Approximability, parameterized complexity and solving strategies of some multidimensional assignment problems

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Approximability, parameterized complexity and solving

strategies of some multidimensional assignment

problems

Guillerme Duvillié

To cite this version:

Guillerme Duvillié. Approximability, parameterized complexity and solving strategies of some mul-tidimensional assignment problems. Other [cs.OH]. Université Montpellier, 2016. English. �NNT : 2016MONTT321�. �tel-01816979�

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■ !"♦❧❡❣♦♠❡♥❛ ✶✺ ✶ ●"❛♣❤& ❛♥❞ ❍②♣❡"❣"❛♣❤& ✶✼ ✶✳✶ ●#❛♣❤' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✷ ❍②♣❡#❣#❛♣❤' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷ ❈♦♠♣✉3❛3✐♦♥❛❧ ❈♦♠♣❧❡①✐3② ✷✶ ✷✳✶ /#♦❜❧❡♠'✱ ❘❡❞✉❝9✐♦♥' ❛♥❞ ❈♦♠♣❧❡①✐9② ❈❧❛''❡' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳✶ ❉❡❝✐'✐♦♥ /#♦❜❧❡♠' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳✷ ❚❤❡ NP ❝♦♠♣❧❡①✐9② ❝❧❛'' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✶✳✸ ❖♣9✐♠✐③❛9✐♦♥ ♣#♦❜❧❡♠' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷ ❆♣♣#♦①✐♠❛9✐♦♥ ❆❧❣♦#✐9❤♠' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✸ ❆♣♣#♦①✐♠❛9✐♦♥ /#❡'❡#✈✐♥❣ ❘❡❞✉❝9✐♦♥' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✹ ●❛♣✲#❡❞✉❝9✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✺ ❖9❤❡# ❝♦♠♣❧❡①✐9② ❤②♣♦9❤❡'❡' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✺✳✶ ❚❤❡ ❯♥✐K✉❡ ●❛♠❡ ❈♦♥❥❡❝9✉#❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✺✳✷ NP * QP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✺✳✸ ❊①♣♦♥❡♥9✐❛❧ ❚✐♠❡ ❍②♣♦9❤❡'✐' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸ 8❛"❛♠❡3❡"✐③❡❞ ❈♦♠♣❧❡①✐3② ✹✶ ✸✳✶ ■♥9✉✐9✐♦♥ ❛♥❞ ❞❡✜♥✐9✐♦♥' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✷ ❑❡#♥❡❧✐③❛9✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ■■ ❚❤❡♦"❡.✐❝❛❧ ❛1♣❡❝.1 ✺✸ ✹ 8"❡❧✐♠✐♥❛"✐❡& ✺✺ ✹✳✶ ▼♦❞❡❧❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✷ ❖❜'❡#✈❛9✐♦♥' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✺ ❘❡❧❛3❡❞ ❲♦"❦ ✻✸ ✺✳✶ ▼✉❧#✐✲❉✐♠❡♥*✐♦♥❛❧ ❆**✐❣♥♠❡♥# ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✺✳✷ ▼❛①✐♠✐③✐♥❣ 9❤❡ ❞✐❡ ②✐❡❧❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✺✳✸ ▼✐♥✐♠✐③✐♥❣ 9❤❡ ❢❛✉❧9' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✸

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✹ ❈❖◆❚❊◆❚❙ ✻ ■♥#❡%♠❡❞✐❛#❡ ♣%♦❜❧❡♠. ✼✶ ✻✳✶ ◆❡❣❛(✐✈❡ ❘❡,✉❧(, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✻✳✶✳✶ O m log(m) ✲✐♥❛♣♣4♦①✐♠❛❜✐❧✐(② ❢♦4 max6=0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✻✳✶✳✷ f(n), p1−ε _{❛♥❞ m}1−ε_{✲✐♥❛♣♣4♦①✐♠❛❜✐❧✐(② ❢♦4 max max 1 ✳ ✳ ✳ ✳ ✳} _✼✹ ✻✳✶✳✸ ■♥❛♣♣4♦①✐♠❛❜✐❧✐(② ❡①(❡♥,✐♦♥ (♦ (max max 1)#0≤1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✻✳✶✳✹ (n − 1 − ε)✲✐♥❛♣♣4♦①✐♠❛❜✐❧✐(② ❢♦4 (min min 0)_#0≤1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✻✳✶✳✺ (1 − ε) log(m)✲✐♥❛♣♣4♦①✐♠❛❜✐❧✐(② ❢♦4 min min 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✻✳✷ @♦,✐(✐✈❡ ❘❡,✉❧(, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✻✳✸ ❈♦♥❝❧✉,✐♦♥ ❛♥❞ ♦♣❡♥ C✉❡,(✐♦♥, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✼ ▼❛①✐♠✐③✐♥❣ #❤❡ ♦✈❡%❛❧❧ ♥✉♠❜❡% ♦❢ ♦♥❡. ✽✼ ✼✳✶ ◆❡❣❛(✐✈❡ ❘❡,✉❧(, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✼✳✶✳✶ O( m log(m))✲✐♥❛♣♣4♦①✐♠❛❜✐❧✐(② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✼✳✶✳✷ f(n), p1−ε_{, m}1−ε_{✲✐♥❛♣♣4♦①✐♠❛❜✐❧✐(② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳} _✾✵ ✼✳✷ ❈♦♥,✐❞❡4✐♥❣ (❤❡ ✈❡❝(♦4, ❞✐♠❡♥,✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✼✳✷✳✶ p_/_r✲❛♣♣4♦①✐♠❛(✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✼✳✷✳✷ ❋❛,(❡4 ❛❧❣♦4✐(❤♠ ❢♦4 ✜①❡❞ p ❢♦4 maxP 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✼✳✷✳✸ ◆♦ ♣♦❧②♥♦♠✐❛❧ ,✐③❡ ❦❡4♥❡❧ ✇✐(❤ ♣❛4❛♠❡(❡4 p ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ✼✳✸ ❘❡,♦❧✉(✐♦♥ ♦❢ ,♣❛4,❡ ✐♥,(❛♥❝❡, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶ ✼✳✸✳✶ EPTAS ❢♦4 (maxP 1)#0≤r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶ ✼✳✸✳✷ ❆ ♣♦❧②♥♦♠✐❛❧✲(✐♠❡ ❛❧❣♦4✐(❤♠ ❢♦4 (maxP 1)_#0≤1 ✇❤❡♥ m ✐, ✜①❡❞ ✶✵✸ ✼✳✹ ❈♦♥❝❧✉,✐♦♥ ❛♥❞ ♦♣❡♥ C✉❡,(✐♦♥, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼ ✽ ▼✐♥✐♠✐③✐♥❣ #❤❡ ♦✈❡%❛❧❧ ♥✉♠❜❡% ♦❢ ③❡%♦. ✶✵✾ ✽✳✶ ❆♣♣4♦①✐♠❛❜✐❧✐(② ♦❢ minP 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵ ✽✳✶✳✶ (n − ε)✲✐♥❛♣♣4♦①✐♠❛❜✐❧✐(② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵ ✽✳✶✳✷ ■♥❛♣♣4♦①✐♠❛❜✐❧✐(② ✇✐(❤♦✉( ❯●❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✸ ✽✳✶✳✸ ❆♣♣4♦①✐♠❛(✐♦♥ ❛❧❣♦4✐(❤♠ ❢♦4 minP 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✻ ✽✳✷ @❛4❛♠❡(❡4✐③❡❞ ❝♦♠♣❧❡①✐(② ♦❢ minP 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✽ ✽✳✷✳✶ O(k2_m)_{❦❡4♥❡❧ ❢♦4 (min}P 0, k) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✽ ✽✳✷✳✷ @♦,✐(✐✈❡ 4❡,✉❧(, ❛❝❝♦4❞✐♥❣ (♦ ζβ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✽✳✷✳✸ ◆❡❣❛(✐✈❡ 4❡,✉❧(, ❛❝❝♦4❞✐♥❣ (♦ ζβ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵ ✽✳✷✳✹ @♦,✐(✐✈❡ 4❡,✉❧(, ❛❝❝♦4❞✐♥❣ (♦ ζp ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✶ ✽✳✷✳✺ ◆❡❣❛(✐✈❡ 4❡,✉❧(, ❛❝❝♦4❞✐♥❣ (♦ ζp ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻ ✽✳✸ ❈♦♥❝❧✉,✐♦♥ ❛♥❞ ♦♣❡♥ C✉❡,(✐♦♥, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✼ ■■■ !"❛❝%✐❝❛❧ ❆)♣❡❝%) ✶✷✾ ✾ <%❡❧✐♠✐♥❛%✐❡. ✶✸✶ ✾✳✶ ❘❡❧❛(❡❞ ✇♦4❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✷ ✾✳✷ ■♥,(❛♥❝❡ ❣❡♥❡4❛(✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸ ✾✳✷✳✶ ❈♦♥,✐❞❡4❡❞ ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸ ✾✳✷✳✷ ▼♦❞❡❧ 4❡,(4✐❝(✐♦♥, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✹

(6)

❈❖◆❚❊◆❚❙ ✺ ✾✳✷✳✸ ❆❧'❡)♥❛'✐✈❡ ♠♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✹ ✾✳✸ ❊①♣❡)✐♠❡♥' ❝❤❛)❛❝'❡)✐8'✐❝8 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✹ ✶✵ ■▲$ ❢♦'♠✉❧❛,✐♦♥/ ✶✸✼ ✶✵✳✶ ▼♦❞❡❧ ❞❡8❝)✐♣'✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✽ ✶✵✳✶✳✶ ▼❉❆ ✐♥8♣✐)❡❞ ❢♦)♠✉❧❛'✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✽ ✶✵✳✶✳✷ ❍✉❜ 8❡' ❢♦)♠✉❧❛'✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✷ ✶✵✳✷ ■▲B ❝♦♠♣❛)✐8♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✸ ✶✵✳✷✳✶ ▲B )❡❧❛①❛'✐♦♥8 ❝♦♠♣❛)✐8♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✸ ✶✵✳✷✳✷ ❈♦♠♣✉'❛'✐♦♥❛❧ ❘❡8✉❧'8 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✽ ✶✵✳✸ ■▲B ❜❛8❡❞ ❤❡✉)✐8'✐❝8 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✾ ✶✵✳✸✳✶ B)✐♥❝✐♣❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✾ ✶✵✳✸✳✷ ❈♦♠♣✉'❛'✐♦♥❛❧ ❘❡8✉❧'8 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✵ ✶✵✳✹ ❙❡'8 ❛❣❣)❡❣❛'✐♦♥ '❡❝❤♥✐H✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✻ ✶✵✳✹✳✶ ▼♦❞✉8 ❖♣❡)❛♥❞✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✻ ✶✵✳✹✳✷ ❈♦♠♣✉'❛'✐♦♥❛❧ )❡8✉❧'8 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✻ ✶✵✳✺ ❈♦♥❝❧✉8✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✼ ✶✶ ▼❛,❝❤✐♥❣ ❜❛/❡❞ ❤❡✉'✐/,✐❝/ ✶✻✾ ✶✶✳✶ ❙❡' 8❡❧❡❝'✐♦♥ 8')❛'❡❣✐❡8 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✵ ✶✶✳✶✳✶ ■'❡)❛'✐✈❡ ▼❛'❝❤✐♥❣8 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✵ ✶✶✳✶✳✷ ▲♦' B❛)'✐'✐♦♥✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✵ ✶✶✳✶✳✸ ❇❛❧❛♥❝❡❞ ❇✐♥❛)② ❚)❡❡ ❙')❛'❡❣② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✷ ✶✶✳✷ ❈♦♠♣✉'❛'✐♦♥❛❧ )❡8✉❧'8 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✸ ✶✶✳✸ ❈♦♥❝❧✉8✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✺ ✶✷ $❡'/♣❡❝,✐✈❡ ❛♥❞ ❝♦♥❝❧✉/✐♦♥ ✶✼✼ ❆ ❈♦♠♣✉,❛,✐♦♥❛❧ '❡/✉❧,/ ✶✽✺ ❆✳✶ ■▲B ❝♦♠♣❛)✐8♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✺

(7)

(8)

