Machine Theory

# Constraint Integer Programming by TobiasAchterberg

By TobiasAchterberg

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Additional resources for Constraint Integer Programming

Example text

Xjk ¸ Djri = [lji , uji ]¸ lji , uji ∈ ÒØ × Ø × Ø ÓÒ ÔÖÓ Ð Ñ ËÈ = (C, D) Û Ø ∈ Ê¸ ÓÖ Dji = {lji , . . , uji }¸ lji , uji ∈ ¸ Ð × Dji = [lji , uji ]¸ lji , uji Ò¸ C × ÐÐ ÓÙÒ ÓÒ× ×Ø ÒØ ∀i ∈ {1, . . , k} ∀xji ∈ {lji , uji } ∃x⋆ ∈ Djr1 × . . × Djrk : x⋆ji = xji ∧ C(x⋆ ) = 1. ËÈ Û Ø ÓÑ ÓÒ×ØÖ ÆÓØ Ø ÓÒ× ×Ø ÒØ Ø ÒØ× Ò ÓÒ Ö Ð¹Ú ÐÙ ÓÙÒ ÓÒ× ×Ø ÒØ Ò× × ÐÐ ÓÙÒ ÓÒ× ×Ø Ò
Ý × Û ËÈ Ò Û Ø ÓÒ× ×Ø ÒØº ÇÒ Ø ÓØ ÓÒ× ×Ø ÒØ¸ ÓÐÐÓÛ Ò × Ø Ü ÑÔÐ ¾º º Ä Ø ÐÐ Ó Ö ÓÒ×ØÖ Ò ¸Ø Ö Ø ÒØ× Ö Ü ÑÔÐ Ö Ú Ö Ð × Ø× ÓÒ×ØÖ Ò ÒØ× Ú Ö Ö Ð × Û Ø ÓÙÒ Ò ÒØ ÖÚ Ð ÓÒ× ×Ø Ò
Ý Ö ÓÙÒ Ò ÓÒ Ö Ð¹Ú ÐÙ ÓÒ× ×Ø ÒØ Ú ÖÝ ÒØ ÖÚ Ð Ú Ö ËÈ× Ø ÒØ ÖÚ Ð ÓÒ× ×Ø ÒØº Ø Ð × × Ö ÓÙÒ ÒÓØ ÒØ ÖÚ Ð ÐÐÙ×ØÖ Ø × C : [0, 1] × [0, 1] × [0, 1] → {0, 1} Ø Ð Ò Ö ÓÒ×ØÖ ÒØ C(x) = 1 ⇔ 2x1 + 2x2 + 2x3 = 3.

Xjk ¸ Djri = [lji , uji ]¸ lji , uji ∈ ÒØ × Ø × Ø ÓÒ ÔÖÓ Ð Ñ ËÈ = (C, D) Û Ø ∈ Ê¸ ÓÖ Dji = {lji , . . , uji }¸ lji , uji ∈ ¸ Ð × Dji = [lji , uji ]¸ lji , uji Ò¸ C × ÐÐ ÓÙÒ ÓÒ× ×Ø ÒØ ∀i ∈ {1, . . , k} ∀xji ∈ {lji , uji } ∃x⋆ ∈ Djr1 × . . × Djrk : x⋆ji = xji ∧ C(x⋆ ) = 1. ËÈ Û Ø ÓÑ ÓÒ×ØÖ ÆÓØ Ø ÓÒ× ×Ø ÒØ Ø ÒØ× Ò ÓÒ Ö Ð¹Ú ÐÙ ÓÙÒ ÓÒ× ×Ø ÒØ Ò× × ÐÐ ÓÙÒ ÓÒ× ×Ø Ò
Ý × Û ËÈ Ò Û Ø ÓÒ× ×Ø ÒØº ÇÒ Ø ÓØ ÓÒ× ×Ø ÒØ¸ ÓÐÐÓÛ Ò × Ø Ü ÑÔÐ ¾º º Ä Ø ÐÐ Ó Ö ÓÒ×ØÖ Ò ¸Ø Ö Ø ÒØ× Ö Ü ÑÔÐ Ö Ú Ö Ð × Ø× ÓÒ×ØÖ Ò ÒØ× Ú Ö Ö Ð × Û Ø ÓÙÒ Ò ÒØ ÖÚ Ð ÓÒ× ×Ø Ò
Ý Ö ÓÙÒ Ò ÓÒ Ö Ð¹Ú ÐÙ ÓÒ× ×Ø ÒØ Ú ÖÝ ÒØ ÖÚ Ð Ú Ö ËÈ× Ø ÒØ ÖÚ Ð ÓÒ× ×Ø ÒØº Ø Ð × × Ö ÓÙÒ ÒÓØ ÒØ ÖÚ Ð ÐÐÙ×ØÖ Ø × C : [0, 1] × [0, 1] × [0, 1] → {0, 1} Ø Ð Ò Ö ÓÒ×ØÖ ÒØ C(x) = 1 ⇔ 2x1 + 2x2 + 2x3 = 3.

