last update: 18.03.06 (SK)         http://www.mathematik.uni-osnabrueck.de/research/OR/class

Open-shop problems without preemption

• maximal polynomially solvable:

  $O \vert p_{ij}=1;tree \vert C_{max}$	Braesel et al. (1994) [4]
  $O2 \vert \vert C_{max}$	Gonzalez & Sahni (1976) [9]
  $O \vert p_{ij}=1;intree \vert L_{max}$	Brucker et al. (1993) [6], Brucker (1998) []
  $O \vert p_{ij}=1;chains;r_i \vert L_{max}$	Baptiste et al. (2004) [3]
  $O2 \vert p_{ij}=1;prec;r_i \vert L_{max}$	Brucker et al. (1993) [6], Brucker (1998) []
  $O \vert p_{ij}=1;outtree \vert \sum C_i$	Braesel et al. (1995) [5]
  $O2 \vert p_{ij}=1;outtree;r_i \vert \sum C_i$	Lushchakova (2006) [15]
  $O2 \vert p_{ij}=1;prec \vert \sum C_i$	Coffman et al. (2003) [8]
  $O \vert p_{ij}=1;r_i \vert \sum C_i$	Brucker & Kravchenko (2004) [7]
  $O \vert p_{ij}=1 \vert \sum w_iC_i$	Brucker et al. (1993) [6]
  $O \vert p_{ij}=1 \vert \sum U_i$	Liu & Bulfin (1988) [14]
  $Om \vert p_{ij}=1;r_i \vert \sum w_iU_i$	Baptiste (2003) [2]
  $O \vert p_{ij}=1 \vert \sum T_i$	Liu & Bulfin (1988) [14]

• minimal NP-hard:

  	$O3 \vert \vert C_{max}$	Gonzalez & Sahni (1976) [9]
  $*$	$O \vert \vert C_{max}$	Lenstra (-) [13]
  $*$	$O \vert p_{ij}=1;outtree;r_i \vert C_{max}$	Timkovsky (2003) [17]
  $*$	$O \vert p_{ij}=1;prec \vert C_{max}$	Timkovsky (2003) [17]
  $*$	$O2 \vert chains \vert C_{max}$	Tanaev et al. (1994) [16]
  $*$	$O2 \vert r_i \vert C_{max}$	Lawler et al. (1981,1982) [11] [12]
  $*$	$O \vert p_{ij}=1;outtree \vert L_{max}$	Timkovsky (2003) [17]
  $*$	$O2 \vert \vert L_{max}$	Lawler et al. (1981,1982) [11] [12]
  $*$	$O2 \vert \vert \sum C_i$	Achugbue & Chin (1982) [1]
  $*$	$O2 \vert p_{ij}=1;chains \vert \sum w_iC_i$	Timkovsky (2003) [17]
  $*$	$O3 \vert p_{ij}=1;chains \vert \sum w_iC_i$	Timkovsky (2003) [17]
  $*$	$O \vert p_{ij}=1;r_i \vert \sum U_i$	Kravchenko (1999) [10]
  $*$	$O2 \vert p_{ij}=1;chains \vert \sum U_i$	Timkovsky (2003) [17]
  $*$	$O3 \vert p_{ij}=1;chains \vert \sum U_i$	Timkovsky (2003) [17]
  $*$	$O2 \vert p_{ij}=1;chains \vert \sum T_i$	Timkovsky (2003) [17]
  $*$	$O3 \vert p_{ij}=1;chains \vert \sum T_i$	Timkovsky (2003) [17]

• minimal open:

  $O3 \vert p_{ij}=1;intree;r_i \vert C_{max}$
  $O3 \vert p_{ij}=1;outtree;r_i \vert C_{max}$
  $O3 \vert p_{ij}=1;prec \vert C_{max}$
  $O3 \vert p_{ij}=1;outtree \vert L_{max}$
  $O2 \vert p_{ij}=1;intree;r_i \vert \sum C_i$

  $O3 \vert p_{ij}=1;chains;r_i \vert \sum C_i$
  $O3 \vert p_{ij}=1;intree \vert \sum C_i$
  $O2 \vert p_{ij}=1;r_i \vert \sum w_iC_i$
  $O3 \vert p_{ij}=1;r_i \vert \sum w_iC_i$
  $O \vert p_{ij}=1 \vert \sum w_iU_i$

  $O2 \vert p_{ij}=1;r_i \vert \sum T_i$
  $O3 \vert p_{ij}=1;r_i \vert \sum T_i$
  $O2 \vert p_{ij}=1 \vert \sum w_iT_i$
  $O3 \vert p_{ij}=1 \vert \sum w_iT_i$

• maximal open:

  $O \vert p_{ij}=1;intree;r_i \vert L_{max}$
  $Om \vert p_{ij}=1;prec;r_i \vert L_{max}$

  $O \vert p_{ij}=1;prec;r_i \vert \sum C_i$
  $O \vert p_{ij}=1 \vert \sum w_iU_i$

  $O \vert p_{ij}=1;r_i \vert \sum w_iT_i$