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# Min-Sum Scheduling Under Precedence Constraints

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## Cite As

Andreas S. Schulz and José Verschae. Min-Sum Scheduling Under Precedence Constraints. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 74:1-74:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ESA.2016.74

## Abstract

In many scheduling situations, it is important to consider non-linear functions of job completions times in the objective. This was already recognized by Smith (1956). Recently, the theory community has begun a thorough study of the resulting problems, mostly on single-machine instances for which all permutations of jobs are feasible. However, a typical feature of many scheduling problems is that some jobs can only be processed after others. In this paper, we give the first approximation algorithms for min-sum scheduling with (nonnegative, non-decreasing) non-linear functions and general precedence constraints. In particular, for 1|prec|sum w_j f(C_j), we propose a polynomial-time universal algorithm that performs well for all functions f simultaneously. Its approximation guarantee is 2 for all concave functions, at worst. We also provide a (non-universal) polynomial-time algorithm for the more general case 1|prec|sum f_j(C_j). The performance guarantee is no worse than 2+epsilon for all concave functions. Our results match the best bounds known for the case of linear functions, a widely studied problem, and considerably extend the results for minimizing sum w_jf(C_j) without precedence constraints.
##### Keywords
• scheduling
• approximation algorithms
• linear programming relaxations
• precedence constraints

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