❘❡♠❡#❝✐❡♠❡♥'(

❙✬✐❧ ❡%& ✉♥ ❡①❡*❝✐❝❡ ,✉✐ *❡✈.& / ♠❡% ②❡✉① ✉♥❡ ✐♠♣♦*&❛♥❝❡ %✐♥❣✉❧✐6*❡ ❛✉ ❝♦✉*% ❞❡ ❧❛ *8❞❛❝&✐♦♥ ❞✬✉♥ ♠❛♥✉%❝*✐& ❞❡ &❤6%❡✱ ❝✬❡%& ❜✐❡♥ ❝❡❧✉✐ ❞❡% *❡♠❡*❝✐❡♠❡♥&%✳ ❈❡% ,✉❡❧,✉❡% ♣❛*❛❣*❛♣❤❡% %♦♥&✱ ❡♥ ❡✛❡&✱ ❧✬♦❝❝❛%✐♦♥ ❞❡ *8♣❛*❡*✱ ❛♣*6% &*♦✐% ❛♥% ❞❡ ❧❛❜❡✉*✱ ✉♥❡ ❝❡*✲ &❛✐♥❡ ❢♦*♠❡ ❞✬✐♥❣*❛&✐&✉❞❡ ♣*♦♣*❡ ❛✉ ❞♦❝&♦*❛♥&✳ ❈♦♠♠❡ ❛❜%♦*❜8 ♣❛* ✉♥❡ &A❝❤❡ ❞♦♥& ❧✬❛♠♣❧❡✉* ❧✉✐ 8❝❤❛♣♣❡ ❥✉%,✉✬/ %♦♥ ❛❝❝♦♠♣❧✐%%❡♠❡♥&✱ ✐❧ ❡♥ ♦✉❜❧✐❡ %♦✉✈❡♥& ❧✬❛❜♥8❣❛&✐♦♥ ❞♦♥& ♣❡✉✈❡♥& ❢❛✐*❡ ♣*❡✉✈❡ ❧❡% ♣❡*%♦♥♥❡% ❞❡ %♦♥ ❡♥&♦✉*❛❣❡✳ ❆✐♥%✐✱ ✐❧ ♠❡ %❡♠❜❧❡ ♥❛&✉*❡❧ ❞✬❡♥✈✐%❛❣❡* ❝❡ ♠❛♥✉%❝*✐&✱ ♥♦♥ ❝♦♠♠❡ ✉♥❡ D✉✈*❡ ♣❡*%♦♥♥❡❧❧❡✱ ♠❛✐% ❝♦♠♠❡ ❧❛ *8%✉❧&❛♥&❡ ❞❡% %❛❝*✐✜❝❡% ❡& ❞❡% ❡✛♦*&% ❞❡ ❧✬❡♥%❡♠❜❧❡ ❞❡% ♣❡*%♦♥♥❡% ,✉✐ ♠✬♦♥& ❝F&♦②8✱ ❞❡ ❧❡✉* ♣❧❡✐♥ ❣*8 ♦✉ ♥♦♥✱ ❞✉*❛♥& ❝❡% &*♦✐% ❛♥♥8❡%✳

❉❛♥% ✉♥ ♣*❡♠✐❡* &❡♠♣%✱ ❥❡ &✐❡♥% / *❡♠❡*❝✐❡* ♠❡% ❡♥❝❛❞*❛♥&%✱ ❘♦❞♦❧♣❤❡ ●✐*♦✉❞❡❛✉ ❡& ▼❛*✐♥ ❇♦✉❣❡*❡&✳ ▼❡*❝✐ ❘♦❞♦❧♣❤❡ ♣♦✉* ❛✈♦✐* ❝*✉ ❡♥ ♠♦✐ ❡& ❝❡✱ %✉✣%❛♠♠❡♥& ♣♦✉* ♠❡ ♣*♦♣♦%❡* ✉♥ %✉❥❡&✳ ▼❡*❝✐ ❞✬❛✈♦✐* ✈✉ ❡♥ ♠♦✐ ❛✉&*❡ ❝❤♦%❡ ,✉✬✉♥ 8&✉❞✐❛♥& ❞8%✐♥✈♦❧&❡ ❡& ❞✬❛✈♦✐* %✉ *8✈❡✐❧❧❡* ✉♥❡ ❝✉*✐♦%✐&8 ❡♥❞♦*♠✐❡ ❞❡♣✉✐% ❜✐❡♥ ❧♦♥❣&❡♠♣%✳ ▼❛✐% %✉*&♦✉& ✉♥ ❣*❛♥❞ ♠❡*❝✐ ♣♦✉* ❛✈♦✐* %✉ &*♦✉✈❡* ❧✬❡♥❝❛❞*❡♠❡♥& ✐❞8❛❧ ♠✬❛②❛♥& ♣❡*♠✐% ❞❡ ♠✬8♣❛♥♦✉✐*✱ ♣♦✉* &♦✉&❡ ❝❡&&❡ 8♥❡*❣✐❡ ❞8♣❡♥%8❡✱ ♣♦✉* &♦✉&❡% ❝❡% ❤❡✉*❡% ♣❛%%8❡% / *❡❧✐*❡ ♣❛*❢♦✐% ❞✐① ❢♦✐% ❧❡% ♠.♠❡% 8❝*✐&%✱ ♣♦✉* ❛✈♦✐* %✉ &*♦✉✈❡* ❧✬8,✉✐❧✐❜*❡ ✐❞8❛❧ ❡♥&*❡ ❛✉&♦♥♦♠✐❡ ❡& ❛✉&♦*✐&8✱ ♦✉ ❡♥❝♦*❡ ♣♦✉* ❧❡ %♦✉&✐❡♥ ❤✉♠❛✐♥ ❞♦♥& ❥✬❛✐ ❡✉ ❜❡%♦✐♥ ❧♦*%,✉✬✐❧ ♠✬❛ ❢❛❧❧✉ ❝♦♥❝✐❧✐❡* ♠❛ ✈✐❡ ❞❡ ❥❡✉♥❡ ♣❛♣❛ ❡& ❝❡❧❧❡ ❞❡ ❞♦❝&♦*❛♥&✳

▼❡*❝✐ ▼❛*✐♥ ♣♦✉* ♥♦% 8❝❤❛♥❣❡% %♣♦♥&❛♥8% ❡& ✭♣*❡%,✉❡✮ &♦✉❥♦✉*% ♣*♦❞✉❝&✐❢% ❡& ❝❡ / ♥✬✐♠♣♦*&❡ ,✉❡❧❧❡ ❤❡✉*❡ ❞✉ ❥♦✉* ♦✉ ❞❡ ❧❛ ♥✉✐&✳ ▼❡*❝✐ ❞❡ ♥❡ ❥❛♠❛✐% ♠✬❛✈♦✐* ❧❛✐%%8 &♦♠❜❡* ❞❛♥% ❧❛ ❢❛❝✐❧✐&8✱ ♣♦✉* &♦✉&❡% ♥♦% %❡%%✐♦♥% &❡❝❤♥✐,✉❡%✱ / ❧❛ ✜♥ ❞❡%,✉❡❧❧❡% ❥✬❛✈❛✐% ❧✬✐♠♣*❡%%✐♦♥ ❞✬❛✈♦✐* ❝♦✉*✉ ✉♥ ♠❛*❛&❤♦♥ ❡& ❛✉①,✉❡❧❧❡% ✉♥ ❣*❛♥❞ ♥♦♠❜*❡ ❞❡ *8%✉❧&❛&% ❞❡ ❝❡ ♠❛♥✉%❝*✐& ❞♦✐✈❡♥& ❧❡✉* ❡①✐%&❡♥❝❡✱ ♣♦✉* ❝❡&&❡ ❡♥✈✐❡ ❞❡ ❝♦♠♣*❡♥❞*❡ ,✉✐ &✬❡%& %✐ ♣❛*&✐❝✉❧✐6*❡ ❡& ,✉✐ ♠✬❛ ♦❜❧✐❣8 / 8❝❧❛✐*❝✐* ❝❤❛,✉❡ ③♦♥❡ ❞✬♦♠❜*❡ ❞❡% &❤8♦*✐❡% %✉* ❧❡%,✉❡❧❧❡% ♥♦✉% ❛✈♦♥% &*❛✈❛✐❧❧8✳ ▼❡*❝✐ 8❣❛❧❡♠❡♥& ❞✬❛✈♦✐* %✉ ♠❡ ❢♦*❝❡* / &♦✉& *❡❝♦♠♠❡♥❝❡* ❧♦*%,✉❡ ❝✬8&❛✐& ♥8❝❡%%❛✐*❡✱ ❛✐♥%✐ ,✉❡ ❞✬❛✈♦✐* 8&8 ♣*8%❡♥& ❞❛♥% ♠❡% ♠♦♠❡♥&% ❞❡ ❞♦✉&❡ ❡& ❞❡ ❧❛%%✐&✉❞❡ ❡& ❡♥✜♥ ♣♦✉* ❝❡&&❡ ❡♥✈✐❡ ❞❡ ♣❛*&❛❣❡* ❝❡ ,✉✐ &✬❡%& ❝❤❡* ✭② ❝♦♠♣*✐% ❧❡% ❧✐❜❡*&8% ♣❛*❢♦✐% ♦✉&*❛♥❝✐6*❡% ,✉❡ ♣*❡♥♥❡♥& ❧❡% ♣✐❛♥✐%&❡% *♦♠❛♥&✐,✉❡% ❛✈❡❝ ❧❛ ♣✉❧%❛&✐♦♥✮✳

❈❡ ❢✉& ✉♥ ✈8*✐&❛❜❧❡ ♣❧❛✐%✐* ❞✬❛✈♦✐* 8✈♦❧✉8 / ✈♦% ❝F&8% &♦✉& ❛✉ ❧♦♥❣ ❞❡ ❝❡% ❛♥♥8❡% ❡& ❥✬❡%♣6*❡ ,✉❡ ✈♦✉% ✈❡**❡③ ❡♥ ❝❡% ❧✐❣♥❡% ❧✬❛✈❡✉ ❞✬✉♥❡ ♠8❧❛♥❝♦❧✐❡ %✐♥❝6*❡ / ❧✬✐❞8❡ ❞❡ ♥❡ ♣❧✉% &*❛✈❛✐❧❧❡* ❡♥%❡♠❜❧❡✳

❏❡ %♦✉❤❛✐&❡*❛✐% *❡♠❡*❝✐❡* ❈*✐%&✐♥❛ ❇❛③❣❛♥ ❡& ❈❤*✐%&♦♣❤❡ Q✐❝♦✉❧❡❛✉ ♣♦✉* ❛✈♦✐* ❛❝✲ ❝❡♣&8 ❞❡ ♣*❡♥❞*❡ ❞❡ ❧❡✉* &❡♠♣% ♣♦✉* ❧✐*❡ ❡& ❝♦**✐❣❡* ❝❡ ♠❛♥✉%❝*✐& ❛✐♥%✐ ,✉❡ ♣♦✉* ❧❡✉*% ❡♥❝♦✉*❛❣❡♠❡♥&% ❡& ❧❡% 8❝❤❛♥❣❡% ,✉❡ ♥♦✉% ❛✈♦♥% ♣✉ ❛✈♦✐* ❡♥ ❝♦♥❢8*❡♥❝❡ ♥♦&❛♠♠❡♥&✳ ❯♥ ❣*❛♥❞ ♠❡*❝✐ 8❣❛❧❡♠❡♥& / ❉✐♠✐&*✐♦% ❚❤✐❧✐❦♦% ❡& ❋*8❞8*✐❝ ❍❛✈❡& ♣♦✉* ❛✈♦✐* ❛❝❝❡♣&8 ❞❡ %✐8❣❡* ❞❛♥% ♠♦♥ ❥✉*②✳

❏❡ &✐❡♥% 8❣❛❧❡♠❡♥& / *❡♠❡*❝✐❡* ❧✬❡♥%❡♠❜❧❡ ❞❡% ♠❡♠❜*❡% ❞❡ ❧✬8,✉✐♣❡ ▼❆❖❘❊ ♣♦✉* ❧❡✉* ❛❝❝✉❡✐❧ ❡& ❧❡✉* %♦❧❧✐❝✐&✉❞❡ ♣❡♥❞❛♥& ♠♦♥ %8❥♦✉* ♣❛*♠✐ ❡✉①✳ ▼❡*❝✐ ❞✬❛✈♦✐* %✉♣♣♦*&8

