On the Complexity of Decentralized Decision Making and

On the Complexity of Decentralized Decision Making and -...

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440 IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-30. NO. 5, MAY 1985 On the Complexity of Decentralized Decision Making and Detection Problems Abstract-We study the computational complexity of the discrete versions of some simple but basic decentralized decision problems. These problems are variationsof the classical ‘‘team decision problem” and inclnde the problem of decentralized detection whereby a central processor is to select one of two hypotheses. based on l-bit messages from two noncommonicating sensors. Our results point to the inherent diff~cdty of decentralized decision making and suggest that optimality may be an elusive goal. T I. INTRODUCTION HE field of decentralized (distributed) decision making has been an active area of research for more than two decades 141, [ 1 11, [ 121. In the meantime, it has been realized that decentralized decision problems are qualitatively different from the correspond- ing decision problems with centralized information. The classical “counterexample” of Witsenhausen [SI, [17] in decentralized stochastic control best illustrates this point. It is safe to conjecture that the prohibitive factor in decentralized problems is not so much the inadequacy of the mathematical tools presently been used, but rather the inherent complexity (in the broad sense of the term) ofthe problems that have usually been formulated. However, there are very few [3], [ 181 precise mathematical results on the nature of this ever-present complexity. The present paper, which follows the line of research of [lo], should be viewed as a contribution in this direction. We focus on finite versions of some simple but fairly typical decentralized decision-making problems and characterize their complexity by using the tools of the theory of computational complexity [2], [9]. Keeping with the tradition this theory, we consider “easy” those problems that can be solved by a polynomial algorithm, whereas we consider NP- complete (or worse) problems to be “hard.” In our opinion, such an approach is: 1) more satisfying intellectually, and 2) given the present state of the theory of decentralized decision making, it will &OW us to systematically identify hard problems ana redirect research efforts to heuristic and approximate algorithms or possibly easier special cases. Overview The main issue of interest in decentralized systems may be loosely phrased as “who should communicate to whom, what, when, etc.” From a purely logical point of view, however, there is a question which precedes the above: “Are there any communications necessary?” Section II addresses the difficulty of the problem of deciding whether any communications are neces- sary for a given decentralized system. We use a formulation of this problem introduced in [ 101.
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On the Complexity of Decentralized Decision Making and -...

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