last update: 06.04.06 (SK)         http://www.mathematik.uni-osnabrueck.de/research/OR/class

Parallel machine problems without preemption

• maximal polynomially solvable:

  $P \vert p_i=p;outtree;r_i \vert C_{max}$	Brucker et al. (1977) [3]
  $P \vert p_i=p;tree \vert C_{max}$	Hu (1961) [18], Davida & Linton (1976) [10]
  $Q \vert p_i=p;r_i \vert C_{max}$	Assignment-problem
  $Q2 \vert p_i=p;chains \vert C_{max}$	Brucker et al. (1999) [5]
  $P \vert p_i=1;chains;r_i \vert L_{max}$	Dror et al. (1998) [12], Baptiste et al. (2004) [1]
  $P \vert p_i=p;intree \vert L_{max}$	Brucker et al. (1977) [3], Monma (1982) [25]
  $P2 \vert p_i=p;prec \vert L_{max}$	Garey & Johnson (1976) [14]
  $P2 \vert p_i=1;prec;r_i \vert L_{max}$	Garey & Johnson (1977) [15]
  $P \vert p_i=1;outtree;r_i \vert \sum C_i$	Brucker et al. (2002) [4], Huo & Leung (2005) [19]
  $P \vert p_i=p;outtree \vert \sum C_i$	Hu (1961) [18]
  $P2 \vert p_i=1;prec;r_i \vert \sum C_i$	Baptiste & Timkovsky (2004) [2]
  $P2 \vert p_i=p;prec \vert \sum C_i$	Coffman & Graham (1972) [9]
  $Pm \vert p_i=p;tree \vert \sum C_i$	Baptiste et al. (2004) [1]
  $Qm \vert p_i=p;r_i \vert \sum C_i$	Dessouky et al. (1990) [11]
  $R \vert \vert \sum C_i$	Horn (1973) [17], Bruno et al. (1974) [8]
  $P \vert p_i=p;r_i \vert \sum w_iC_i$	Brucker & Kravchenko (2008) [7]
  $P \vert p_i=1;r_i \vert \sum w_iU_i$	Networkflowproblem
  $Pm \vert p_i=p;r_i \vert \sum w_iU_i$	Baptiste et al. (2004) [1]
  $Q \vert p_i=p \vert \sum w_iU_i$	Assignment-problem
  $P \vert p_i=p;r_i \vert \sum T_i$	Brucker & Kravchenko (2005) [6]
  $P \vert p_i=1;r_i \vert \sum w_iT_i$	Networkflowproblem
  $Q \vert p_i=p \vert \sum w_iT_i$	Assignment-problem

• maximal pseudopolynomially solvable:

  $Qm \vert r_i \vert C_{max}$	Lawler et al. (1989) [21]
  $Qm \vert \vert \sum w_iC_i$	Lawler et al. (1989) [21]
  $Qm \vert \vert \sum w_iU_i$	Lawler et al. (1989) [21]

• minimal NP-hard:

  	$P2 \vert \vert C_{max}$	Lenstra et al. (1977) [24]
  $*$	$P \vert \vert C_{max}$	Garey & Johnson (1978) [16]
  $*$	$P \vert p_i=1;intree;r_i \vert C_{max}$	Brucker et al. (1977) [3]
  $*$	$P \vert p_i=1;prec \vert C_{max}$	Ullman (1975) [27]
  $*$	$P2 \vert chains \vert C_{max}$	Du et al. (1991) [13]
  $*$	$Q \vert p_i=p;chains \vert C_{max}$	Kubiak (1988) [20]
  $*$	$P \vert p_i=1;outtree \vert L_{max}$	Brucker et al. (1977) [3]
  $*$	$P \vert p_i=1;intree;r_i \vert \sum C_i$	Lenstra (-) [22]
  $*$	$P \vert p_i=1;prec \vert \sum C_i$	Lenstra & Rinnooy Kan (1978) [23]
  $*$	$P2 \vert chains \vert \sum C_i$	Du et al. (1991) [13]
  $*$	$P2 \vert r_i \vert \sum C_i$	Single-machine problem
  	$P2 \vert \vert \sum w_iC_i$	Bruno et al. (1974) [8]
  $*$	$P \vert \vert \sum w_iC_i$	Lenstra (-) [22]
  $*$	$P2 \vert p_i=1;chains \vert \sum w_iC_i$	Timkovsky (2003) [26]
  $*$	$P2 \vert p_i=1;chains \vert \sum U_i$	Single-machine problem
  $*$	$P2 \vert p_i=1;chains \vert \sum T_i$	Single-machine problem

• minimal open:

  $P2 \vert p_i=p;intree;r_i \vert C_{max}$
  $Pm \vert p_i=1;intree;r_i \vert C_{max}$
  $Pm \vert p_i=1;prec \vert C_{max}$
  $Q2 \vert p_i=p;chains;r_i \vert C_{max}$
  $Q2 \vert p_i=p;intree \vert C_{max}$
  $Q2 \vert p_i=p;outtree \vert C_{max}$
  $Qm \vert p_i=p;chains \vert C_{max}$

  $P2 \vert p_i=p;chains;r_i \vert L_{max}$
  $Pm \vert p_i=1;outtree \vert L_{max}$
  $Q2 \vert p_i=p;chains \vert L_{max}$
  $Q2 \vert p_i=p;r_i \vert L_{max}$
  $P \vert p_i=1;intree \vert \sum C_i$
  $P2 \vert p_i=p;chains;r_i \vert \sum C_i$
  $Pm \vert p_i=1;intree;r_i \vert \sum C_i$

  $Pm \vert p_i=1;prec \vert \sum C_i$
  $Q \vert p_i=p;r_i \vert \sum C_i$
  $Q2 \vert p_i=p;chains \vert \sum C_i$
  $Q2 \vert p_i=p;r_i \vert \sum w_iC_i$
  $P \vert p_i=p;r_i \vert \sum U_i$
  $P2 \vert p_i=p;r_i \vert \sum w_iT_i$

• maximal open:

  $P \vert p_i=p;chains;r_i \vert L_{max}$
  $Qm \vert p_i=p;prec;r_i \vert L_{max}$
  $Q \vert p_i=p;outtree;r_i \vert \sum C_i$

  $Q \vert p_i=p;tree \vert \sum C_i$
  $Qm \vert p_i=p;prec;r_i \vert \sum C_i$

  $Q \vert p_i=p;r_i \vert \sum w_iU_i$
  $Q \vert p_i=p;r_i \vert \sum w_iT_i$