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❈❖◆❚❊◆❚❙ ✾ ♠❡#❝✐ ❞✬❛✈♦✐# +✉♣♣♦#./ ♠❡+ ❧♦♥❣✉❡+ ❥♦✉#♥/❡+ ❞✬❡#♠✐.❡ #❡❝❧✉+ +✉# ♠♦♥ ♦#❞✐♥❛.❡✉# ❡. ❝❡ ♠❛❧❣#/ ♠❡+ ♠❛♥4✉❡♠❡♥.+ /✈✐❞❡♥.+ ❛✉① #6❣❧❡+ ❞❡ ❧✬❤♦+♣✐.❛❧✐./ ❡. ❞✉ +❛✈♦✐#✲✈✐✈#❡ ❡. ❡♥✜♥ ✉♥ ♠❡#❝✐ ♣♦✉# ✈♦.#❡ ❣❡♥.✐❧❧❡++❡ ❡. ✈♦.#❡ ❜✐❡♥✈❡✐❧❧❛♥❝❡ ; ♠♦♥ /❣❛#❞✳ ❯♥ ♠❡#❝✐ .♦✉. ♣❛#.✐❝✉❧✐❡# ; ♠❡+ ❛♠✐+ ❞✉ ◆♦#❞ ✭❱✐❝.♦#✱ ❆❧❡①✐+✱ ❏/#/♠✐❡ ❍✳✱ E❛✉❧✲ ❘♦❜❡#.✱ ■+❛❜❡❧❧❡✱ ❊❧+❛ ❡. ◆✐❝♦❧❛+✮ ♣♦✉# ❛✈♦✐# /./ ♣#/+❡♥. ❡. ❛✈♦✐# ♣#✐+ ❧❡ .❡♠♣+ ❞❡ ♠❡ #❡♥❞#❡ ✈✐+✐.❡ ♠❛❧❣#/ ❞❡+ ❡♠♣❧♦✐+ ❞✉ .❡♠♣+ ❛✉ ♠♦✐♥+ ❛✉++✐ ❝❤❛#❣/+ 4✉❡ ❧❡ ♠✐❡♥✳ ❏❡ #❡❣#❡..❡ ❥✉+.❡ ❞❡ ♥❡ ♣❛+ ❛✈♦✐# ♣✉ ✈♦✉+ #❡♥❞#❡ ❧❛ ♣❛#❡✐❧❧❡✳ ❏❡ .✐❡♥+ /❣❛❧❡♠❡♥. ; #❡♠❡#❝✐❡# +✐♥❣✉❧✐6#❡♠❡♥. ❏/#/♠✐❡ ❋✳ ❡. ❈❧/♠❡♥. ♣♦✉# ❧❡+ ✐♥✲ ♥♦♠❜#❛❜❧❡+ +❡#✈✐❝❡+ 4✉❡ ✈♦✉+ ❛✈❡③ ♣✉ ♠❡ #❡♥❞#❡ ❡. +✉#.♦✉. ♣♦✉# ❛✈♦✐# +✉ ❞♦♥♥❡# ; ❝❡+ .❡##❡+ ♠♦♥.♣❡❧❧✐/#❛✐♥❡+✱ +✐ ❤♦+.✐❧❡+ ; ♠♦♥ ❛##✐✈/❡✱ ✉♥❡ +❛✈❡✉# ❞♦✉❝❡ ❡. +✐♥❝6#❡ 4✉✐ ♠❡ ♠❛♥4✉❡ ; ♣#/+❡♥.✳ ▼❡+ ❞❡#♥✐❡#+ #❡♠❡#❝✐❡♠❡♥.+✱ ❡. ♥♦♥ ❧❡+ ♠♦✐♥❞#❡+✱ +♦♥. ❞❡+.✐♥/+ ; ❙❛❧♦♠/ 4✉✐ ♠✬❛ ❛❝❝♦♠♣❛❣♥/ ❛✉ 4✉♦.✐❞✐❡♥ ❞❛♥+ ❝❡..❡ ❛✈❡♥.✉#❡✳ ❈✬❡+. ❡♥ ❡✛❡. ❡❧❧❡ 4✉✐ ❛ ❞P +❛✈♦✐# ❣/#❡# ♠❡+ ❥♦✉#♥/❡+ +❛♥+ ✜♥✱ ♠❡+ ♥✉✐.+ ❞❡ .#❛✈❛✐❧✱ ♠❛ ❢❛.✐❣✉❡✱ ♠❡+ +❛✉.❡+ ❞✬❤✉♠❡✉# ❡. ♠♦♥ ✐##✐.❛❜✐❧✐./✳ ❈✬❡+. ❡❧❧❡ 4✉✐ ❛ /./ ❧✬♦#❡✐❧❧❡ ❛..❡♥.✐✈❡ ❡. ❧✬/♣❛✉❧❡ #/❝♦♥❢♦#.❛♥.❡ ❞♦♥. ❥✬❛✈❛✐+ ❜❡+♦✐♥ ❛♣#6+ ❞❡+ ❥♦✉#♥/❡+ ♣❛#.✐❝✉❧✐6#❡♠❡♥. ❞✐✣❝✐❧❡+✳ ▼❡#❝✐ ♣♦✉# .❛ +♦❧❧✐❝✐.✉❞❡✱ .♦♥ +♦✉.✐❡♥ ✐♥❝♦♥❞✐.✐♦♥♥❡❧ ❡. #/❝♦♥❢♦#.❛♥.✱ ♠❡#❝✐ ❞✬❛✈♦✐# .❡♥✉ ♥♦.#❡ ♣❡.✐.❡ ❢❛♠✐❧❧❡ ; ❜♦✉. ❞❡ ❜#❛+ ♣❡♥❞❛♥. ❧❛ ♣/#✐♦❞❡ ❞❡ #/❞❛❝.✐♦♥✱ ❞✬❛✈♦✐# .♦❧/#/ ❝❡+ ❧♦♥❣✉❡+ ❥♦✉#♥/❡+ ♦S✱ ♦❜♥✉❜✐❧/ ♣❛# ✉♥ ♣#♦❜❧6♠❡✱ ❥✬/.❛✐+ ♣#/+❡♥. ♣❤②+✐4✉❡♠❡♥. ♠❛✐+ ❛❜+❡♥. ♠❡♥.❛❧❡♠❡♥.✱ ❞✬❛✈♦✐# +✉ ♣#❡♥❞#❡ +♦✐♥ ❞❡ ♥♦.#❡ ♠❛❣♥✐✜4✉❡ ♣❡.✐.❡ ✜❧❧❡ .♦✉. ❡♥ ♣#❡♥❛♥. +♦✐♥ ❞❡ ♠♦✐ ❡. +✉#.♦✉. ❞✬❛✈♦✐# +✉♣♣♦#./ ♣❡♥❞❛♥. .#♦✐+ ❛♥+ ❧❡+ ❛♣♣❡❧+ ❞❡ ▼❛#✐♥ ❛✉ ♠✐❧✐❡✉ ❞❡ ♥♦+ #❛#❡+ #❡♣❛+ ❡♥ ❢❛♠✐❧❧❡✳ ❙❛♥+ .♦✐ #✐❡♥ ❞❡ .♦✉. ❝❡❧❛ ♥✬❛✉#❛✐. /./ ♣♦++✐❜❧❡ ❡. ❥✬❡+♣6#❡ ❛✈♦✐# ❧✬♦❝❝❛+✐♦♥✱ ✉♥ ❥♦✉#✱ ❞❡ .❡ #❡♥❞#❡ ❧❛ ♣❛#❡✐❧❧❡✳ ❊♥ ❛..❡♥❞❛♥.✱ +❛❝❤❡ 4✉❡ ❥❡ +✉✐+ ♦♥ ♥❡ ♣❡✉. ♣❧✉+ ❝♦♠❜❧/ ❞✬❡①♣❧♦#❡# ❞❡ ♥♦✉✈❡❛✉① ❤♦#✐③♦♥+ ; .❡+ ❝U./+✳

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■♥"#♦❞✉❝"✐♦♥

❚❤❡ ❞❡✈❡❧♦♣♠❡♥* ♦❢ *❤❡ ✐♥*❡❣.❛*❡❞ ❝✐.❝✉✐* ✭■❈✮ ♠❛♥✉❢❛❝*✉.✐♥❣ 6✐♥❝❡ ✐*6 ✜.6* ♣.♦*♦*②♣❡ ♦♥ ✶✷ ❙❡♣*❡♠❜❡. ✶✾✺✽ ❜✉✐❧* ❜② ❏❛❝❦ ❑✐❧❜②✱ ❧❡❞ ●♦.❞♦♥ ▼♦♦.❡ *♦ ❢♦.♠✉❧❛*❡✱ ✐♥ ✶✾✻✺✱ ❛ ♣.❡❞✐❝*✐♦♥ ♦♥ *❤❡ ❢✉*✉.❡ ♦❢ 6❡♠✐❝♦♥❞✉❝*♦. ✐♥❞✉6*.②✿ ❚❤❡ ❝♦♠♣❧❡①✐*② ❢♦. ♠✐♥✐♠✉♠ ❝♦♠♣♦♥❡♥* ❝♦6*6 ❤❛6 ✐♥❝.❡❛6❡❞ ❛* ❛ .❛*❡ ♦❢ .♦✉❣❤❧② ❛ ❢❛❝*♦. ♦❢ *✇♦ ♣❡. ②❡❛.✳ ❈❡.*❛✐♥❧② ♦✈❡. *❤❡ 6❤♦.* *❡.♠ *❤✐6 .❛*❡ ❝❛♥ ❜❡ ❡①♣❡❝*❡❞ *♦ ❝♦♥*✐♥✉❡✱ ✐❢ ♥♦* *♦ ✐♥❝.❡❛6❡✳ ❖✈❡. *❤❡ ❧♦♥❣❡. *❡.♠✱ *❤❡ .❛*❡ ♦❢ ✐♥❝.❡❛6❡ ✐6 ❛ ❜✐* ♠♦.❡ ✉♥❝❡.*❛✐♥✱ ❛❧*❤♦✉❣❤ *❤❡.❡ ✐6 ♥♦ .❡❛6♦♥ *♦ ❜❡❧✐❡✈❡ ✐* ✇✐❧❧ ♥♦* .❡♠❛✐♥ ♥❡❛.❧② ❝♦♥6*❛♥* ❢♦. ❛* ❧❡❛6* ✶✵ ②❡❛.6✳ ■♥ ✶✾✼✺✱ ▼♦♦.❡ .❡✈✐6❡❞ ✐*6 ♣.❡❞✐❝*✐♦♥ ❞✉.✐♥❣ *❤❡ ■❊❊❊ ■♥*❡.♥❛*✐♦♥❛❧ ❊❧❡❝*.♦♥ ❉❡✲ ✈✐❝❡6 ▼❡❡*✐♥❣✳ ❍❡ 6*❛*❡❞ *❤❛* ❢.♦♠ ✶✾✽✵✱ ✏*❤❡ .❛*❡ ♦❢ ✐♥❝.❡❛6❡ ❝♦♠♣❧❡①✐*② ❝❛♥ ❜❡ ❡①♣❡❝*❡❞ *♦ ❝❤❛♥❣❡ 6❧♦♣❡ ❬✳ ✳ ✳ ❪✳ ❚❤❡ ♥❡✇ 6❧♦♣❡ ♠✐❣❤* ❛♣♣.♦①✐♠❛*❡ ❛ ❞♦✉❜❧✐♥❣ ❡✈❡.② *✇♦ ②❡❛.6✱ .❛*❤❡. *❤❛♥ ❡✈❡.② ②❡❛.❬✳ ✳ ✳ ❪✳✑ ❚❤✐6 ❝♦♥❥❡❝*✉.❡ ❤❛6 ❜❡❡♥ ❧❛*❡. ♣♦♣✉❧❛.✐③❡❞ ❛6 ❤❡ #❡❝♦♥❞ ▼♦♦)❡✬# ▲❛✇✱ ❛♥❞ ❜② ❡①*❡♥6✐♦♥ *❤❡ ✶✾✻✺ ❝♦♥❥❡❝*✉.❡ ♣♦♣✉❧❛.✐③❡❞ ❛6 ❤❡ ✜)# ▼♦♦)❡✬# ▲❛✇✳ ❚❤✐6 ❝♦♥❥❡❝*✉.❡ ♣.♦✈❡❞ ❛❝❝✉.❛*❡ ❢♦. 6❡✈❡.❛❧ ❞❡❝❛❞❡6✳ ❍♦✇❡✈❡.✱ ▼♦♦.❡ ❛♥❞ ♥✉♠❜❡. ♦❢ ✐♥❞✉6*.② ❢♦.❡❝❛6*✲ ❡.6 ❡①♣❡❝* ▼♦♦.❡✬6 ▲❛✇ ✇✐❧❧ ❡♥❞ ♦♥ ✷✵✷✺✳ ◆♦✇❛❞❛②6✱ *❤❡ ✐♥❝.❡❛6❡ ♦❢ *❤❡ ♠✐❝.♦♣.♦❝❡66♦.6 ♣♦✇❡. ❝♦♠♣✉*❛*✐♦♥ ✐6 ♣❛.*❧② ❡♥6✉.❡❞ ❜② ❛♥ ✐♥❝.❡❛6❡ ♦❢ *❤❡ ♥✉♠❜❡. ♦❢ ❝♦.❡6✳ ❚❤✐6 ✐♥❝.❡❛6❡ ✐6 6✉❝❤ *❤❛* 6♦♠❡ ❡✈❡♥ ♣.❡❞✐❝* ✐* *♦ ❢♦❧❧♦✇ ▼♦♦.❡✬6 ▲❛✇ ❬❑✉♠✰✵✺❪✳ ◆❡✈❡.*❤❡❧❡66✱ *❤❡ ✐♥❝.❡❛6❡ ♦❢ *❤❡ ♥✉♠❜❡. ♦❢ ❝♦.❡6 ❧❡❛❞6 *♦ ❛♥ ✐♥❝.❡❛6❡ ♦❢ ❞✐6*❛♥❝❡ ❜❡*✇❡❡♥ *❤❡♠ ❛♥❞ *❤✉6 ❛♥ ✐♥❝.❡❛6❡ ♦❢ *❤❡ ❝♦♠♠✉♥✐❝❛*✐♦♥ *✐♠❡6✳ ❚❤✐6 ❧✐♠✐*6 *❤❡ ❡✣❝✐❡♥❝② ♦❢ 6✉❝❤ 6♦❧✉*✐♦♥✳ ❆ ❧❡❛❞ *♦ ♦✈❡.❝♦♠❡ 6✉❝❤ ✐66✉❡6 ✐6 *♦ ♣.♦❞✉❝❡ *❤.❡❡✲❞✐♠❡♥6✐♦♥❛❧ ♠✐❝.♦♣.♦❝❡66♦.6✳ ■♥6*❡❛❞ ♦❢ ❡♥❝❧♦6✐♥❣ 6❡✈❡.❛❧ ❛❞❥❛❝❡♥* ❝♦.❡6✱ ❛ *❤.❡❡✲❞✐♠❡♥6✐♦♥❛❧ ♠✐❝.♦♣.♦❝❡66♦. ❛✐♠6 ❛* 6✉♣❡.✐♠♣♦6✐♥❣ *❤❡ ❧❛**❡.✳ ❚❤❡ ❞✐6*❛♥❝❡ ❜❡*✇❡❡♥ ❡❛❝❤ ❞✐❡ ✐6 *❤❡♥ ❣.❡❛*❧② .❡❞✉❝❡❞ *♦ .♦✉❣❤❧② *❤❡ *❤✐❝❦♥❡66 ♦❢ *❤❡ ❞✐❡✳ ❚❤❡ ♠❛♥✉❢❛❝*✉.✐♥❣ ♣.♦❝❡66 .❡♠❛✐♥6 ^✉✐*❡ ✐❞❡♥*✐❝❛❧ ❛6 *❤❡ ♦♥❡ ❢♦. ❝❧❛##✐❝❛❧ ♠✐❝.♦♣.♦❝❡66♦.6✳ ■♥ ❛ ✜.6* *✐♠❡✱ *❤❡ ❞✐❡6 ❛.❡ ❡♥❣.❛✈❡❞ ♦♥ ❛ ❝✐.❝✉❧❛. 6❧✐❝❡ ♦❢ 6✐❧✐❝♦♥ ❝❛❧❧❡❞ ✇❛❢❡)✳ ❉❡♣❡♥❞✐♥❣ ♦♥ *❤❡ ❝♦♠♣❧❡①✐*② ♦❢ *❤❡ ❡♥❣.❛✈❡❞ ❞✐❡ ❛♥❞ ♦♥ *❤❡ 6✐③❡ ♦❢ *❤❡ ✇❛❢❡.✱ ♦♥❡ ❝❛♥ ❡♥❣.❛✈❡❞ ✉♣ *♦ 6❡✈❡.❛❧ *❤♦✉6❛♥❞6 ♦❢ ❞✐❡6 ♦♥ ❛ 6✐♥❣❧❡ 6❧✐❝❡✳ ❖♥❝❡ *❤✐6 6*❡♣ ✐6 ♣❡.❢♦.♠❡❞✱ ❞✐❡6 ❛.❡ *❡6*❡❞ *♦ 6♣♦* ❢❛✉❧*② ♦♥❡6 ❛♥❞ *❤❡♥ ❝✉* ♦✉*✳ ❱✐❛❜❧❡ ❞✐❡6 ❛.❡ *❤❡♥ ❡♥❝❛♣6✉❧❛*❡❞ ✐♥*♦ ❝❤✐♣6 *♦ ❣❡* ♠✐❝.♦♣.♦❝❡66♦.6 ❛6 ✇❡ ✉6✉❛❧❧② ❦♥♦✇ *❤❡♠✳ ❆♥ ❛❞❞✐*✐♦♥❛❧ 6*❡♣ ✐6 .❡^✉✐.❡❞ *♦ ♠❛♥✉❢❛❝*✉.❡ *❤.❡❡✲❞✐♠❡♥6✐♦♥❛❧ ♠✐❝.♦♣.♦❝❡66♦.6✿ *❤❡ ✐♥*❡❣.❛*✐♦♥✳ ❉✐❡6 ♥❡❡❞ ✐♥❞❡❡❞ *♦ ❜❡ 6✉♣❡.✐♠♣♦6❡❞✳ ❚❤✐6 ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ❜② ✈❛.✐♦✉6 ♣.♦❝❡66❡6✿ ❞✐❡✲$♦✲❞✐❡ *❤❡ ❞✐❡6 ❛.❡ 6✉♣❡.✐♠♣♦6❡❞ ❛♠♦♥❣ *❤❡♠ ♦♥❝❡ *❤❡② ❤❛✈❡ ❜❡❡♥ ❝✉* ♦✉*✳ ✶✶

