On the Approximability of Single-Machine Scheduling with Precedence Constraints

Authors:
Christoph Ambühl;Monaldo Mastrolilli;Nikolaus Mutsanas;Ola Svensson
Affiliations:
Department of Computer Science, The University of Liverpool, Liverpool L69 3BX, United Kingdom;Instituto Dalle Molle di Studi sull'Intelligenza Artificiale, 6928 Manno-Lugano, Switzerland;Université Libre de Bruxelles, Département de Mathématique, 1150 Ixelles-Bruxelles, Belgium;School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
Venue:
Mathematics of Operations Research
Year:
2011

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Abstract

We consider the single-machine scheduling problem to minimize the weighted sum of completion times under precedence constraints. In a series of recent papers, it was established that this scheduling problem is a special case of minimum weighted vertex cover. In this paper, we show that the vertex cover graph associated with the scheduling problem is exactly the graph of incomparable pairs defined in the dimension theory of partial orders. Exploiting this relationship allows us to present a framework for obtaining (2-2/f)-approximation algorithms, provided that the set of precedence constraints has fractional dimension of at most f. Our approach yields the best-known approximation ratios for all previously considered special classes of precedence constraints, and it provides the first results for bounded degree and orders of interval dimension 2. On the negative side, we show that the addressed problem remains NP-hard even when restricted to the special case of interval orders. Furthermore, we prove that the general problem, if a fixed cost present in all feasible schedules is ignored, becomes as hard to approximate as vertex cover. We conclude by giving the first inapproximability result for this problem, showing under a widely believed assumption that it does not admit a polynomial-time approximation scheme.