By Satoru Iwata, Satoko Moriguchi, Kazuo Murota

This paper provides a quicker set of rules for the M-convex submodular How challenge, that is a generalization of the minimum-cost How challenge with an M-convex price functionality for the How-boundary, the place an M-convex functionality is a nonlinear nonseparable cliserete convex functionality on integer issues. The set of rules extends the means sealing method lor the submodular How challenge by way of Fleischer. Iwata and MeCormiek (2002) by using a unique means of altering the capability via fixing greatest submodular How difficulties.

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✴ ✭✤❄✌ ✭✘✤✙✮❑✎✡✜✣✤✍✺€✜ ✙ ✭ ✝ ✂ ☛✞ ✁ ✂ ✂ ✁ ✂ ✁ ✡ ✂ ☛ ✠☛✡ ✂ ✒✓✡ ✂ ✁ ✁ ✁✞ ✡ ✡ ✁ ✂ ✡ ✂ ✂ ✡ ✂ ✁ ✡ ✒✓✡ ✁ ✡ ✂ ✠☛✡ ✂ ✂ ✂ ✁ ✂ ✁ ✁ ✂ ✁ ✁ ✂ ✂ ✁ ✑✥✤✛✺✣✄✘✤❄✕✒✁✪✭✘✮✡✑✥✺€✕✒✑✥✄✘✤✵✕☎✄❑✕✒✎✡✜■✭✳✓✒✓✒✸✡❱❑✗●✕☎✑✲✄✳✤❦✞ ✞ ✘★✩✎✡✖✜✻✩✧✡ s❬✜✣✆ ✑✥✽✘❇✢✶ ✎❄✕❘✧✖✄✝✠✢✟ ✆❬✖✕✒✠✎✡✜ ✗✖❡❩✸ ✾❶✕✒✎ ❯ ✤❙❱✻✸✡✽✘✭✘✽✳✤✢✜€✕❘✎✙✭✘✑✥✓ ✓✿✑✲✤❙✱✳✄✳✜✧✸✡✓✖✕✒✤✙✜✧✺✣✮ ✜✧✓✧✭✝✞✵✦✥✦❈❛✡✎✡✄✘✑✔✁■✓☛✓✒❱♦✑✥❱♦✄✳✤✡✗✙✜✣✦✲✑✔✴✛✺€✑✲✕✟✑✥✤✴✹✌❈❀✝s✏❥q✜✻✽✘✭✘✄✳✓☎✦✥✮✛✓✒✸✡✤❙❱❑✸✡✜⑥✽✳✽✘✕☎✜€✎✙✕✪✭❜✓✣✕ ✞ ✞ ✌✑ ✞ ❾ ✭ ✮ ❾ ✣✴✣✴✣ ❾ ✭✡✠ ✜ ✭ ★ ✚ ☞ ✔✌ ✘✭ ✁✒✜❬✤✙✭❜✕☎✸✡✁✪✭✝✦❄✤❄✸✙❱❇✬▲✜✣✁✪✓✣✞ ✎✡✜✣✤⑥✕✒✎✙✜❬❱✻✭✘✤♦✮●✑✥✜✧✓ ❜✎✡✑✔✓✶✦✥✭✳✓✟✕✶s❖✑✥✦✲✦✡✺✣✄✘✤❄✕☎✭✘✑✲✤✙✓ ✕✒✎✙✜✹✆➛✄✘✦✥✦✲✄❜s❖✑✥✤✡✽ ✺✣✄✘✤✙✮●✑✲✕✒✑✥✄✘✤✷♥✢❩❉✆❘✎✡✑✥✓q✁✒✑✔✺✪✎✡✜☞✓✻✺✣✭✝✤ ✬▲✜✢✓✒✗✡✦✲✑✲✕q✑✥✤❄✕✒✄ ✕✟s❬✄ ✗✙✭✝✁✒✕☎✓✻✄✘✆❘✜ ❄✸✙✭✝✦ s❬✜✣✑✥✽✘✎❄✕ ❜✕☎✎✡✜✣✤♦✜✧✭✳✺✪✎❇✄✘✆✙✎✡✑✔✓✼✮✡✭✝✸✡✽✳✎❄✕✒✜✣✁✪✓◗✓✖✎✡✄✳✸✡✦✔✮❇✄✳✬●✕☎✭✘✑✲✤⑥✄✘✤✡✜✩✄✝✆✙✕✒✎✡✜☞✓✖✜✩✗✙✭✘✁✖✕✪✓✣✞❈❩❉✆✍✑✥✤✙✓✟✕☎✜✧✭✳✮ ✕✒✎✙✜✣✁☎✜❘✑✥✓✩✤✡✄✵✓✒✸✙✺✪✎r✗✙✭✝✁✒✕✒✑✲✕✒✑✥✄✘✤ ❙✕✒✎✡✜✧✤✛✭✘✦✲✦➆✎✙✑✥✓❖✽✳✄✘✦✔✮✫✓✒✎✡✄✳✸✡✦✥✮✫✽✘✄♦✕☎✄⑥✕☎✎✡✜■✺✪✎❙✸✡✁✪✺✪✎❦✞ ★✩✎✡✜♦✽✘✜✧✤✡✜✣✁✪✭✝✦ ✳✸✙✜✧✓✖✕✒✑✥✄✘✤✹✕☎✎✡✜✣✁☎✜€✆➛✄✳✁✒✜⑥✑✥✓✧♥✰❩✟✓☛✑➀✕❏✗✍✄❄✓✒✓✒✑✥✬✡✦✲✜❇✕✒✄✛✮●✑✥✱❄✑✔✮●✜♦✭q✓✒✜€✕❏✄✝✆ ✂ ✤✙✭✝✕✒✸✡✁✪✭✝✦ ✤❙✸✡❱❇✬▲✜✣✁✪✓ ✎ ✭ ✞ ❾ ❾✳✭ ✠ ✖☛✑✥✤✻✕✟s❬✄❇✓✖✸✙✬✙✓✖✜✣✕☎✓❬✓✖✸✍✺✪✎✻✕✒✎✙✭✝✕♠✕☎✎✡✜☛✓✒✸✡❱✻✓♠✄❜✱✳✜✣✁✂✜☞✭✘✺✪✎✵✓✒✸✡✬✙✓✒✜€✕✩✭✝✁☎✜ ✜ ❄✸✙✭✘✆ ✦ ☛ ★✩✎✙✑✥✓♦✗✡✁☎✄✘✬✡✦✥✜✣❱ ✑✔✓✻✺✣✭✘✦✲✦✥✜✧✮ ✁ ✭✘✤✙✮ ✑✔✓ ✁ ❫✾❉✺✣✄✘❱❑✗✡✦✥✜€✕☎✜✘✞ ★✩✎✡✜ ✝ ✝ ✓✖✕✒✁✪✭✝✑✥✽✘✎❄✕✖✆➛✄✳✁✒s✩✭✝✁✪✮ ✕☎✁✒✑✔✭✝✦✲✾❁✭✝✤✙✮❙✾❉✜✣✁☎✁☎✄✘✁❑❱❑✜€✕✒✎✙✄❙✮ ✑✥✓✫✄✘✤✡✦✥✴◆✗✡✁☎✭✳✺❽✕☎✑✥✺✧✭✝✦❖✆➛✄✳✁✫✓✒❱❑✭✘✦✲✦ ✂ ✿✭✘✓q✑➀✕ ✁☎✜ ❄✸✡✑✥✁✒✜☞✓✂✕☎✜✧✓✖✕✒✑✥✤✡✽❇✄✘✆❈✭✝✦✥✦ ❃ ✠ ✗✍✄❄✓✒✓✒✑✥✬✡✑✲✦✥✑✲✕✒✑✥✜✧✓✏✑✲✤q✕✒✎✡✜❫s✏✄✳✁☎✓✖✕❬✺✧✭✘✓✒✜✘✞ ❣ ✄✘✕✒✜✰✕☎✎✙✭❜✕✩✑✲✆❈✓✖✄✳❱❑✜✣✄✘✤✙✜ ✽✘✑✥✱✘✜☞✓✩✴✘✄✳✸r✭❑✓✒✄✘✦✥✸●✕✒✑✥✄✘✤ ●✴✳✄✘✸✛✺✣✭✘✤✫✱✳✜✣✁☎✑➀✆➛✴✻✕☎✎✡✑✔✓✰✓✖✄✳✦✲✸✡✕✒✑✥✄✘✤r✱✘✜✣✁☎✴✻✜✧✭✳✓✖✑✥✦✥✴✘✞ ❃ ★✩✎✡✜❑✆➛✄✘✦✥✦✥✄❜s❖✑✲✤✡✽✹✗✙✁✒✄✳✬✡✦✲✜✧❱ ◗✺✧✭✝✦✥✦✲✜☞✌ ✮ ☞ ✁ ✝ ✝ ✁ ✁ ✚ ✝ ❈✑✔✓■✭✝✦✔✓✒✄ ✁ ❫✾❁✺€✄✘❱❑✗✡✦✥✜€✕☎✜✘✞⑥❍❫✑✥✱✘✜✧✤ ✭✘✤ ✸✡✤✍✮●✑✲✁☎✜✧✺€✕✒✜☞✮✢✽✘✁✪✭✝✗✙✎ s❖✑✲✕✒✎ ✂ ✤✡✄●✮●✜✧✓ ❦✮●✄❙✜✧✓☛✕☎✎✡✜✣✁☎✜⑥✜ ●✑✥✓✖✕■✭ ✺€✴●✺✣✦✲✑✔✺❇✗✍✭❜✕✒✎ ✄✝✆♠✦✥✜✣✤✙✽✝✕✒✎ ✂ ✕✒✎✡✁☎✄✘✸✙✽✘✎✹✕☎✎✡✜⑥✽✳✁☎✭✘✗✡✎✢s❖✎✡✑✔✺✪✎✢✕☎✁☎✭❋✱✳✜✣✁✪✓✖✜☞✓❖✜✧✭✳✺✪✎✢✤✡✄●✮●✜♦✑✲✤ ✕✒✎✡✜ ✽✘✁✪✭✝✗✙✎✫✜ ●✭✳✺❽✕☎✦✲✴✵✄✳✤✙✺€✜✍☛ ✺ ✭ ✔✌ ✔✓ ✔✓ ✗ ✑✸ ✭ ✓ ✣✴✣✴✣ ✌ ✭ ✌ ✍✎ ✔✓ ✽✌ ✖ ✄ ☛ ☛✒ ✭ ✖ ☛ ✣✒ ☛✩✖ ✔✁✌ ☛✩✄ ✁ ☛ ☎✌ ✌ ✍✎ ❼ ➓✩➁✣✇❁✉❄✈①➈❄❻ ➔❽→ ✯✰✶✕✏ ✌✞✛❀✎✒✛❃✓❉✌ ✧✖✠✢✟✡✟❉✠ ✘✵❊ ✱ ✙✦✓ ✡ ✆ ❇✢✶☛✡ ✶✷✌✏✎✒✩✫✑ ✥ ✴✩✬✜✛✹✣✍✯✖✱✏✢ ✛ ✾✝★❍✺✹✱■✙✏✥▲✶✭✥✮✛✟✦✘✴✵✬❅✛✹✣✍✯✖✱✍✢ ✬❅✱✏✶✭❂❄✯ ✪✮✶☎✾■✬❅✛✟✢ ✣✏❏ ✦✵✱❃✻❉★✰✪❃✥✧✦❉✻ ★✰✯✧✯ ★✰✦ ✶☞☛◗✱✍✬■✶ ✍ ✺✹✱■✙✍✥▲✶■✥✮✛✟✦❙✴✩✬❅✛✑✣✏✯ ✱✏✢ ✥▲✶ ✙■★✰✯✧✯ ✱✜✺ ✙✥✓✮③ ✭ ③ ✙✦✓ ✡ ✆ ❇✢✶ ✎✒✶ ❯✒✥ ✾ ✪❋❊ ✱☛✙■✛✰✢✝✴✩✯✖✱✍✢ ✱✏✦ ✪✫★✟✬✭❏✲✴✵✬❅✛✹✣ ✌ ✯✖✱✍✢ ✥▲✶ ✭ ✌✿✙■✛✟✢ ✴✵✯✖✱✏✪ ✱ ✘ ✖✧✩ ✡ ✆ ❇✢✶ ✧✖✠✢✟✫✧✾✠ ★✩✎✙✜❏✆➛✄✘✦✥✦✲✄❜s❖✑✥✤✡✽❑✗✡✁☎✄✘✬✡✦✥✜✣❱ ✑✔✓ ✙✦✓❥③ ✭❫✾❁✺€✄✳❱❑✗✡✦✲✜✣✕✒✜✳♥ ☛✍✌ ✆✞✗✘✎ ✔ ❯ ✤r✸✙✤✙✮●✑✥✁✒✜☞✺❽✕✒✜☞✮r✽✘✁✪✭✝✗✡✎✫s❖✑➀✕☎✎ ✂ ✤✡✄●✮●✜☞✓✣✞ ✒✿✗✘✎ ✆ ✗✘✎ ✔ ✆ ✜✧✓✞✝ ✍✌ ✑➀✆✼✕☎✎✡✜✣✁☎✜ ✺✑✛✔✱✏✶✘✦✵✛✰❖✪ ✜✏●✎ ✑✔✓✟✕❫✭✵✺€✴●✺€✦✥✑✔✺❘✗✙✭❜✕☎✎✢✄✝✆✼✦✥✜✣✤✡✽✘✕✒✎ ✂ s❖✎✡✑✔✺✪✎✛✕✒✁✪✭❜✺ ✾ ✱✘✜✧✁☎✓✒✜✧✓❬✜☞✭✘✺✪✎r✤✡✄●✮●✜❘✑✲✤r✕☎✎✡✜❘✽✘✁✪✭✝✗✡✎✫✜✏✡✎ ✭✘✺€✕✒✦✥✴✵✄✘✤✍✺€✜✘✞ ✆ ❣ ✄ ✝✒●✌ ✄✝✕☎✎✡✜✣✁☎s❖✑✥✓✒✜✘✞ ✂ ✂ ✂ ✂ ✂ ✡ ✁ ✁ ✡ ✁ ✞ ✁ ❛❝❜✄✂✆☎ r✧♥ ☛ ♦✕q ✛ ✩✫✪✜❁✘✛❃✌✘❄✘✠ ✹✌ ❁✘✩✻✪✒❁✞✛✢✌✞❄ ☞ ★✩✎✡✜❬✕☎✜✣✁☎❱ ✮●✜✣✤✡✄✘✕✒✜☞✓◗✮✙✭❜✕☎✭❏✓✖✕✒✄✳✁☎✭✘✽✘✜✏✭✘✤✙✮■✁☎✜€✕☎✁✒✑✥✜✣✱❜✭✘✦❄❱❑✜☞✺✪✎✙✭✝✤✡✑✔✓✒❱❑✓ s❖✎✡✑✔✺✪✎ ✺€✄✳❱♦✗✙✸●✕✒✜❑✕☎✎✡✜q✭✘✮✙✮●✁✒✜☞✓✒✓✒✜✧✓❏✄✘✆✩✕✒✎✡✜q✮✙✭❜✕☎✭✛✁☎✜✧✺✣✄✘✁✪✮✡✓☛✆➛✁✒✄✳❱➎✺✣✄✘✁☎✁✒✜☞✓✖✗▲✄✘✤✍✮●✑✲✤✙✽✫❚✳✜✣✴●✓✣✞ ❛✡✄✳✁✒❱✻✭✘✦✲✦✥✴ ✡✭♦✎✙✭✳✓✖✎✫✆➛✸✡✤✙✺€✕✒✑✥✄✘✤ ✝ ✏❳ ☎ ⑤ ♥✟✞ ✎✙✭✳✓❑✕✒✄ ✬▲✜✢✮●✜ ✙✤✙✜✧✮◆s❖✎✡✑✔✺✪✎◆❱✻✭✘✗✙✓♦✕✒✎✡✜✢✓✒✜€✕✵✄✘✆❫❚✳✜✣✴●✓✠✞ ✕☎✄ ✭ ✓✖✜✣✕✵✄✝✆❘✭✘✮✡✮●✁☎✜✧✓☎✓✒✜✧✓❇⑤■✞ ❯ ✮✡✭✝✕☎✭✛✁☎✜✧✺✣✄✘✁✪✮✢s❖✑✲✕✒✎ ✭✛❚✘✜✧✴ ✳✙ ✞ ✑✔✓❏✕☎✎✡✜✣✤ ✓✖✕✒✄✳✁✒✜☞✮ ✭✝✕❘✕✒✎✡✜q✭✳✮✡✮●✁☎✜✧✓☎✓ ✝ €✞✫❧♠✭✳✺✪✎ ✓✒✸✡✬✙✓✒✜ ❄✸✡✜✣✤❄✕✩✁✒✜✣✕✒✁☎✑✲✜✧✱❜✭✝✦✙✁☎✜ ❄✸✡✜☞✓✟✕✏✆➛✄✘✁✏✕✒✎✡✑✔✓✩✮✡✭❜✕✪✭■✁☎✜✧✺✣✄✘✁✪✮✻✺€✄✘✤✍✓✖✑✔✓✟✕✪✓✏✓✒✄✘✦✥✜✣✦✥✴❑✄✝✆✷✺✣✄✘❱❑✗✡✸●✕☎✑✲✤✡✽ ✕✒✎✙✜❘✎✙✭✘✓✒✎✛✱❋✭✘✦✲✸✙✜ ✝ ❖✭✝✤✍✮✫✓✒✜✧✭✘✁☎✺✪✎✡✑✥✤✡✽❑✭❜✕✩✕✒✎✙✑✥✓✰✭✳✮✡✮●✁☎✜✧✓☎✓✣✞ ❩❁✤⑥✕✒✎✡✜✩✮●✜✧✓✒✑✥✽✘✤❇✄✘✆✙✎✙✭✳✓✖✎❘✕☎✜✧✺✪✎✡✤✙✑ ❄✸✡✜☞✓✷✕✟s❬✄✿✑✥❱❑✗✍✄✳✁✖✕✪✭✝✤❄✕✼✭✘✓✒✗✍✜☞✺❽✕☎✓◗✎✙✭❋✱✳✜✂✕☎✄☛✬▲✜❬✺✣✄✘✤✙✓✒✑✔✮●✜✣✁☎✜✧✮➆♥ ✔✓ ✔✓ ✗ ✸ ✗ ✸ ✓ ✁ ✓❉❇❃❇✢✛❃✪✒✭✝✛❃✓❉✤✍✌✞✮q✪ ✑✔✔ ✮●✜✣❩❉✕☛✤❄✕☎✎✙✑✥✺✧✭✘✭✝✓✩✦❦✕☎✎✙✄✵✭✳✬▲✓✖✎r✜❇✱❜✮●✭✝✜€✦✥✕☎✸✡✜✣✜✁☎❱❑✑✲✗ ✤✡✜☞✸✲✮✛✴ ✎✡✄❜s➋✗ ✸❬✕✒✎✡✑✥✜❇✓✩✎✍✺✧✭✘✭✝✓✒✤✙✜❘✮●✦✥✄✝✜✧✆✶✮❦✕✟✞ s❬✄✻❚✘✜✣✴●✓ ✭✝✤✙✮ ✄ s❖✑➀✕☎✎ ✁✭✝✤✍❁✘✮✫✓✡✓✒✛✢✎✡✙✥✶✄✘✸✙✓✡✦✥✮r✸ ❁✘✮●✑✔✩✫✓✟✪✜✕☎✁✒❁ ✑✥✬✡✸✺✸●✗✘✕☎✌✞✜❏✙✚✕✒✎✜✎✡✛✢✜■✓✡✌ ✓✒✜€✔ ✕❖★✩✄✝✎✡✆◗✜☛❚✘✜✧ ✎✙✴❙✭✘✓ ✓✒✎❑✆➛✸✡✭✘✄✤✙✓❬✺€✜✣✕✒✱✳✑✥✄✘✜✣✤✤✛✭✝✤✙✓✖✮r✎✡✄✳✭✘✸✡✓✩✦✔✮❑✁✪✭✝✬✍✤✍✜☛✮●✄✘✜✧❱❑✭✘✓✒✦✥✴■✴✵✕☎✭✘✄❇✓✩✺€✗▲✄✳✄✳❱❑✓☎✗✡✓✖✑✥✸●✬✡✕✒✦✥✜✜ ✄✴ ✂ ✄ ✝ ✝ ✝ ✗➛✑✥✤✛✄✘✁✪✮●✜✣✁✩✕☎✄❑❚✘✜✣✜✧✗q✕☎✎✡✜❘✤❙✸✡❱❇✬▲✜✣✁✿✄✝✆◗✺✣✄✘✦✥✦✲✑✔✓✒✑✲✄✳✤✙✓❖✓✖❱✻✭✘✦✲✦✧✸❽✞ ✁ ✓❉❇❃❇✢✛❃✪✒✛❃✓❉✌ ❇❃✛✢✪✒✎✦✠ ❯ ✤ ✜✧✭✳✓✖✴✛s✩✭❋✴✫✕☎✄q✕☎✁✒✜☞✭❜✕❏✺€✄✘✦✥✦✥✑✥✓✒✑✲✄✳✤✙✌ ✓☛✕✒✑✥✎✡✓☛✜✹✕✒✄r✓✒✄✝✦✥✾❁✑✲✺✣✤✡✭✝❚✢✦✥✦✥✜✧✮✙✮ ✭❜✕☎✙✥✭✫✓❉✄✘❇❃❇✢✬ ✛✢❱✟✪✜✜✧✛✢✺❽✓✡✕✪✌✂✓❫s❖❇✢✛✢✑➀✪✒✕☎✎ ✎ ✞ ✑✔✮●✜✣✤❄✕✒✑✔✺✣✭✘✦❖✎✙✭✘✓✒✎ ✱❜✭✘✦✲✸✡✜☞✓♦✕✒✄✳✽✘✜€✕☎✎✡✜✣✁♦✑✥✤◆✭ ✦✥✑✲✤✡✜☞✭✝✁❑✦✥✑✥✓✖✕✔♠ ❲❙✜☞✭✝✁✪✺✪✎✡✑✲✤✙✽✛✆➛✄✘✁❇✭✢✮✙✭❜✕☎✭✛✁☎✜✧✺✣✄✘✁✪✮ s❖✑➀✕☎✎ ❚✘✜✣✴ ✕☎✎✡✜✣✤ ✁☎✜✔❄✓ ✸✡✑✥✁✒✜☞✓■✺€✄✳❱❑✗✡✸●✕✒✑✥✤✡✽ ✗ ✸✭✼✌ ✭✝✤✍✮ ✓✒✸✡✬✙✓✒✜✔❄✓ ✸✡✜✣✤❄✕☎✦✲✴ ✓✖✜☞✭✝✁✪✺✪✎✡✑✲✤✙✽r✓✒✜✔❄✓ ✸✡✜✧✤✳✕☎✑✥✭✘✦✲✦✥✴✹✆➛✄✳✁ ✑✲✤ ✕☎✎✡✜q✺€✄✳✦✲✦✥✑✔✓✖✑✥✄✘✤ ✦✲✑✔✓✖✕❇✺€✄✘✁☎✁☎✜✧✓✒✗✍✄✳✤✙✮●✑✥✤✡✽ ✕✒✄ ✗ ❽✸ ✞ ✘✭✻✖✓✖✕✒✧✩ ✁☎✡ ✑✲✤✡✽✆ ❇✢✶ ✴ ✧✖✠✢✟✱❆✖✣✒✠ ✣✴❨✷✣ ✜€✕✸ ✡✌ s❬✜❘✬✍✜✰✮●✜✏✕✒✎✡✍❳ ✜✰✤✡✜❏✓✒✜€✕✒✕✂✎✡✄✘✜■✆▲✺✣✭✝✄✘✦✥✁☎✦✙✁✒✓✖✜☞✕✒✓✖✁☎✗▲✑✲✤✙✄✘✽✳✤✍✓✔✮●✘✌ ✑✲✭✝✤✙✤✙✽⑥✮⑥✎✙✦✥✭✳✜€✓✖✕❍✎r⑤ ✱❜✭✝✴ ✦✥✸✡✎ ✜❏❥❥✬❙❾ ✴✣✴✣✒✣ ❾✒❢✘❢ ✖✳✞◗❛✡✄✘✁ ✟✞ ✸ ✗ ✲✸ ✴ ✗ ✟✞ ✣✴✣✴✣ ✸ ✶✸ ✴ ✭✘✓☎✺✟✗ ✑✚ ✸ ➑✗ ❱❑✄❙✮ ❡☞❥✘❥✑€✸ ❾ ✡✞ ✝ ✝ ☛✞ ✞ ✝ ✝ ☞ ✚✍✌✧✞ ❼❖➒◗➁✧✇❁✈①❷✪✇ 0 .

17 18 . .

S ✕✒✎✡✜❏✆➛✄✳✦✲✦✥✄❜s❖✑✥✤✡✽❑✦✲✜✧❱♦❱✻✭⑥✆➛✄✘✦✥✦✥✄❜s✰✓✩✜✧✭✘✓✒✑✥✦✲✴ ♥ ✞ ☛✶ ✡✪✡ ✩ ✧✾✠ ★ ✠ ✘❋✾ ✆✆☎ ✁ ★✰✦✵✺ ✁ ✙ ✭ ✪❋❊ ✱✍✦ ✙ ✭ ✯✰✶✕✏ ✌✞✛❀✎✒✛❃✓❉✌ ✧✖✠ ✵✠ ✍ ✴✩✬❅✛✑✣✏✯✖✱✍✢ ✥▲✶ ✙■★✟✯✧✯✖✱■✺ ✭ ③ ❁✞✩✱✑✔✕ ✥ ✾❙★✟✯✧✯ ✁ ✙ ✭ ✶✍★✰✪❃✥▲✶❋✾■❏ ✁ ✆✆☎ ✍ ✴✩✬✜✛✹✣✍✯✖✱✏✢ ✥▲✶❆✙✜★✟✯✧✯✖✱■✺ ✭ ③ ✙✦✓ ✡ ✆✞❇❃✶ ✎✒✶ ❯ ✥ ✾ ✣■✛✰✪❋❊ ✥▲✶ ✭ ✌❁❊ ★✟✬❅✺ ★✰✦✩✺ ✙ ✭ ✝ ❁✞✶✷✓✻✑✔☛ ✶ ✡☞✧✖✠ ✵✠ ✔ ✱✏✪ ✣■✱ ★✟✦ ✭ ✌✿✙■✛✰✢✝✴✩✯✖✱✍✪✫✱ ✴✵✬❅✛✹✣✍✯✖✱✍✢ ✘✵❊ ✱✍✦ ✪✧❊✒✱ ✾✏✛✟✯✧✯✖✛ ☛☎✥✧✦●✻ ❊✒✛✰✯✖✺✰✶ ✘❋✾ ✙ ✭ ✪✧❊✒✱✏✦✤✭ ✴ ✭ ✘❋✾ ✙ ✭ ✪✧❊✒✱✏✦✚★✟✯✧✯ ✭ ✌✿✙■✛✰✢✝✴✩✯✖✱✍✪ ✱ ✴✵✬❅✛✹✣✍✯✖✱✍✢✘✶ ✁ ✶✏★✟✪❁✥▲✶❋✾■❏ ✁ ✙ ✭ ✟ ✬❅✛❩✛ ✾ ❨❦✜✣✕ ✙ ✭✤❄✌ ✭✝✤✙✮♦✦✥✜€✕ ✁ ✬▲✜✰✭✝✤❑✭✝✁☎✬✡✑✲✕✒✁✪✭✝✁☎✴❘✗✡✁✒✄✳✬✡✦✥✜✣❱ ✑✥✤ ✭❑✞ ❯ ✓✼✑✥✤❑✗✙✭✝✁✒✕✒✑✔✺€✸✙✦✥✭✘✁ ✑✥✓ ❫ ✭ ✾❉✎✙✭✝✁✪✮❲❄✌ s✏✜❫✎✙✭❋✱✳✜ ✁ ✆✆☎ ✻✞✂★✩✎✡✜✧✤✫❨❦✜✧❱❑❱❑✭❑❃●✞①❊■✑✲❱❑✗✡✦✥✑✲✜☞✓ ✁ ✙ ❑✭ ✞ ❯ ✓ ✁ s✩✭✘✓ ✺✪✎✡✄❄✓✖✜✧✤r✭✘✁✒✬✡✑✲✕✒✁✪✭✝✁☎✑✥✦✲✴❑✆➛✁☎✄✘❱ ✭✤✙✌ s✏✜❏✎✙✭❋✱✳✜ ✭ ✴ ❑✭ ✞ ❣ ✄❜s ✦✲✜✣✕ ✙ ✭✤✡✌ ✭✝✤✙✮✻✦✥✜€✕ ✁ ✬✍✜❏✭✝✤ ❫ ✭ ✾❉✺✣✄✘❱❑✗✡✦✥✜€✕✒✜☛✗✡✁☎✄✘✬✡✦✥✜✣❱ s❖✑✲✕✒✎ ✁ ✙ ❑✭ ✞ ❯ ✺✧✺€✄✳✁☎✮❙✾ ✑✥✤✡✽❘✕✒✄❇✕✒✎✡✜ ✙❳ ✁☎✓✖✕✏✓✖✕☎✭✝✕✒✜✣❱❑✜✧✤✳✕❬✄✝✆ ✕✒✎✙✜✰✕✒✎✙✜✣✄✘✁☎✜✣❱ s✏✜✿✎✙✭❋✱✘✜ ✭ ✴ ✭✤❄✌ ✭✘✤✙✮❑✎✡✜✣✤✍✺€✜ ✙ ✭ ✝ ✂ ☛✞ ✁ ✂ ✂ ✁ ✂ ✁ ✡ ✂ ☛ ✠☛✡ ✂ ✒✓✡ ✂ ✁ ✁ ✁✞ ✡ ✡ ✁ ✂ ✡ ✂ ✂ ✡ ✂ ✁ ✡ ✒✓✡ ✁ ✡ ✂ ✠☛✡ ✂ ✂ ✂ ✁ ✂ ✁ ✁ ✂ ✁ ✁ ✂ ✂ ✁ ✑✥✤✛✺✣✄✘✤❄✕✒✁✪✭✘✮✡✑✥✺€✕✒✑✥✄✘✤✵✕☎✄❑✕✒✎✡✜■✭✳✓✒✓✒✸✡❱❑✗●✕☎✑✲✄✳✤❦✞ ✞ ✘★✩✎✡✖✜✻✩✧✡ s❬✜✣✆ ✑✥✽✘❇✢✶ ✎❄✕❘✧✖✄✝✠✢✟ ✆❬✖✕✒✠✎✡✜ ✗✖❡❩✸ ✾❶✕✒✎ ❯ ✤❙❱✻✸✡✽✘✭✘✽✳✤✢✜€✕❘✎✙✭✘✑✥✓ ✓✿✑✲✤❙✱✳✄✳✜✧✸✡✓✖✕✒✤✙✜✧✺✣✮ ✜✧✓✧✭✝✞✵✦✥✦❈❛✡✎✡✄✘✑✔✁■✓☛✓✒❱♦✑✥❱♦✄✳✤✡✗✙✜✣✦✲✑✔✴✛✺€✑✲✕✟✑✥✤✴✹✌❈❀✝s✏❥q✜✻✽✘✭✘✄✳✓☎✦✥✮✛✓✒✸✡✤❙❱❑✸✡✜⑥✽✳✽✘✕☎✜€✎✙✕✪✭❜✓✣✕ ✞ ✞ ✌✑ ✞ ❾ ✭ ✮ ❾ ✣✴✣✴✣ ❾ ✭✡✠ ✜ ✭ ★ ✚ ☞ ✔✌ ✘✭ ✁✒✜❬✤✙✭❜✕☎✸✡✁✪✭✝✦❄✤❄✸✙❱❇✬▲✜✣✁✪✓✣✞ ✎✡✜✣✤⑥✕✒✎✙✜❬❱✻✭✘✤♦✮●✑✥✜✧✓ ❜✎✡✑✔✓✶✦✥✭✳✓✟✕✶s❖✑✥✦✲✦✡✺✣✄✘✤❄✕☎✭✘✑✲✤✙✓ ✕✒✎✙✜✹✆➛✄✘✦✥✦✲✄❜s❖✑✥✤✡✽ ✺✣✄✘✤✙✮●✑✲✕✒✑✥✄✘✤✷♥✢❩❉✆❘✎✡✑✥✓q✁✒✑✔✺✪✎✡✜☞✓✻✺✣✭✝✤ ✬▲✜✢✓✒✗✡✦✲✑✲✕q✑✥✤❄✕✒✄ ✕✟s❬✄ ✗✙✭✝✁✒✕☎✓✻✄✘✆❘✜ ❄✸✙✭✝✦ s❬✜✣✑✥✽✘✎❄✕ ❜✕☎✎✡✜✣✤♦✜✧✭✳✺✪✎❇✄✘✆✙✎✡✑✔✓✼✮✡✭✝✸✡✽✳✎❄✕✒✜✣✁✪✓◗✓✖✎✡✄✳✸✡✦✔✮❇✄✳✬●✕☎✭✘✑✲✤⑥✄✘✤✡✜✩✄✝✆✙✕✒✎✡✜☞✓✖✜✩✗✙✭✘✁✖✕✪✓✣✞❈❩❉✆✍✑✥✤✙✓✟✕☎✜✧✭✳✮ ✕✒✎✙✜✣✁☎✜❘✑✥✓✩✤✡✄✵✓✒✸✙✺✪✎r✗✙✭✝✁✒✕✒✑✲✕✒✑✥✄✘✤ ❙✕✒✎✡✜✧✤✛✭✘✦✲✦➆✎✙✑✥✓❖✽✳✄✘✦✔✮✫✓✒✎✡✄✳✸✡✦✥✮✫✽✘✄♦✕☎✄⑥✕☎✎✡✜■✺✪✎❙✸✡✁✪✺✪✎❦✞ ★✩✎✡✜♦✽✘✜✧✤✡✜✣✁✪✭✝✦ ✳✸✙✜✧✓✖✕✒✑✥✄✘✤✹✕☎✎✡✜✣✁☎✜€✆➛✄✳✁✒✜⑥✑✥✓✧♥✰❩✟✓☛✑➀✕❏✗✍✄❄✓✒✓✒✑✥✬✡✦✲✜❇✕✒✄✛✮●✑✥✱❄✑✔✮●✜♦✭q✓✒✜€✕❏✄✝✆ ✂ ✤✙✭✝✕✒✸✡✁✪✭✝✦ ✤❙✸✡❱❇✬▲✜✣✁✪✓ ✎ ✭ ✞ ❾ ❾✳✭ ✠ ✖☛✑✥✤✻✕✟s❬✄❇✓✖✸✙✬✙✓✖✜✣✕☎✓❬✓✖✸✍✺✪✎✻✕✒✎✙✭✝✕♠✕☎✎✡✜☛✓✒✸✡❱✻✓♠✄❜✱✳✜✣✁✂✜☞✭✘✺✪✎✵✓✒✸✡✬✙✓✒✜€✕✩✭✝✁☎✜ ✜ ❄✸✙✭✘✆ ✦ ☛ ★✩✎✙✑✥✓♦✗✡✁☎✄✘✬✡✦✥✜✣❱ ✑✔✓✻✺✣✭✘✦✲✦✥✜✧✮ ✁ ✭✘✤✙✮ ✑✔✓ ✁ ❫✾❉✺✣✄✘❱❑✗✡✦✥✜€✕☎✜✘✞ ★✩✎✡✜ ✝ ✝ ✓✖✕✒✁✪✭✝✑✥✽✘✎❄✕✖✆➛✄✳✁✒s✩✭✝✁✪✮ ✕☎✁✒✑✔✭✝✦✲✾❁✭✝✤✙✮❙✾❉✜✣✁☎✁☎✄✘✁❑❱❑✜€✕✒✎✙✄❙✮ ✑✥✓✫✄✘✤✡✦✥✴◆✗✡✁☎✭✳✺❽✕☎✑✥✺✧✭✝✦❖✆➛✄✳✁✫✓✒❱❑✭✘✦✲✦ ✂ ✿✭✘✓q✑➀✕ ✁☎✜ ❄✸✡✑✥✁✒✜☞✓✂✕☎✜✧✓✖✕✒✑✥✤✡✽❇✄✘✆❈✭✝✦✥✦ ❃ ✠ ✗✍✄❄✓✒✓✒✑✥✬✡✑✲✦✥✑✲✕✒✑✥✜✧✓✏✑✲✤q✕✒✎✡✜❫s✏✄✳✁☎✓✖✕❬✺✧✭✘✓✒✜✘✞ ❣ ✄✘✕✒✜✰✕☎✎✙✭❜✕✩✑✲✆❈✓✖✄✳❱❑✜✣✄✘✤✙✜ ✽✘✑✥✱✘✜☞✓✩✴✘✄✳✸r✭❑✓✒✄✘✦✥✸●✕✒✑✥✄✘✤ ●✴✳✄✘✸✛✺✣✭✘✤✫✱✳✜✣✁☎✑➀✆➛✴✻✕☎✎✡✑✔✓✰✓✖✄✳✦✲✸✡✕✒✑✥✄✘✤r✱✘✜✣✁☎✴✻✜✧✭✳✓✖✑✥✦✥✴✘✞ ❃ ★✩✎✡✜❑✆➛✄✘✦✥✦✥✄❜s❖✑✲✤✡✽✹✗✙✁✒✄✳✬✡✦✲✜✧❱ ◗✺✧✭✝✦✥✦✲✜☞✌ ✮ ☞ ✁ ✝ ✝ ✁ ✁ ✚ ✝ ❈✑✔✓■✭✝✦✔✓✒✄ ✁ ❫✾❁✺€✄✘❱❑✗✡✦✥✜€✕☎✜✘✞⑥❍❫✑✥✱✘✜✧✤ ✭✘✤ ✸✡✤✍✮●✑✲✁☎✜✧✺€✕✒✜☞✮✢✽✘✁✪✭✝✗✙✎ s❖✑✲✕✒✎ ✂ ✤✡✄●✮●✜✧✓ ❦✮●✄❙✜✧✓☛✕☎✎✡✜✣✁☎✜⑥✜ ●✑✥✓✖✕■✭ ✺€✴●✺✣✦✲✑✔✺❇✗✍✭❜✕✒✎ ✄✝✆♠✦✥✜✣✤✙✽✝✕✒✎ ✂ ✕✒✎✡✁☎✄✘✸✙✽✘✎✹✕☎✎✡✜⑥✽✳✁☎✭✘✗✡✎✢s❖✎✡✑✔✺✪✎✢✕☎✁☎✭❋✱✳✜✣✁✪✓✖✜☞✓❖✜✧✭✳✺✪✎✢✤✡✄●✮●✜♦✑✲✤ ✕✒✎✡✜ ✽✘✁✪✭✝✗✙✎✫✜ ●✭✳✺❽✕☎✦✲✴✵✄✳✤✙✺€✜✍☛ ✺ ✭ ✔✌ ✔✓ ✔✓ ✗ ✑✸ ✭ ✓ ✣✴✣✴✣ ✌ ✭ ✌ ✍✎ ✔✓ ✽✌ ✖ ✄ ☛ ☛✒ ✭ ✖ ☛ ✣✒ ☛✩✖ ✔✁✌ ☛✩✄ ✁ ☛ ☎✌ ✌ ✍✎ ❼ ➓✩➁✣✇❁✉❄✈①➈❄❻ ➔❽→ ✯✰✶✕✏ ✌✞✛❀✎✒✛❃✓❉✌ ✧✖✠✢✟✡✟❉✠ ✘✵❊ ✱ ✙✦✓ ✡ ✆ ❇✢✶☛✡ ✶✷✌✏✎✒✩✫✑ ✥ ✴✩✬✜✛✹✣✍✯✖✱✏✢ ✛ ✾✝★❍✺✹✱■✙✏✥▲✶✭✥✮✛✟✦✘✴✵✬❅✛✹✣✍✯✖✱✍✢ ✬❅✱✏✶✭❂❄✯ ✪✮✶☎✾■✬❅✛✟✢ ✣✏❏ ✦✵✱❃✻❉★✰✪❃✥✧✦❉✻ ★✰✯✧✯ ★✰✦ ✶☞☛◗✱✍✬■✶ ✍ ✺✹✱■✙✍✥▲✶■✥✮✛✟✦❙✴✩✬❅✛✑✣✏✯ ✱✏✢ ✥▲✶ ✙■★✰✯✧✯ ✱✜✺ ✙✥✓✮③ ✭ ③ ✙✦✓ ✡ ✆ ❇✢✶ ✎✒✶ ❯✒✥ ✾ ✪❋❊ ✱☛✙■✛✰✢✝✴✩✯✖✱✍✢ ✱✏✦ ✪✫★✟✬✭❏✲✴✵✬❅✛✹✣ ✌ ✯✖✱✍✢ ✥▲✶ ✭ ✌✿✙■✛✟✢ ✴✵✯✖✱✏✪ ✱ ✘ ✖✧✩ ✡ ✆ ❇✢✶ ✧✖✠✢✟✫✧✾✠ ★✩✎✙✜❏✆➛✄✘✦✥✦✲✄❜s❖✑✥✤✡✽❑✗✡✁☎✄✘✬✡✦✥✜✣❱ ✑✔✓ ✙✦✓❥③ ✭❫✾❁✺€✄✳❱❑✗✡✦✲✜✣✕✒✜✳♥ ☛✍✌ ✆✞✗✘✎ ✔ ❯ ✤r✸✙✤✙✮●✑✥✁✒✜☞✺❽✕✒✜☞✮r✽✘✁✪✭✝✗✡✎✫s❖✑➀✕☎✎ ✂ ✤✡✄●✮●✜☞✓✣✞ ✒✿✗✘✎ ✆ ✗✘✎ ✔ ✆ ✜✧✓✞✝ ✍✌ ✑➀✆✼✕☎✎✡✜✣✁☎✜ ✺✑✛✔✱✏✶✘✦✵✛✰❖✪ ✜✏●✎ ✑✔✓✟✕❫✭✵✺€✴●✺€✦✥✑✔✺❘✗✙✭❜✕☎✎✢✄✝✆✼✦✥✜✣✤✡✽✘✕✒✎ ✂ s❖✎✡✑✔✺✪✎✛✕✒✁✪✭❜✺ ✾ ✱✘✜✧✁☎✓✒✜✧✓❬✜☞✭✘✺✪✎r✤✡✄●✮●✜❘✑✲✤r✕☎✎✡✜❘✽✘✁✪✭✝✗✡✎✫✜✏✡✎ ✭✘✺€✕✒✦✥✴✵✄✘✤✍✺€✜✘✞ ✆ ❣ ✄ ✝✒●✌ ✄✝✕☎✎✡✜✣✁☎s❖✑✥✓✒✜✘✞ ✂ ✂ ✂ ✂ ✂ ✡ ✁ ✁ ✡ ✁ ✞ ✁ ❛❝❜✄✂✆☎ r✧♥ ☛ ♦✕q ✛ ✩✫✪✜❁✘✛❃✌✘❄✘✠ ✹✌ ❁✘✩✻✪✒❁✞✛✢✌✞❄ ☞ ★✩✎✡✜❬✕☎✜✣✁☎❱ ✮●✜✣✤✡✄✘✕✒✜☞✓◗✮✙✭❜✕☎✭❏✓✖✕✒✄✳✁☎✭✘✽✘✜✏✭✘✤✙✮■✁☎✜€✕☎✁✒✑✥✜✣✱❜✭✘✦❄❱❑✜☞✺✪✎✙✭✝✤✡✑✔✓✒❱❑✓ s❖✎✡✑✔✺✪✎ ✺€✄✳❱♦✗✙✸●✕✒✜❑✕☎✎✡✜q✭✘✮✙✮●✁✒✜☞✓✒✓✒✜✧✓❏✄✘✆✩✕✒✎✡✜q✮✙✭❜✕☎✭✛✁☎✜✧✺✣✄✘✁✪✮✡✓☛✆➛✁✒✄✳❱➎✺✣✄✘✁☎✁✒✜☞✓✖✗▲✄✘✤✍✮●✑✲✤✙✽✫❚✳✜✣✴●✓✣✞ ❛✡✄✳✁✒❱✻✭✘✦✲✦✥✴ ✡✭♦✎✙✭✳✓✖✎✫✆➛✸✡✤✙✺€✕✒✑✥✄✘✤ ✝ ✏❳ ☎ ⑤ ♥✟✞ ✎✙✭✳✓❑✕✒✄ ✬▲✜✢✮●✜ ✙✤✙✜✧✮◆s❖✎✡✑✔✺✪✎◆❱✻✭✘✗✙✓♦✕✒✎✡✜✢✓✒✜€✕✵✄✘✆❫❚✳✜✣✴●✓✠✞ ✕☎✄ ✭ ✓✖✜✣✕✵✄✝✆❘✭✘✮✡✮●✁☎✜✧✓☎✓✒✜✧✓❇⑤■✞ ❯ ✮✡✭✝✕☎✭✛✁☎✜✧✺✣✄✘✁✪✮✢s❖✑✲✕✒✎ ✭✛❚✘✜✧✴ ✳✙ ✞ ✑✔✓❏✕☎✎✡✜✣✤ ✓✖✕✒✄✳✁✒✜☞✮ ✭✝✕❘✕✒✎✡✜q✭✳✮✡✮●✁☎✜✧✓☎✓ ✝ €✞✫❧♠✭✳✺✪✎ ✓✒✸✡✬✙✓✒✜ ❄✸✡✜✣✤❄✕✩✁✒✜✣✕✒✁☎✑✲✜✧✱❜✭✝✦✙✁☎✜ ❄✸✡✜☞✓✟✕✏✆➛✄✘✁✏✕✒✎✡✑✔✓✩✮✡✭❜✕✪✭■✁☎✜✧✺✣✄✘✁✪✮✻✺€✄✘✤✍✓✖✑✔✓✟✕✪✓✏✓✒✄✘✦✥✜✣✦✥✴❑✄✝✆✷✺✣✄✘❱❑✗✡✸●✕☎✑✲✤✡✽ ✕✒✎✙✜❘✎✙✭✘✓✒✎✛✱❋✭✘✦✲✸✙✜ ✝ ❖✭✝✤✍✮✫✓✒✜✧✭✘✁☎✺✪✎✡✑✥✤✡✽❑✭❜✕✩✕✒✎✙✑✥✓✰✭✳✮✡✮●✁☎✜✧✓☎✓✣✞ ❩❁✤⑥✕✒✎✡✜✩✮●✜✧✓✒✑✥✽✘✤❇✄✘✆✙✎✙✭✳✓✖✎❘✕☎✜✧✺✪✎✡✤✙✑ ❄✸✡✜☞✓✷✕✟s❬✄✿✑✥❱❑✗✍✄✳✁✖✕✪✭✝✤❄✕✼✭✘✓✒✗✍✜☞✺❽✕☎✓◗✎✙✭❋✱✳✜✂✕☎✄☛✬▲✜❬✺✣✄✘✤✙✓✒✑✔✮●✜✣✁☎✜✧✮➆♥ ✔✓ ✔✓ ✗ ✸ ✗ ✸ ✓ ✁ ✓❉❇❃❇✢✛❃✪✒✭✝✛❃✓❉✤✍✌✞✮q✪ ✑✔✔ ✮●✜✣❩❉✕☛✤❄✕☎✎✙✑✥✺✧✭✘✭✝✓✩✦❦✕☎✎✙✄✵✭✳✬▲✓✖✎r✜❇✱❜✮●✭✝✜€✦✥✕☎✸✡✜✣✜✁☎❱❑✑✲✗ ✤✡✜☞✸✲✮✛✴ ✎✡✄❜s➋✗ ✸❬✕✒✎✡✑✥✜❇✓✩✎✍✺✧✭✘✭✝✓✒✤✙✜❘✮●✦✥✄✝✜✧✆✶✮❦✕✟✞ s❬✄✻❚✘✜✣✴●✓ ✭✝✤✙✮ ✄ s❖✑➀✕☎✎ ✁✭✝✤✍❁✘✮✫✓✡✓✒✛✢✎✡✙✥✶✄✘✸✙✓✡✦✥✮r✸ ❁✘✮●✑✔✩✫✓✟✪✜✕☎✁✒❁ ✑✥✬✡✸✺✸●✗✘✕☎✌✞✜❏✙✚✕✒✎✜✎✡✛✢✜■✓✡✌ ✓✒✜€✔ ✕❖★✩✄✝✎✡✆◗✜☛❚✘✜✧ ✎✙✴❙✭✘✓ ✓✒✎❑✆➛✸✡✭✘✄✤✙✓❬✺€✜✣✕✒✱✳✑✥✄✘✜✣✤✤✛✭✝✤✙✓✖✮r✎✡✄✳✭✘✸✡✓✩✦✔✮❑✁✪✭✝✬✍✤✍✜☛✮●✄✘✜✧❱❑✭✘✓✒✦✥✴■✴✵✕☎✭✘✄❇✓✩✺€✗▲✄✳✄✳❱❑✓☎✗✡✓✖✑✥✸●✬✡✕✒✦✥✜✜ ✄✴ ✂ ✄ ✝ ✝ ✝ ✗➛✑✥✤✛✄✘✁✪✮●✜✣✁✩✕☎✄❑❚✘✜✣✜✧✗q✕☎✎✡✜❘✤❙✸✡❱❇✬▲✜✣✁✿✄✝✆◗✺✣✄✘✦✥✦✲✑✔✓✒✑✲✄✳✤✙✓❖✓✖❱✻✭✘✦✲✦✧✸❽✞ ✁ ✓❉❇❃❇✢✛❃✪✒✛❃✓❉✌ ❇❃✛✢✪✒✎✦✠ ❯ ✤ ✜✧✭✳✓✖✴✛s✩✭❋✴✫✕☎✄q✕☎✁✒✜☞✭❜✕❏✺€✄✘✦✥✦✥✑✥✓✒✑✲✄✳✤✙✌ ✓☛✕✒✑✥✎✡✓☛✜✹✕✒✄r✓✒✄✝✦✥✾❁✑✲✺✣✤✡✭✝❚✢✦✥✦✥✜✧✮✙✮ ✭❜✕☎✙✥✭✫✓❉✄✘❇❃❇✢✬ ✛✢❱✟✪✜✜✧✛✢✺❽✓✡✕✪✌✂✓❫s❖❇✢✛✢✑➀✪✒✕☎✎ ✎ ✞ ✑✔✮●✜✣✤❄✕✒✑✔✺✣✭✘✦❖✎✙✭✘✓✒✎ ✱❜✭✘✦✲✸✡✜☞✓♦✕✒✄✳✽✘✜€✕☎✎✡✜✣✁♦✑✥✤◆✭ ✦✥✑✲✤✡✜☞✭✝✁❑✦✥✑✥✓✖✕✔♠ ❲❙✜☞✭✝✁✪✺✪✎✡✑✲✤✙✽✛✆➛✄✘✁❇✭✢✮✙✭❜✕☎✭✛✁☎✜✧✺✣✄✘✁✪✮ s❖✑➀✕☎✎ ❚✘✜✣✴ ✕☎✎✡✜✣✤ ✁☎✜✔❄✓ ✸✡✑✥✁✒✜☞✓■✺€✄✳❱❑✗✡✸●✕✒✑✥✤✡✽ ✗ ✸✭✼✌ ✭✝✤✍✮ ✓✒✸✡✬✙✓✒✜✔❄✓ ✸✡✜✣✤❄✕☎✦✲✴ ✓✖✜☞✭✝✁✪✺✪✎✡✑✲✤✙✽r✓✒✜✔❄✓ ✸✡✜✧✤✳✕☎✑✥✭✘✦✲✦✥✴✹✆➛✄✳✁ ✑✲✤ ✕☎✎✡✜q✺€✄✳✦✲✦✥✑✔✓✖✑✥✄✘✤ ✦✲✑✔✓✖✕❇✺€✄✘✁☎✁☎✜✧✓✒✗✍✄✳✤✙✮●✑✥✤✡✽ ✕✒✄ ✗ ❽✸ ✞ ✘✭✻✖✓✖✕✒✧✩ ✁☎✡ ✑✲✤✡✽✆ ❇✢✶ ✴ ✧✖✠✢✟✱❆✖✣✒✠ ✣✴❨✷✣ ✜€✕✸ ✡✌ s❬✜❘✬✍✜✰✮●✜✏✕✒✎✡✍❳ ✜✰✤✡✜❏✓✒✜€✕✒✕✂✎✡✄✘✜■✆▲✺✣✭✝✄✘✦✥✁☎✦✙✁✒✓✖✜☞✕✒✓✖✁☎✗▲✑✲✤✙✄✘✽✳✤✍✓✔✮●✘✌ ✑✲✭✝✤✙✤✙✽⑥✮⑥✎✙✦✥✭✳✜€✓✖✕❍✎r⑤ ✱❜✭✝✴ ✦✥✸✡✎ ✜❏❥❥✬❙❾ ✴✣✴✣✒✣ ❾✒❢✘❢ ✖✳✞◗❛✡✄✘✁ ✟✞ ✸ ✗ ✲✸ ✴ ✗ ✟✞ ✣✴✣✴✣ ✸ ✶✸ ✴ ✭✘✓☎✺✟✗ ✑✚ ✸ ➑✗ ❱❑✄❙✮ ❡☞❥✘❥✑€✸ ❾ ✡✞ ✝ ✝ ☛✞ ✞ ✝ ✝ ☞ ✚✍✌✧✞ ❼❖➒◗➁✧✇❁✈①❷✪✇ 0 .

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A capacity scaling algorithm for M-convex submodular flow by Satoru Iwata, Satoko Moriguchi, Kazuo Murota


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4.3

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