(13)

✶✷ ❈❖◆❚❊◆❚❙

✭❛✮ ❇❛❞ ❛%%✐❣♥♠❡♥+ ✭❜✮ ●♦♦❞ ❛%%✐❣♥♠❡♥+

❋✐❣✉&❡ ✶✿ ■❧❧✉+,&❛,✐♦♥ ♦❢ ,❤❡ ✐♥✢✉❡♥❝❡ ♦❢ ,❤❡ ❛++✐❣♥♠❡♥, ♦♥ ,❤❡ ✜♥❛❧ ❞✐❡ ②✐❡❧❞✳ ❞✐❡✲$♦✲✇❛❢❡) ❛ ♣❛&, ♦❢ ❡♥❣&❛✈❡❞ ✇❛❢❡&+ ❛&❡ ✉+❡❞ ❛+ +✉♣♣♦&, ♦♥ ✇❤✐❝❤ ❝✉, ♦✉, ❞✐❡+ ❛&❡

+✉♣❡&✐♠♣♦+❡❞✳

✇❛❢❡)✲$♦✲✇❛❢❡) ,❤❡ ✇❛❢❡&+ ❛&❡ +✉♣❡&✐♠♣♦+❡❞ ❜❡❢♦&❡ ❜❡✐♥❣ ❝✉, ♦✉,✳

■♥ ❛ ♥✉,+❤❡❧❧✱ ❞✐❡✲$♦✲❞✐❡ ❛♥❞ ❞✐❡✲$♦✲✇❛❢❡) ♣&♦❝❡++❡+ ♦✛❡& ❛ ❣♦♦❞ ②✐❡❧❞ ❜✉, ❣&❡❛,❧② ✐♥❝&❡❛+❡ ♠❛♥✉❢❛❝,✉&✐♥❣ ❝♦+,✳ ❖♥ ,❤❡ ♦,❤❡& ❤❛♥❞✱ ,❤❡ ✇❛❢❡)✲$♦✲✇❛❢❡) ♣&❡+❡♥, +❡✈❡&❛❧ ❛❞✈❛♥,❛❣❡+ ✭,❤❡ ✐♥,❡❣&❛,✐♦♥ ❝❛♥ ❜❡ ♣❡&❢♦&♠❡❞ ❛, &❡❛+♦♥❛❜❧❡ ❝♦+, ✇✐,❤ ♠♦&❡ ❛❝❝✉&❛❝② ❛♥❞ ❡✣❝✐❡♥❝②✱ ,❤❡ ,❤✐❝❦♥❡++ ♦❢ ,❤❡ ✇❛❢❡&+ ❝❛♥ ❜❡ &❡❞✉❝❡❞✱ ✳ ✳ ✳ ✮ ❜✉, ❝❛♥ ❧❡❛❞ ,♦ ✈❡&② ♣♦♦& ②✐❡❧❞✳ ■♥❞❡❡❞✱ ❛ +,❛❝❦ ♦❢ ❞✐❡+ ✐+ ❝♦♥+✐❞❡&❡❞ ,♦ ❜❡ ✈✐❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ,❤❡ ❞✐❡+ ❝♦♠♣♦+✐♥❣ ,❤❡ +,❛❝❦ ❛&❡ ✈✐❛❜❧❡ ♦♥❡+✳ ❋✐❣✉&❡ ✶ ❣✐✈❡+ ❝❧✉❡+ ♦♥ ,❤❡ ♣♦++✐❜❧❡ ②✐❡❧❞ ❞❡❣&❛❞❛,✐♦♥ ✇❤❡♥ +✉♣❡&✐♠♣♦+✐♥❣ ✇❛❢❡&+✳

❚❤✐+ ,❤❡+✐+ ❢♦❝✉+ ♦♥ ,❤❡ ✇❛② ,♦ ❤❛♥❞❧❡ +✉❝❤ ❛ ❝♦♠❜✐♥❛,♦&✐❝+ ✐♥ ,❤❡+❡ ♣&♦❜❧❡♠+✳ ❚❤❡ ♠❛♥✉+❝&✐♣, ✐+ +♣❧✐, ✐♥,♦ ,❤&❡❡ ♠❛✐♥ ♣❛&,+✳ ❚❤❡ ✜&+, ♣❛&, ✐+ ❞❡✈♦,❡❞ ,♦ ,❤❡ ✐♥,&♦❞✉❝,✐♦♥ ♦❢ ,❤❡ ,❤❡♦&❡,✐❝❛❧ ❢&❛♠❡✇♦&❦+ ,❤❛, ❤❛✈❡ ❜❡❡♥ ✉+❡❞ ,♦ ❛♣♣&♦❛❝❤ ,❤❡ ♣&♦❜❧❡♠✳ ■♥ ❛ ✜&+, ,✐♠❡ ✇❡ ♣&❡+❡♥, ❛ +❧✐❣❤, ♦✈❡&✈✐❡✇ ♦♥ ❣&❛♣❤+ ❛♥❞ ❤②♣❡&❣&❛♣❤+ ❛♥+ ,❤❡♥ ✇❡ ❜&✐❡✢② ✐♥,&♦❞✉❝❡ ,❤❡ ❆❇❈ ♦❢ ❝♦♠♣✉,❛,✐♦♥❛❧ ❝♦♠♣❧❡①✐,② ❛♥❞ ✜①❡❞ ♣❛&❛♠❡,❡& ❝♦♠♣❧❡①✐,② ,❤❡♦&②✳

■♥ ,❤❡ +❡❝♦♥❞ ♣❛&,✱ ✇❡ ❝♦♥+✐❞❡& ,❤❡ ,❤❡♦&❡,✐❝❛❧ ❛+♣❡❝,+ ♦❢ ,❤❡ ♣&♦❜❧❡♠✳ ❚♦ ❞♦ +♦✱ ,❤❡ ✜&+, ❝❤❛♣,❡& ♦❢ ,❤✐+ ♣❛&, ✐+ ❞❡✈♦,❡❞ ,♦ ,❤❡ ♠♦❞❡❧✐③❛,✐♦♥ ♦❢ ,❤❡ ❧❛,,❡& ❛♥❞ ,♦ ,❤❡ ✐♥,&♦❞✉❝,✐♦♥ ♦❢ +❡✈❡&❛❧ ♦❢ ✐,+ ✈❛&✐❛♥,+✱ ❞✐✛❡&✐♥❣ ♠❛✐♥❧② ♦♥ ,❤❡ ✇❛② ,♦ ❡✈❛❧✉❛,❡ ,❤❡ J✉❛❧✐,② ♦❢ ❛♥ ❛++✐❣♥♠❡♥,✳ ❆ ❝❤❛♣,❡& ✐+ ,❤❡♥ ❞❡✈♦,❡❞ ,♦ ✇❤❛, ✇❡ ✇✐❧❧ ❝❛❧❧ ✐♥$❡)♠❡❞✐❛$❡ ♣)♦❜❧❡♠/✱ ✐✳❡✳ ♣&♦❜❧❡♠+ ♣&❡+❡♥,✐♥❣ ♠❛✐♥❧② ,❤❡♦&❡,✐❝❛❧ ✐♥,❡&❡+, ❛+ ,❤❡② ♣&♦✈✐❞❡ ✉+❡❢✉❧ ,♦♦❧+ ,♦ ❤❛♥❞❧❡ ❝♦♠♣❧❡①✐,② ♦❢ ,❤❡ ♠❛✐♥ ♣&♦❜❧❡♠+✳ ▲❛+,✱ ✇❡ ❝♦♥+❡❝&❛,❡ ♦♥❡ ❝❤❛♣,❡& ❢♦& ❡❛❝❤ ♦❢ ,❤❡ ❜♦,❤ ♠❛✐♥ ♣&♦❜❧❡♠+ ✇❡ ❝♦♥+✐❞❡&✳ ■♥ ❡❛❝❤ ❝❤❛♣,❡&✱ ✇❡ ,&② ,♦ &❡♥❞❡& ❛+ ♣&❡❝✐+❡❧② ❛+ ♣♦++✐❜❧❡ ,❤❡ ❝♦♠♣❧❡①✐,② ♦❢ ,❤❡ ❝♦♥+✐❞❡&❡❞ ✈❛&✐❛♥,✳

