Lower bounds on the approximation ratios of leading heuristics for the single machine total tardiness problem

Texte intégral

(1)Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision CNRS UMR 7024. CAHIER DU LAMSADE 193 Avril 2002. Lower bounds on the approximation ratios of leading heuristics for the single machine total tardiness problem F. Della Croce, A. Grosso, V. Th. Paschos.

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(180) SEDD = (1, . . . , 2m + 1) F ! C1 (r) r m+r−1−ε r = 1, . . . , m + 1 C1 (r) = 2m + (r − m − 2)ε r = m + 2, . . . , 2m + 1. 7 r = 2, . . . , m + 1 # dr + pr > C1 (r); r = m + 2, . . . , 2m # C1 (r) > dr+1 F 2m + 1 " µ = (1, . . . , 2m + 1) θ = (2, . . . , 2m + 1, 1) T (N, µ) = (1 − ε) + εm(m − 1)/2 − (m − 2)ε + (m + 1) T (N, θ) = m + (m − 2)ε ? T (N, µ) > T (N, θ) 2m + 1 m + (m − 2)ε ! " (2, . . . , 2m) # # # 8 SDEC/EDD = θ > " S ∗ = (1, m + 2, . . . , 2m + 1, 2, . . . , m + 1) T (N, S ∗ ) = 1 + m(m − 1)ε ? ε rDEC/EDD (N, SDEC/EDD ) = (m + (m − 2)ε)/(1 + m(m − 1)ε) ≈ m ≈ n/2. ! rDEC/MDD = rDEC/PSK = rDEC/WI n/3.

(181) ) E6 : n = 3m/2 N = {1, 2, . . . , 3m/2} p1 = m2 pi = 2m i = 2, . . . , m/2 pj = 2 j = (m/2) + 1, . . . , 3m/2 d1 = m2 di = m2 + 2(i − 1)m i = 2, . . . , (m/2) − 1 dm/2 = 2m2 − 2m − ε dj = 2m2 − 2m + ε j = (m/2) + 1, . . . , 3m/2 F 3m/2 # ( + 6 " (1, 2, . . . , 3m/2)

(182) m(m + 1) − (m − 1)ε = m2 + m − (m − 1)ε ( m2 # m2 ? ε ( SDEC/MDD = (2, . . . , 3m/2, 1) T (N, SDEC/MDD ) = m2 ?

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