaptive modulation and coding over correlated fading channel is considered, where run-time power adjustment is not available. Higher layer data packets are enqueued into a finite size buffer space before being released into the time-varying wireless channel. Without fixing the physical layer error probability, the objective is to minimize the average joint packet loss rate due to both erroneous transmission and buffer overflow. Two optimization techniques are incorporated to achieve the best solution. The first is policy domain optimization that formulates the data rate adaptation design as classical Markov decision problem. The second is channel domain optimization that appropriately partitions the channel variation based on particular fading environment and carried traffic pattern. The derived policy domain analytical model can precisely map any policy design into various QoS performance metrics with finite buffer setup. We then propose a tractable suboptimization framework to produce different two-dimensional suboptimal solutions with scalable complexity-optimality tradeoff for practical implementations.

Extended TOPSISs for Belief Group Decision Making

Multiple attribute decision analysis (MADA) problems in the situation of belief group decision making (BGDM) are a special class of decision problems, where the attribute evaluations of each decision maker (DM) are represented by belief functions. In order to solve these special problems, in this paper, TOPSIS (technique for order preference by similarity to ideal solution) model is extended by three approaches, by which group preferences are aggregated in different manners. Corresponding to the three approaches, three extended TOPSIS models, the pre-model, post-model, and inter-model, are developed and their procedures are elaborated step by step. Aggregating group preferences in the three extended models respectively depends on Dempster’s rule or its modifications, some social choice functions, and some mean approaches. Furthermore, a numerical example clearly illustrates the procedures of the three extended models for BGDM.

An Approximation Algorithm for the Discrete Team Decision Problem

In this paper we study a discrete version of the classical team decision problem. It has been shown previously that the general discrete team decision problem is NP-hard. Here we present an efficient approximation algorithm for this problem. For the maximization version of this problem with nonnegative rewards, this algorithm computes decision rules which are guaranteed to be within a fixed bound of optimal.

A measure of risk and a decision-making model based on expected utility and entropy

In this paper we extend the classical decision model under risk to a more general case, in which the state of nature corresponding to each risky action may have its own distribution. More specifically, we propose an expected utility-entropy (EU-E) measure of risk and a decision-making model based on expected utility and entropy. The EU-E measure of risk reflects an individual's intuitive attitude toward risk. The decision model incorporates the expected utility decision criterion as a special case. Using this decision model, a class of decision problems, which cannot be dealt with the expected utility or mean–variance criterion reasonably, can be solved. Besides, some famous decision paradoxes can be interpreted. This decision model can either serve as a descriptive or a normative decision model involving risk.

Abstract Combinatorial Programs and Efficient Property Testers

Property testing is a relaxation of classical decision problems which aims at distinguishing between functions having a predetermined property and functions being far from any function having the property. In this paper we present a novel framework for analyzing property testing algorithms. Our framework is based on a connection of property testing and a new class of problems which we call abstract combinatorial programs. We show that if the problem of testing a property can be reduced to an abstract combinatorial program of small dimension, then the property has an efficient tester. We apply our framework to a variety of problems. We present efficient property testing algorithms for geometric clustering problems, for the reversaldistance problem, and for graph and hypergraph coloring problems. We also prove that, informally, any hereditary graph property can be efficiently tested if and only if it can be reduced to an abstract combinatorial program of small size. Our framework allows us to analyze all our testers in a unified way, and the obtained complexity bounds either match or improve the previously known bounds. Furthermore, even if the asymptotic complexity of the testers is not improved, the obtained proofs are significantly simpler than the previous ones. We believe that our framework will help to understand the structure of efficiently testable properties.

The impact of problem size on decision processes: an experimental investigation on very large choice problems with support of decision support systems

Abstract: Choice problems as a class of decision problems have attracted great attention for the last couple of decades. Among the frameworks and supporting theories used in their study, two have had the greatest impact: bounded rationality and cost–benefit. Both theories could find support from past empirical studies under different conditions or problem environments. In the past studies, problem size has been shown to play an important role in decision‐making. As problem size increases, a decision process may be detoured and the decision outcome may be different. In this paper we investigate the impact of problem size on three important aspects of the computer‐aided decision process – strategy selection, decision time/effort, and decision quality – through very large choice problems.