❚❤❡ ,❤✐&❞ ♣❛&, ♣&❡+❡♥, &❡+✉❧,+ ♦❜,❛✐♥❡❞ ❢&♦♠ +❡✈❡&❛❧ ❝❛♠♣❛✐❣♥+ ♦❢ ❡①♣❡&✐♠❡♥,❛,✐♦♥✳ ❚❤❡ ♦❜❥❡❝,✐✈❡ ♦❢ ,❤✐+ ♣❛&, ✐+ ,♦ ❧❛② ,❤❡ ❜❛+✐+ ♦❢ ,❤❡ ♣&❛❝,✐❝❛❧ &❡+♦❧✉,✐♦♥ ♦❢ ,❤❡ ♣&♦❜❧❡♠ ❜② ❡①♣❧♦&✐♥❣ ❞✐✛❡&❡♥, +♦❧✈✐♥❣ ♠❡,❤♦❞+ +✉❝❤ ❛+ ■♥,❡❣❡& ▲✐♥❡❛& M&♦❣&❛♠♠✐♥❣ ✭■▲M✮ ❢♦&♠✉❧❛,✐♦♥+✱ ❈♦♥+,&❛✐♥,+ M&♦❣&❛♠♠✐♥❣ ✭❈M✮ ❢♦&♠✉❧❛,✐♦♥+ ♦& ❛❞ ❤♦❝ ❤❡✉&✐+,✐❝+✳

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❈❖◆❚❊◆❚❙ ✶✸ ❬❇❉●✶✻❪ ▼❛*✐♥ ❇♦✉❣❡*❡1✱ ●✉✐❧❧❡*♠❡ ❉✉✈✐❧❧✐6✱ ❛♥❞ ❘♦❞♦❧♣❤❡ ●✐*♦✉❞❡❛✉✳ ✏❆♣♣*♦①✲ ✐♠❛❜✐❧✐1② ❛♥❞ ❊①❛❝1 ❘❡D♦❧✉1✐♦♥ ♦❢ 1❤❡ ▼✉❧1✐❞✐♠❡♥D✐♦♥❛❧ ❇✐♥❛*② ✈❡❝1♦* ❆DD✐❣♥♠❡♥1 F*♦❜❧❡♠✑✳ ■♥✿ !♦❝❡❡❞✐♥❣) ♦❢ +❤❡ ✹+❤ ■♥+❡!♥❛+✐♦♥❛❧ ❙②♠♣♦)✐✉♠ ♦♥ ❈♦♠❜✐♥❛+♦!✐❛❧ ❖♣+✐♠✐③❛+✐♦♥✱ ■❙❈❖ ✷✵✶✻✱ ❱✐❡+!✐✲❙✉❧✲▼❛!❡✱ ■+❛❧②✱ ▼❛② ✶✼✲✶✾✱ ✷✵✶✻✳ ✷✵✶✻✱ 1♦ ❜❡ ♣✉❜❧✐D❤❡❞✳ ❬❇♦✉✰✶✺❪ ▼❛*✐♥ ❇♦✉❣❡*❡1 ❡1 ❛❧✳ ✏▼✉❧1✐❞✐♠❡♥D✐♦♥❛❧ ❇✐♥❛*② ❱❡❝1♦* ❆DD✐❣♥♠❡♥1 F*♦❜✲ ❧❡♠✿ ❙1❛♥❞❛*❞✱ ❙1*✉❝1✉*❛❧ ❛♥❞ ❆❜♦✈❡ ●✉❛*❛♥1❡❡ F❛*❛♠❡1❡*✐③❛1✐♦♥D✑✳ ■♥✿ !♦❝❡❡❞✐♥❣) ♦❢ +❤❡ ✷✵+❤ ■♥+❡!♥❛+✐♦♥❛❧ ❙②♠♣♦)✉♠ ♦♥ ❋✉♥❞❛♠❡♥+❛❧) ♦❢ ❈♦♠✲ ♣✉+❛+✐♦♥ ❚❤❡♦!②✱ ❋❈❚ ✷✵✶✺✱ ●❞❛➠)❦✱ ♦❧❛♥❞✱ ❆✉❣✉)+ ✶✼✲✶✾✱ ✷✵✶✺✳ ✷✵✶✺✱ ♣♣✳ ✶✽✾✕✷✵✶✳ ❬❇♦✉✰✶✻❪ ▼❛*✐♥ ❇♦✉❣❡*❡1 ❡1 ❛❧✳ ✏❖♥ 1❤❡ ❈♦♠♣❧❡①✐1② ♦❢ ❲❛❢❡*✲1♦✲❲❛❢❡* ■♥1❡❣*❛1✐♦♥✑✳ ■♥✿ ❉✐)❝!❡+❡ ❖♣+✐♠✐③❛+✐♦♥ ✭✷✵✶✻✮✱ 1♦ ❜❡ ♣✉❜❧✐D❤❡❞✳ ❬❉✉✈✰✶✺❪ ●✉✐❧❧❡*♠❡ ❉✉✈✐❧❧✐6 ❡1 ❛❧✳ ✏❖♥ 1❤❡ ❈♦♠♣❧❡①✐1② ♦❢ ❲❛❢❡*✲1♦✲❲❛❢❡* ■♥1❡✲ ❣*❛1✐♦♥✑✳ ■♥✿ ❆❧❣♦!✐+❤♠) ❛♥❞ ❈♦♠♣❧❡①✐+② ✲ ✾+❤ ■♥+❡!♥❛+✐♦♥❛❧ ❈♦♥❢❡!❡♥❝❡✱ ❈■❆❈ ✷✵✶✺✱ ❛!✐)✱ ❋!❛♥❝❡✱ ▼❛② ✷✵✲✷✷✱ ✷✵✶✺✳ !♦❝❡❡❞✐♥❣)✳ ✷✵✶✺✱ ♣♣✳ ✷✵✽✕ ✷✷✵✳

(15)

(16)

!♦❧❡❣♦♠❡♥❛

❛"# ■

(17)

(18)

✶

●"❛♣❤& ❛♥❞ ❍②♣❡"❣"❛♣❤&

❈♦♥#❡♥#%

✶✳✶ ●#❛♣❤' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✷ ❍②♣❡#❣#❛♣❤' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

❚❤✐# #❤♦%& ❝❤❛♣&❡% ✐# ❞❡✈♦&❡❞ &♦ &❤❡ ✐♥&%♦❞✉❝&✐♦♥ ♦❢ ❛♥ ✐♠♣♦%&❛♥& ❞❛&❛ #&%✉❝&✉%❡✱ &❤❡ ❣%❛♣❤#✳ ❚❤❡ ❤②♣❡%❣%❛♣❤#✱ ❛ ❣❡♥❡%❛❧✐③❛&✐♦♥ ♦❢ ❣%❛♣❤#✱ ❛%❡ ❛❧#♦ ✐♥&%♦❞✉❝❡❞ ❛# &❤❡② %❡♣%❡#❡♥& ❛ ❦❡② ♥♦&✐♦♥ ✐♥ ♦✉% ✇♦%❦✳ ▲❡& ✉# %❡♠❛%❦ &❤❛& &❤✐# ❝❤❛♣&❡% ♦♥❧② ♣%❡#❡♥&# &%✐✈✐❛ ♦♥ ●%❛♣❤ ❚❤❡♦%②✳ ❚❤❡%❡❢♦%❡✱ ✇❡ ✐♥✈✐&❡ ✇❡❧❧✲✈❡%#❡❞ %❡❛❞❡% ✐♥ ●%❛♣❤ ❚❤❡♦%② &♦ #&❛%& &❤❡ %❡❛❞✐♥❣ ♦❢ &❤✐# ♠❛♥✉#❝%✐♣& ❛& ❈❤❛♣&❡% ✷✳

✶✳✶ ●#❛♣❤'

❉❡✜♥✐%✐♦♥ ✶✳✶ ✭●%❛♣❤✮✳ ❆ ❣"❛♣❤ G ✐' ❛ ♣❛✐" (U, E) ✇❤❡"❡ U ✐' ❛ ♥♦♥ ❡♠♣-② '❡- ♦❢ ✈❡%&✐❝❡# ❛♥❞ E ✐' ❛ '❡- ♦❢ ❡❞❣❡#✱ ❛♥ ❡❞❣❡ ❜❡✐♥❣ ❛ ♣❛✐" ♦❢ ✈❡"-✐❝❡' ♦❢ U✳

■♥&✉✐&✐✈❡❧②✱ ❛ ❣%❛♣❤ ✐# ❛ ❝♦♥✈❡♥✐❡♥& ✇❛② ♦❢ %❡♣%❡#❡♥&✐♥❣ ❜✐♥❛%② %❡❧❛&✐♦♥#❤✐♣# ❛♠♦♥❣ ❛ #❡& ♦❢ ❡♥&✐&✐❡#✳ ❆# #✉❝❤ ❣%❛♣❤# ❛%❡ ✇✐❞❡❧② ✉#❡❞ ✐♥ ❧♦&# ♦❢ ❛%❡❛#✳ ❋♦% ✐♥#&❛♥❝❡✱ &❤❡② ❝❛♥ ❜❡ ✉#❡❞ &♦ %❡♣%❡#❡♥& ♥❡&✇♦%❦#✿

)♦❝✐❛❧ ♥❡%✇♦.❦✿ ♦♥❡ ❝❛♥ &❤✐♥❦ ♦❢ ❛ ♣❡%#♦♥ ❛# ❛ ✈❡%&❡①✳ ❚✇♦ ✈❡%&✐❝❡# ❛%❡ ❝♦♥♥❡❝&❡❞ ❜② ❛♥ ❡❞❣❡ ✐❢ &❤❡ ♣❡%#♦♥ &❤❡② %❡♣%❡#❡♥& ❛%❡ ✐♥ ❝♦♥&❛❝&✳

♣❤②)✐❝❛❧ ♥❡%✇♦.❦✿ ❛ ✈❡%&❡① %❡♣%❡#❡♥&# ❛ ♠❛❝❤✐♥❡ ✐♥ &❤❡ ♥❡&✇♦%❦ ❛♥❞ ❛♥ ❡❞❣❡ ❛ ♣❤②#✲ ✐❝❛❧ ❧✐♥❦ ❜❡&✇❡❡♥ &✇♦ ♠❛❝❤✐♥❡#✳