Djrk → {0, 1}, ÓÒ×ØÖ ÒØ Ê¸ i = 1, . . , k¸ Û ÒØ ÖÚ Ð ÓÑ i = 1, . . , k º Ò× Ì Ò ÓÒ Ö Ð¹Ú ÐÙ × Ô ÖØ Ó Ú Ö ÓÒ×ØÖ xj1 , . . , xjk ¸ Djri = [lji , uji ]¸ lji , uji ∈ ÒØ × Ø × Ø ÓÒ ÔÖÓ Ð Ñ ËÈ = (C, D) Û Ø ∈ Ê¸ ÓÖ Dji = {lji , . . , uji }¸ lji , uji ∈ ¸ Ð × Dji = [lji , uji ]¸ lji , uji Ò¸ C × ÐÐ ÓÙÒ ÓÒ× ×Ø ÒØ ∀i ∈ {1, . . , k} ∀xji ∈ {lji , uji } ∃x⋆ ∈ Djr1 × . . × Djrk : x⋆ji = xji ∧ C(x⋆ ) = 1. ËÈ Û Ø ÓÑ ÓÒ×ØÖ ÆÓØ Ø ÓÒ× ×Ø ÒØ Ø ÒØ× Ò ÓÒ Ö Ð¹Ú ÐÙ ÓÙÒ ÓÒ× ×Ø ÒØ Ò× × ÐÐ ÓÙÒ ÓÒ× ×Ø Ò
Ý × Û ËÈ Ò Û Ø ÓÒ× ×Ø ÒØº ÇÒ Ø ÓØ ÓÒ× ×Ø ÒØ¸ ÓÐÐÓÛ Ò × Ø Ü ÑÔÐ ¾º º Ä Ø ÐÐ Ó Ö ÓÒ×ØÖ Ò ¸Ø Ö Ø ÒØ× Ö Ü ÑÔÐ Ö Ú Ö Ð × Ø× ÓÒ×ØÖ Ò ÒØ× Ú Ö Ö Ð × Û Ø ÓÙÒ Ò ÒØ ÖÚ Ð ÓÒ× ×Ø Ò
Ý Ö ÓÙÒ Ò ÓÒ Ö Ð¹Ú ÐÙ ÓÒ× ×Ø ÒØ Ú ÖÝ ÒØ ÖÚ Ð Ú Ö ËÈ× Ø ÒØ ÖÚ Ð ÓÒ× ×Ø ÒØº Ø Ð × × Ö ÓÙÒ ÒÓØ ÒØ ÖÚ Ð ÐÐÙ×ØÖ Ø × C : [0, 1] × [0, 1] × [0, 1] → {0, 1} Ø Ð Ò Ö ÓÒ×ØÖ ÒØ C(x) = 1 ⇔ 2x1 + 2x2 + 2x3 = 3.