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✶✽ ❈❍❆#❚❊❘ ✶✳ ●❘❆#❍❙ ❆◆❉ ❍❨#❊❘●❘❆#❍❙ ❛✐❧✇❛② ♥❡(✇♦ ❦✿ ❞✉$✐♥❣ (❤❡ ❈♦❧❞ ❲❛$✱ ❯❙ ❆$♠② ✉6❡❞ ❣$❛♣❤6 (♦ $❡♣$❡6❡♥( (❤❡ ❙♦✈✐❡( $❛✐❧✇❛② ♥❡(✇♦$❦ ✐♥ ♦$❞❡$ (♦ ✈✐6✉❛❧✐③❡ ✇❡❛❦♥❡66❡6 ✐♥ (❤❡ ❙♦✈✐❡( $❛✐❧✇❛② ($❛✣❝ ✢♦✇✳ ✳ ✳ ✳ ◆❛(✉$❛❧❧②✱ ❣$❛♣❤6 ❛$❡ ♥♦( $❡6($✐❝(❡❞ (♦ (❤❡ ♠♦❞❡❧✐③❛(✐♦♥ ♦❢ ♥❡(✇♦$❦6✱ (❤❡② ♦✛❡$ ❛ ♣♦✇❡$❢✉❧ (❤❡♦$❡(✐❝❛❧ ❢$❛♠❡✇♦$❦ (♦ ♠♦❞❡❧✐③❡ ♣$♦❜❧❡♠6 ♦❢ ❛❧❧ 6♦$(6✳ ❖♥❡ ❝❛♥ ❞❡✜♥❡ 6♦♠❡ ✐♥(❡$❡6(✐♥❣ ❛♥❞ ✉6❡❢✉❧ ♥♦(✐♦♥6 ♦♥ ❛ ❣$❛♣❤✳ ❚❤✐6 6❡❝(✐♦♥ ✐6 (❤✉6 ❞❡✈♦(❡❞ (♦ (❤❡ ✐♥($♦❞✉❝(✐♦♥ ♦❢ 6♦♠❡ ♦❢ (❡$♠6 ❛♥❞ ♥♦(✐♦♥6 (❤❛( ✇✐❧❧ ❜❡ ✉6❡❞ (❤$♦✉❣❤♦✉( (❤✐6 ♠❛♥✉6❝$✐♣(✳ ❖❜✈✐♦✉6❧②✱ ✐( ❞♦❡6 ♥♦( ❛✐♠ ❛( ❜❡✐♥❣ ❛♥ ❡①❤❛✉6(✐✈❡ ♦✈❡$✈✐❡✇ ♦❢ (❤❡ ❣$❛♣❤ ❜❛6✐❝6✳ ❲❡ $❡❢❡$ (❤❡ $❡❛❞❡$ (♦ (❤❡ ✇♦$❦6 ❣✐✈❡♥ ❜② ❘❡✐♥❤❛$❞( ❉✐❡6(❡❧ ❬❉✐❡✶✷❪ ❛♥❞ ❜② ❆❞$✐❛♥ ❇♦♥❞② ❛♥❞ ❯♣♣❛❧✉$✐ ▼✉$(② ❬❇▼✶✶❪✳ ■♥ (❤❡ ❢♦❧❧♦✇✐♥❣✱ ✇❡ ♣$❡6❡♥( (❤❡ ❣$❛♣❤ ♥♦(✐♦♥6 (❤❛( ✇✐❧❧ ❜❡ ✉6❡❞ (❤$♦✉❣❤♦✉( (❤✐6 ♠❛♥✉6❝$✐♣(✳ ❉❡✜♥✐(✐♦♥ ✶✳✷ ✭❈♦♠♣❧❡(❡ ❣$❛♣❤✮✳ ❆ ❣"❛♣❤ G = (U, E) ✐' '❛✐❞ ❝♦♠♣❧❡.❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∀u, v ∈ U, u 6= v, {u, v} ∈ E✳

❉❡✜♥✐(✐♦♥ ✶✳✸ ✭■♥❞✉❝❡❞ 6✉❜❣$❛♣❤ ❬❉✐❡✶✷❪✮✳ ●✐✈❡♥ ❛ ❣"❛♣❤ G = (U, E) ❛♥❞ ❛ '✉❜'❡. ♦❢ ✈❡".✐❝❡' S ⊆ U✱ .❤❡ '✉❜❣"❛♣❤ G[S] ✐♥❞✉❝❡❞ ❜② S ✐' .❤❡ ❣"❛♣❤ ✇❤♦'❡ ✈❡".❡① '❡. ✐' S ❛♥❞ ✇❤♦'❡ ❡❞❣❡ '❡. ❝♦♥'✐'.' ♦❢ ❛❧❧ .❤❡ ❡❞❣❡' ✐♥ E .❤❛. ❤❛✈❡ ❜♦.❤ ❡♥❞♣♦✐♥.' ✐♥ S✳ ▼♦"❡ ❢♦"♠❛❧❧② G[S] = (S, {{u, v} ∈ E : u, v ∈ S})✳ ❲❡ ❝❛♥ ✉6❡ (❤❡6❡ ❞❡✜♥✐(✐♦♥ (♦ ❞❡✜♥❡ ❛ ❝❧✐;✉❡ ✐♥ ❛ ❣$❛♣❤✳ ❉❡✜♥✐(✐♦♥ ✶✳✹ ✭❈❧✐R✉❡✮✳ ●✐✈❡♥ ❛ ❣"❛♣❤ G = (U, E)✱ ❛ ❝❧✐;✉❡ ♦❢ G ✐' ❛ '✉❜'❡. ♦❢ ✈❡".✐❝❡' C ⊆ U '✉❝❤ .❤❛. G[C] ✐' ❝♦♠♣❧❡.❡✳ ❆♥♦(❤❡$ ✐♥(❡$❡6(✐♥❣ ❛♥❞ ✐♠♣♦$(❛♥( ♥♦(✐♦♥ ✐♥ ❛ ❣$❛♣❤ ✐6 (❤❡ ♥♦(✐♦♥ ♦❢ ✐♥❞❡♣❡♥❞❡♥. '❡.✳ ❉❡✜♥✐(✐♦♥ ✶✳✺ ✭■♥❞❡♣❡♥❞❡♥( ❙❡(✮✳ ●✐✈❡♥ ❛ ❣"❛♣❤ G = (U, E)✱ ❛♥ ✐♥❞❡♣❡♥❞❡♥. '❡. ♦❢ G ✐' ❛ '✉❜'❡. ♦❢ ✈❡".✐❝❡' IS ⊆ U '✉❝❤ .❤❛. G[IS] = (IS, E′) '❛.✐'✜❡' E′ = ∅✳ ■♥ ♦.❤❡" ✇♦"❞'✱ .❤❡"❡ ❞♦❡' ♥♦. ❡①✐'. u, v ∈ IS '✉❝❤ .❤❛. {u, v} ∈ E✳ ❙✉❝❤ ❛ 6✉❜6❡( ♦❢ ✈❡$(✐❝❡6 ✐6 ❛❧6♦ ❝❛❧❧❡❞ ❛ '.❛❜❧❡ '❡.✳ ❆ 6($♦♥❣❧② $❡❧❛(❡❞ ♥♦(✐♦♥ ✐6 (❤❡ ✈❡".❡① ❝♦✈❡"✳ ❉❡✜♥✐(✐♦♥ ✶✳✻ ✭❱❡$(❡① ❈♦✈❡$✮✳ ●✐✈❡♥ ❛ ❣"❛♣❤ G = (U, E)✱ ❛ ✈❡".❡① ❝♦✈❡" ♦❢ G ✐' ❛ '✉❜'❡. ♦❢ ✈❡".✐❝❡' V C ⊆ U '✉❝❤ .❤❛. ❢♦" ❛❧❧ {u, v} ∈ E, u ∈ V C ♦" v ∈ V C✳ ■♥ ♦.❤❡" ✇♦"❞'✱ V C ❝♦♥.❛✐♥' ❛. ❧❡❛'. ♦♥❡ ❡♥❞♣♦✐♥. ♦❢ ❡❛❝❤ ❡❞❣❡✳ 5 ♦♣❡ (② ✶✳✶✳ ●✐✈❡♥ ❛ ❣"❛♣❤ G = (U, E) ❛♥❞ V C ❛ ✈❡".❡① ❝♦✈❡" ♦❢ ●✱ .❤❡ '❡. IS = U \V C ✐' ❛♥ ✐♥❞❡♣❡♥❞❡♥. '❡.✳ ❉❡✜♥✐(✐♦♥ ✶✳✼ ✭●$❛♣❤ ❝♦❧♦$✐♥❣✮✳ ❆ k✲✈❡".❡① ❝♦❧♦"✐♥❣ ✭❛ k✲❝♦❧♦"✐♥❣ ❢♦" '❤♦".✮ ♦❢ ❛ ❣"❛♣❤ G = (U, E) ✐' ♠❛♣ c : U → [k]✳ ❆ k✲❝♦❧♦"✐♥❣ c ✐' '❛✐❞ .♦ ❜❡ ♣"♦♣❡" ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∀{u, v} ∈ E, c(u) 6= c(v)✳

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✶✳✷✳ ❍❨%❊❘●❘❆%❍❙ ✶✾

✶✳✷ ❍②♣❡'❣'❛♣❤+

❉❡✜♥✐%✐♦♥ ✶✳✽ ✭❍②♣❡'❣'❛♣❤✮✳ ❆ ❤②♣❡%❣%❛♣❤ H ✐) ❛ ❝♦✉♣❧❡ (U, E) ✇❤❡%❡ U ✐) ❛ ♥♦♥ ❡♠♣1② )❡1 ♦❢ ✈❡%1✐❝❡) ❛♥❞ E ❛ )❡1 ♦❢ ❤②♣❡%❡❞❣❡)✱ ❛♥ ❤②♣❡%❡❞❣❡ ❜❡✐♥❣ ❛ ♥♦♥ ❡♠♣1② )✉❜)❡1 ♦❢ U✳ ❲❡ ❛❧.♦ ❞❡✜♥❡ .❡✈❡'❛❧ ♥♦4✐♦♥. ♦♥ 4❤❡.❡ ❤②♣❡'❣'❛♣❤.✳ ❉❡✜♥✐%✐♦♥ ✶✳✾ ✭k✲✉♥✐❢♦'♠✮✳ ●✐✈❡♥ ❛♥ ✐♥1❡❣❡% k✱ ❛ ❤②♣❡%❣%❛♣❤ H = (U, E) ✐) )❛✐❞ 1♦ ❜❡ k✲✉♥✐❢♦%♠ ✐❢ ❛♥❞ ♦♥❧② ✐❢ E ⊆!U k✱ ! U k ❞❡♥♦1✐♥❣ 1❤❡ )✉❜)❡1) ♦❢ U ♦❢ )✐③❡ k✳ ■♥ ♦4❤❡' ✇♦'❞.✱ ❛ ❤②♣❡'❣'❛♣❤ ✐. .❛✐❞ k✲✉♥✐❢♦'♠ ✐❢ ❡✈❡'② ❤②♣❡'❡❞❣❡ ❝♦♥4❛✐♥. ❡①❛❝4❧② k✈❡'4✐❝❡.✳ ❉❡✜♥✐%✐♦♥ ✶✳✶✵ ✭k✲♣❛'4✐4❡✮✳ ●✐✈❡♥ ❛♥ ✐♥1❡❣❡% k✱ ❛ ❤②♣❡%❣%❛♣❤ H = (U, E) ✐) )❛✐❞ 1♦ ❜❡ k✲♣❛%1✐1❡ ✐❢ ✐1) ✈❡%1❡① )❡1 ❝❛♥ ❜❡ ♣❛%1✐1✐♦♥❡❞ ✐♥1♦ k ❞✐)❥♦✐♥1 )✉❜)❡1) U1, . . . , Uk )✉❝❤ 1❤❛1 ❢♦% ❛❧❧ i ∈ [k] ❛♥❞ ❢♦% ❛❧❧ ❝♦✉♣❧❡ ♦❢ ✈❡%1✐❝❡) u, v ∈ Ui✱ ♥♦ ❤②♣❡%❡❞❣❡ e ∈ E ❝♦♥1❛✐♥✐♥❣ ❜♦1❤ u ❛♥❞ v ❡①✐)1)✳ ■♥ ♦1❤❡% ✇♦%❞)✱ 1✇♦ ✈❡%1✐❝❡) ♦❢ ❛ )❛♠❡ )✉❜)❡1 ❝❛♥♥♦1 ❜❡ ❡❧❡♠❡♥1) ♦❢ ❛ )❛♠❡ ❤②♣❡%✲ ❡❞❣❡✳ ❲❡ ❝❛♥ ❛❧.♦ ♥❛4✉'❛❧❧② ❡①4❡♥❞ .♦♠❡ ❣'❛♣❤ ♥♦4✐♦♥. 4♦ 4❤❡ ❤②♣❡'❣'❛♣❤.✳ ❉❡✜♥✐%✐♦♥ ✶✳✶✶ ✭■♥❞✉❝❡❞ .✉❜❤②♣❡'❣'❛♣❤✮✳ ●✐✈❡♥ ❛ ❤②♣❡%❣%❛♣❤ H = (U, E) ❛♥❞ ❛ )✉❜)❡1 ♦❢ ✈❡%1✐❝❡) S ⊆ U✱ 1❤❡ )✉❜❤②♣❡%❣%❛♣❤ H[S] ✐♥❞✉❝❡❞ ❜② S ✐) 1❤❡ ❤②♣❡%❣%❛♣❤ ✇❤♦)❡ ✈❡%1❡① )❡1 ✐) S ❛♥❞ ✇❤♦)❡ ❤②♣❡%❡❞❣❡ )❡1 ❝♦♥)✐)1) ♦❢ ❛❧❧ 1❤❡ ❤②♣❡%❡❞❣❡) ✐♥ E 1❤❛1 ❤❛✈❡ ❛❧❧ ❡♥❞♣♦✐♥1) ✐♥ S✳ ▼♦%❡ ❢♦%♠❛❧❧② H[S] = (S, {e ∈ E : e ∩ S = e})✳ ❉❡✜♥✐%✐♦♥ ✶✳✶✷ ✭■♥❞❡♣❡♥❞❡♥4 ❙❡4✮✳ ●✐✈❡♥ ❛ ❤②♣❡%❣%❛♣❤ H = (U, E)✱ ❛♥ ✐♥❞❡♣❡♥❞❡♥1 )❡1 ♦❢ H ✐) ❛ )✉❜)❡1 ♦❢ ✈❡%1✐❝❡) IS ⊆ U )✉❝❤ 1❤❛1 H[IS] = (IS, E′₎ _{)❛1✐)✜❡) E}′ _{= ∅}_✳ ■♥ ♦1❤❡% ✇♦%❞)✱ 1❤❡%❡ ❞♦❡) ♥♦1 ❡①✐)1 e ⊂ IS )✉❝❤ 1❤❛1 e ∈ E✳ ❉❡✜♥✐%✐♦♥ ✶✳✶✸ ✭❱❡'4❡① ❈♦✈❡'✮✳ ●✐✈❡♥ ❛ ❤②♣❡%❣%❛♣❤ H = (U, E)✱ ❛ ✈❡%1❡① ❝♦✈❡% ♦❢ H ✐) ❛ )✉❜)❡1 ♦❢ ✈❡%1✐❝❡) V C ⊆ U )✉❝❤ 1❤❛1 ∀e ∈ E, e ∩ V C 6= ∅✳ ■♥ ♦1❤❡% ✇♦%❞)✱ V C ❝♦♥1❛✐♥) ❛1 ❧❡❛)1 ♦♥❡ ❡❧❡♠❡♥1 ♦❢ ❡❛❝❤ ❤②♣❡%❡❞❣❡✳ ❆. ❢♦' ❣'❛♣❤.✱ 4❤❡ ❢♦❧❧♦✇✐♥❣ ♣'♦♣❡'4② ❤♦❧❞. ❢♦' ❤②♣❡'❣'❛♣❤.✳ ./♦♣❡/%② ✶✳✷✳ ●✐✈❡♥ ❛ ❤②♣❡%❣%❛♣❤ H = (U, E) ❛♥❞ V C ❛ ✈❡%1❡① ❝♦✈❡% ♦❢ ❍✱ 1❤❡ )❡1 IS = U \V C ✐) ❛♥ ✐♥❞❡♣❡♥❞❡♥1 )❡1✳ ❉❡✜♥✐%✐♦♥ ✶✳✶✹ ✭❍②♣❡'❣'❛♣❤ ♣'♦♣❡' ❝♦❧♦'✐♥❣✮✳ ❆ k✲❝♦❧♦%✐♥❣ ♦❢ 1❤❡ ✈❡%1✐❝❡) ♦❢ ❛ ❤②✲ ♣❡%❣%❛♣❤ H = (U, E) ✐) ♠❛♣ c : U → [k]✳

❆ k✲❝♦❧♦%✐♥❣ ✐) )❛✐❞ 1♦ ❜❡ ♣%♦♣❡% ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∀e ∈ E, ∃u, v ∈ e )✉❝❤ 1❤❛1 c(u) 6= c(v)✳

❉❡✜♥✐%✐♦♥ ✶✳✶✺ ✭❍②♣❡'❣'❛♣❤ '❛✐♥❜♦✇ ❝♦❧♦'✐♥❣✮✳ ❆ k✲❝♦❧♦%✐♥❣ ✐) )❛✐❞ 1♦ ❜❡ ❛ %❛✐♥❜♦✇ ❝♦❧♦%✐♥❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∀e ∈ E, ∀u, v ∈ e, c(u) 6= c(v)✳

(21)

(22)

✷

❈♦♠♣✉&❛&✐♦♥❛❧ ❈♦♠♣❧❡①✐&②

❈♦♥#❡♥#% ✷✳✶ #$♦❜❧❡♠*✱ ❘❡❞✉❝0✐♦♥* ❛♥❞ ❈♦♠♣❧❡①✐0② ❈❧❛**❡* ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳✶ ❉❡❝✐'✐♦♥ *+♦❜❧❡♠' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳✷ ❚❤❡ NP ❝♦♠♣❧❡①✐3② ❝❧❛'' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✶✳✸ ❖♣3✐♠✐③❛3✐♦♥ ♣+♦❜❧❡♠' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷ ❆♣♣$♦①✐♠❛0✐♦♥ ❆❧❣♦$✐0❤♠* ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✸ ❆♣♣$♦①✐♠❛0✐♦♥ #$❡*❡$✈✐♥❣ ❘❡❞✉❝0✐♦♥* ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✹ ●❛♣✲$❡❞✉❝0✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✺ ❖0❤❡$ ❝♦♠♣❧❡①✐0② ❤②♣♦0❤❡*❡* ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✺✳✶ ❚❤❡ ❯♥✐<✉❡ ●❛♠❡ ❈♦♥❥❡❝3✉+❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✺✳✷ NP * QP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✺✳✸ ❊①♣♦♥❡♥3✐❛❧ ❚✐♠❡ ❍②♣♦3❤❡'✐' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ■♥ "❤✐% ❝❤❛♣"❡*✱ ✇❡ ✐♥"*♦❞✉❝❡ "❤❡ *✉❞✐♠❡♥"% ✐♥ ❝♦♠♣❧❡①✐"② "❤❡♦*② *❡4✉✐*❡❞ ❢♦* ❛ ❝♦♥✲ ✈❡♥✐❡♥" *❡❛❞✐♥❣ ♦❢ "❤✐% ♠❛♥✉%❝*✐♣"✳ ❚❤❡ ❞❡✜♥✐"✐♦♥ ❛♥❞ *❡%✉❧"% ✐♥"*♦❞✉❝❡❞ ❤❡*❡ ❛*❡ ❝♦♥%✐❞❡*❡❞ "♦ ❜❡ %"❛♥❞❛*❞ ❛♥❞ "❤❡*❡❢♦*❡ ❛*❡ %✉❝❝✐♥❝"❧② ♣*❡%❡♥"❡❞✳ ❍♦✇❡✈❡*✱ ✇❡ ❛✐♠ ❛" ♣*♦✈✐❞✐♥❣ ❛ ❜✉♥❝❤ ♦❢ *❡❢❡*❡♥❝❡% ✐♥ ✇❤✐❝❤ "❤❡%❡ *❡%✉❧"% ❛*❡ ♠♦*❡ ❞❡❡♣❧② %"✉❞✐❡❞✳ ❋✉*✲ "❤❡*♠♦*❡✱ "❤✐% %❡❝"✐♦♥ ❛❧❧♦✇% ✉% "♦ ✐♥"*♦❞✉❝❡ ♥♦"❛"✐♦♥% "❤❛" ✇✐❧❧ ❜❡ ✉%❡❞ "❤*♦✉❣❤♦✉" "❤✐% ♠❛♥✉%❝*✐♣"✳ ■♥ ❛ ✜*%" "✐♠❡ ✇❡ ♣*❡%❡♥" "❤❡ ♥♦"✐♦♥ ♦❢ ❞❡❝✐$✐♦♥ ♣(♦❜❧❡♠✱ ✇❤✐❝❤ ✐% ❛♥ ✐♠♣♦*"❛♥" ♥♦"✐♦♥ ✐♥ ❝♦♠♣❧❡①✐"② "❤❡♦*②✳ ❲❡ "❤❡♥ %❤♦✇ ❤♦✇ ❛❧❣♦*✐"❤♠% ❝❛♥ ❜❡ ✉%❡❞ "♦ ❤✐❣❤❧✐❣❤" *❡❧❛"✐♦♥%❤✐♣% ❜❡"✇❡❡♥ ♣*♦❜❧❡♠%✱ ❜② ❞❡✜♥✐♥❣ (❡❞✉❝-✐♦♥$✳ ❚❤❡ ❧❛""❡* ❧❛② "❤❡ ❢♦✉♥❞❛"✐♦♥% ❢♦* ❛ ❝❧❛%%✐✜❝❛"✐♦♥ ♦❢ ❞❡❝✐%✐♦♥ ♣*♦❜❧❡♠% ✐♥ "❡*♠% ♦❢ ❝♦♠♣❧❡①✐"②✳ ❲❡ ✐♥"*♦❞✉❝❡ "❤❡♥ "✇♦ ❝♦♠♣❧❡①✐-② ❝❧❛$$❡$ P ❛♥❞ NP✳ ✷✶

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✷✷ ❈❍❆#❚❊❘ ✷✳ ❈❖▼#❯❚❆❚■❖◆❆▲ ❈❖▼#▲❊❳■❚❨ ■♥ ❛ $❡❝♦♥❞ )✐♠❡✱ ✇❡ )❛❦❡ ♦♣"✐♠✐③❛"✐♦♥ ♣(♦❜❧❡♠, ✐♥)♦ ❝♦♥$✐❞❡/❛)✐♦♥✳ ❚❤❡ ❝♦♠♣❧❡①✐)② ❝❧❛$$ NPO ✐$ )❤❡♥ ❞❡✜♥❡❞✳ ❚❤❡ ✐$$✉❡ ♦❢ ♦♣)✐♠✐③❛)✐♦♥ ♣/♦❜❧❡♠ $♦❧✈✐♥❣ ✐$ )❤❡♥ ❞✐$❝✉$$❡❞ )♦ ✐♥)/♦❞✉❝❡ ❛ ♥❡✇ ❢❛♠✐❧② ♦❢ ❛❧❣♦/✐)❤♠$✱ ❝❛❧❧❡❞ ❛♣♣(♦①✐♠❛"✐♦♥ ❛❧❣♦(✐"❤♠,✳ ❲❡ $❤♦✇ ❤♦✇ )❤❡$❡ ❛❧❣♦/✐)❤♠$ ❛/❡ ✉$❡❞ )♦ ❞❡$✐❣♥ ♥❡✇ ❝♦♠♣❧❡①✐)② ❝❧❛$$❡$ $✉❝❤ ❛$ APX✱ PTAS ♦/ FPTAS✳ ❆♥❞ ✜♥❛❧❧② ✇❡ ♣/❡$❡♥) ❤♦✇ ♣♦❧②♥♦♠✐❛❧ /❡❞✉❝)✐♦♥$ ❝❛♥ ❜❡ )✉♥❡❞ )♦ ❞❡$✐❣♥ ❛♣♣(♦①✐♠❛"✐♦♥ ♣(❡,❡(✈✐♥❣ (❡❞✉❝"✐♦♥,✱ ❤✐❣❤❧✐❣❤)✐♥❣ ❛) )❤❡ $❛♠❡ )✐♠❡✱ $♦♠❡ /❡❧❛)✐♦♥$❤✐♣$ ❜❡)✇❡❡♥ ♦♣)✐♠✐③❛)✐♦♥ ♣/♦❜❧❡♠$✱ ✐♥ )❡/♠$ ♦❢ ❛♣♣/♦①✐♠❛❜✐❧✐)②✳

✷✳✶ #$♦❜❧❡♠✱ ❘❡❞✉❝0✐♦♥ ❛♥❞ ❈♦♠♣❧❡①✐0② ❈❧❛**❡*

✷✳✶✳✶ ❉❡❝✐'✐♦♥ *+♦❜❧❡♠' ❚❤❡ ❞❡❝✐,✐♦♥ ♣(♦❜❧❡♠, /❡♣/❡$❡♥) )❤❡ ♠♦$) ♥❛)✉/❛❧ ♣/♦❜❧❡♠$ ♦♥❡ ❝❛♥ ❜❡ ❡①♣♦$❡❞ )♦✳ ■) $✐♠♣❧② ❝♦♥$✐$)$ ✐♥ ❛ ②❡$✲♦/✲♥♦ A✉❡$)✐♦♥✳ ●✐✈❡♥ ❛ ♣/♦❜❧❡♠ Π ❛♥❞ ❛♥ ✐♥♣✉) I ∈ {0, 1}∗_✱ )❤❡ ♦❜❥❡❝)✐✈❡ ✐$ )♦ ❞❡)❡/♠✐♥❡ ✇❤❡)❤❡/ I ✐$ ❛ ♣♦$✐)✐✈❡ ✐♥$)❛♥❝❡ ✭)❤❡ ❛♥$✇❡/ )♦ )❤❡ A✉❡$)✐♦♥ ✐$ ②❡,✮ ♦/ ❛ ♥❡❣❛)✐✈❡ ✐♥$)❛♥❝❡ ✭)❤❡ ❛♥$✇❡/ )♦ )❤❡ A✉❡$)✐♦♥ ✐$ ♥♦✮✳ ❆$ $✉❝❤✱ ❛ ❞❡❝✐$✐♦♥ ♣/♦❜❧❡♠ ❝❛♥ ❜❡ $✐♠♣❧② )❛❦❡♥ ❢♦/ ✐)$ $❡) ♦❢ ♣♦$✐)✐✈❡ ✐♥$)❛♥❝❡✳ ❚❤✐$ ♥♦)✐♦♥ ✐$ ❢♦/♠❛❧❧② ❞❡✜♥❡❞ ✐♥ ❉❡✜♥✐)✐♦♥ ✷✳✶✳ ❉❡✜♥✐%✐♦♥ ✷✳✶ ✭❉❡❝✐$✐♦♥ H/♦❜❧❡♠ ❬H❛♣✾✹❪✮✳ ❆ ❞❡❝✐$✐♦♥ ♣/♦❜❧❡♠ Π ✐, ❛ ,✉❜,❡" D+ Π ⊆ {0, 1}∗ _{♦❢ ♣♦,✐"✐✈❡ ✐♥,"❛♥❝❡,✳ ❚❤✉, "❤❡ ,❡" ♦❢ ♥❡❣❛"✐✈❡ ✐♥,"❛♥❝❡, ✐, ❞❡✜♥❡❞ ❛, D}− Π = {0, 1}∗\D+_Π✳ ■❢ ✇❡ ❝♦♥$✐❞❡/ )❤❡ ❢♦❧❧♦✇✐♥❣ ♣/♦❜❧❡♠✿ ❉❡❝✐+✐♦♥ ,-♦❜❧❡♠ ✶ ❍❛♠✐❧%♦♥✐❛♥ ❈②❝❧❡ ■♥♣✉% ❆ ❣/❛♣❤ G = (U, E)✳ ◗✉❡+%✐♦♥ ❉♦❡$ G ❛❞♠✐) ❛♥ ❍❛♠✐❧%♦♥✐❛♥ ❈②❝❧❡✱ ✐✳❡✳ ❛ ❝②❝❧❡ )❤❛) ✐♥❝❧✉❞❡$ ❡✈❡/② ✈❡/)❡① ♦❢ U ❡①❛❝)❧② ♦♥❝❡❄ ❍❛♠✐❧%♦♥✐❛♥ ❈②❝❧❡ ✐$ ❛ ❝❧❛$$✐❝❛❧ ❡①❛♠♣❧❡ ♦❢ ❞❡❝✐$✐♦♥ ♣/♦❜❧❡♠ ❛♥❞ ❡①❛♠♣❧❡$ ♦❢ ♣♦$✐)✐✈❡ ❛♥❞ ♥❡❣❛)✐✈❡ ✐♥$)❛♥❝❡$ ❛/❡ ❞❡♣✐❝)❡❞ ✐♥ ❋✐❣✉/❡ ✷✳✶ ●✐✈❡♥ ❛♥ ❛❧❣♦/✐)❤♠ A✱ ✇❡ ❝❛♥ ❞❡✜♥❡ $❡✈❡/❛❧ ♠❡)/✐❝$ ✐♥ ♦/❞❡/ )♦ ❞❡)❡/♠✐♥❡ ✐)$ ❝♦♠♣❧❡①✐"②✳ ❚✇♦ ♠❛✐♥ ❦✐♥❞$ ♦❢ ♠❡)/✐❝$ ❛/❡ ✇✐❞❡❧② ✉$❡❞✱ )❤❡ "✐♠❡ ❝♦♠♣❧❡①✐"② )❤❛) ❛✐♠$ ❛) ❣✐✈✐♥❣ ❛♥ ✐❞❡❛ ♦♥ ❤♦✇ ♠✉❝❤ )✐♠❡ ✇✐❧❧ ❜❡ ♥❡❡❞❡❞ ❜② A )♦ /❡❛❝❤ ✐)$ ❡♥❞ ❝♦♥❞✐)✐♦♥$✱ ❛♥❞ )❤❡ ,♣❛❝❡ ❝♦♠♣❧❡①✐"② ❣✐✈✐♥❣ $♦♠❡ ❝❧✉❡$ ♦♥ )❤❡ ❛♠♦✉♥) ♦❢ ♠❡♠♦/② /❡A✉✐/❡❞ ❜② A✳ ❚❤❡$❡ ♠❡)/✐❝$ ❛/❡ ♦❢)❡♥ ❡①♣/❡$$❡❞ ✐♥ ❢✉♥❝)✐♦♥ ♦❢ )❤❡ ,✐③❡ ♦❢ "❤❡ ✐♥,"❛♥❝❡✳ ◆♦%❛%✐♦♥ ✷✳✶✳ ●✐✈❡♥ Π ❛ ♣(♦❜❧❡♠ ❛♥❞ ❛ Π✲✐♥,"❛♥❝❡ I ∈ {0, 1}∗_{✱ |I| ❞❡♥♦"❡, "❤❡ ,✐③❡} ♦❢ "❤❡ ✐♥,"❛♥❝❡✳ ■" (❡♣(❡,❡♥", "❤❡ ❧❡♥❣"❤ ♦❢ "❤❡ ✇♦(❞ ♦♥ "❤❡ ❛❧♣❤❛❜❡" {0, 1} "❤❛" ❡♥❝♦❞❡, "❤❡ ✐♥,"❛♥❝❡✳

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✷✳✶✳ #❘❖❇▲❊▼❙✱ ❘❊❉❯❈❚■❖◆❙ ❆◆❉ ❈❖▼#▲❊❳■❚❨ ❈▲❆❙❙❊❙ ✷✸ 1 2 3 4 5 1 2 3 4 5 ❋✐❣✉&❡ ✷✳✶✿ ❖♥ -❤❡ ❧❡❢- ❛ ❣&❛♣❤ ❛❞♠✐--✐♥❣ ❛♥ ❍❛♠✐❧-♦♥✐❛♥ ❈②❝❧❡ ✭♣♦;✐-✐✈❡ ✐♥;-❛♥❝❡✮✱ ♦♥ -❤❡ &✐❣❤- ❛ ❣&❛♣❤ ✇✐-❤♦✉- ❍❛♠✐❧-♦♥✐❛♥ ❈②❝❧❡ ✭♥❡❣❛-✐✈❡ ✐♥;-❛♥❝❡✮✳

❚❤&♦✉❣❤♦✉- -❤✐; ♠❛♥✉;❝&✐♣-✱ ✇❡ ❢♦❝✉; ♦♥ -❤❡ -✐♠❡ ❝♦♠♣❧❡①✐-② ❛♥❞ ♠♦&❡ ♣&❡❝✐;❡❧② ♦♥ -❤❡ ✇♦"#$ ❝❛#❡ $✐♠❡ ❝♦♠♣❧❡①✐$②✳ ■♥-✉✐-✐✈❡❧②✱ -❤✐; ❝♦♠♣❧❡①✐-② ❣✐✈❡; ❛♥ ♦&❞❡& ♦❢ ♠❛❣♥✐✲ -✉❞❡ ♦❢ -❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡& ♦❢ ❛-♦♠✐❝ ♦♣❡&❛-✐♦♥; -❤❛- ❛♥ ❛❧❣♦&✐-❤♠ ♣❡&❢♦&♠; ❜❡❢♦&❡ &❡❛❝❤✐♥❣ ✐-; ❡♥❞ ❝♦♥❞✐-✐♦♥;✳ ❉❡✜♥✐%✐♦♥ ✷✳✷✳ ●✐✈❡♥ ❛ ❞❡❝✐#✐♦♥ ♣"♦❜❧❡♠ Π ❛♥❞ ❛♥ ❛❧❣♦"✐$❤♠ A✱ ✇❡ #❛② $❤❛$ A ❞❡❝✐❞❡# ♦" #♦❧✈❡# Π ✐♥ $✐♠❡ f(n) ✐❢ ❣✐✈❡♥ ❛♥② ✐♥#$❛♥❝❡ I ∈ {0, 1}∗_{✱ A ❞❡$❡"♠✐♥❡#} ✇❤❡$❤❡" I ∈ D+ Π ♦" ♥♦$ ✐♥ ❛$ ♠♦#$ f(n) ♦♣❡"❛$✐♦♥#✱ ✇✐$❤ n = |I|✳ ❚❤✐; ♦&❞❡& ♦❢ ♠❛❣♥✐-✉❞❡ ✐; ❝♦♠♠♦♥❧② ❡①♣&❡;;❡❞ ✐♥ ❢✉♥❝-✐♦♥ ♦❢ -❤❡ ;✐③❡ ♦❢ -❤❡ ❛❧❣♦✲ &✐-❤♠ ✐♥♣✉- ❛♥❞ ✉;✐♥❣ -❤❡ ❇✐❣✲❖✲◆♦-❛-✐♦♥✳ ✷✳✶✳✷ ❚❤❡ NP ❝♦♠♣❧❡①✐-② ❝❧❛00 ❉❡✜♥✐%✐♦♥ ✷✳✸ ✭P✮✳ ❚❤❡ ❝❧❛## P ✐# $❤❡ #❡$ ♦❢ ❛❧❧ ❞❡❝✐#✐♦♥ ♣"♦❜❧❡♠# $❤❛$ ❝❛♥ ❜❡ #♦❧✈❡❞ ✇✐$❤ ❛ ❞❡$❡"♠✐♥✐#$✐❝ ❛❧❣♦"✐$❤♠ "✉♥♥✐♥❣ ✐♥ ♣♦❧②♥♦♠✐❛❧ $✐♠❡✳

■♥ -❤❡ ❧✐❣❤- ♦❢ -❤❡ ❛❜♦✈❡✱ P ❝♦♥-❛✐♥; -❤❡ ❡❛#② ♣&♦❜❧❡♠;✳ ❍♦✇❡✈❡&✱ P ✐; ❛ &❡;-&✐❝-✐✈❡ ❝❧❛;; ;✐♥❝❡✱ ❢♦& ❛ ❣&❡❛- ♥✉♠❜❡& ♦❢ ♥❛-✉&❛❧ ♣&♦❜❧❡♠; ♥♦ ❡✣❝✐❡♥- ❛❧❣♦&✐-❤♠ ✐; ❦♥♦✇♥✳ ❋♦& ✐♥;-❛♥❝❡✱ ❍❛♠✐❧%♦♥✐❛♥ ❈②❝❧❡ ♣&❡✈✐♦✉;❧② ❞❡✜♥❡❞ ✐; ♥♦- ❦♥♦✇♥ -♦ ❜❡ ✐♥ P✳ ◆❡✈❡&-❤❡✲ ❧❡;;✱ ❣✐✈❡♥ G = (U, E)✱ ❛♥ ✐♥;-❛♥❝❡ ♦❢ ❍❛♠✐❧%♦♥✐❛♥ ❈②❝❧❡ ❛♥❞ ❛ ;❡- ♦❢ ❡❞❣❡; T ⊆ E✱ ❞❡-❡&♠✐♥✐♥❣ ✇❤❡-❤❡& T ❞❡✜♥❡; ❛♥ ❍❛♠✐❧-♦♥✐❛♥ ❝②❝❧❡ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ -✐♠❡✳ ❚❤✐; ♠♦-✐✈❛-❡; -❤❡ ✐♥-&♦❞✉❝-✐♦♥ ♦❢ -❤❡ ❝❧❛;; NP✳ ❉❡✜♥✐%✐♦♥ ✷✳✹ ✭NP ❬❲✐❧✵✷❪✮✳ ❚❤❡ ❝❧❛## NP ✐# ❞❡✜♥❡❞ ❛# $❤❡ #❡$ ♦❢ ❞❡❝✐#✐♦♥ ♣"♦❜❧❡♠# Π $❤❛$ ❛❞♠✐$ ❛ ♣♦❧②♥♦♠✐❛❧ ✈❡"✐✜❡" A✱ ✐✳❡✳ ❛♥ ❛❧❣♦"✐$❤♠ ✇✐$❤ $❤❡ ❢♦❧❧♦✇✐♥❣ ♣"♦♣❡"$✐❡#✿ ✶✳ A "✉♥# ✐♥ ♣♦❧②♥♦♠✐❛❧ $✐♠❡✱ ✷✳ ❣✐✈❡♥ ❛ ♣♦#✐$✐✈❡ ✐♥#$❛♥❝❡ I ∈ {0, 1}∗ _{♦❢ Π✱ $❤❡"❡ ❡①✐#$# y ∈ {0, 1}}∗ _{#✉❝❤ $❤❛$}

|y| = O(poly(|I|)) ❛♥❞ A ✇✐$❤ ✐♥♣✉$ (I, y) "❡$✉"♥# $"✉❡✱

✸✳ ❣✐✈❡♥ ❛ ♥❡❣❛$✐✈❡ ✐♥#$❛♥❝❡ I ∈ {0, 1}∗ _{♦❢ Π✱ ❢♦" ❡✈❡"② y ∈ {0, 1}}∗ _{#✉❝❤ $❤❛$ |y| =}

O(poly(|I|))✱ A ✇✐$❤ ✐♥♣✉$ (I, y) "❡$✉"♥# ❢❛❧#❡✳