k“ dominating set D ”>rF H H &ñ ë H]j\ ¦ ˜Ð . G\ ñ\ ¦ '' #Œ

1. G= (V, E)\ |D|6k“dominating set D”>rFHH &ñëH]j\¦ ˜Ð.G\  ñ\¦ '‘ #Œ ¢-a„ ñ ÕªAáԖРëߖ[þt."é¶A  ñ\H 1, Dh–Ð '‘)a  ñ\H 2_ q6 x`¦ ïr.Õªo“¦ s

ÕªAáÔ\"f &ñ_)a BjàÔaË: k-G'p' ëH]j\(2−)-HKZO`¦ &h6 xôÇ.

² ú

šs “\V”“ ëH]jH 3lq&h†<Êú þj@/ (2−) “ K\¦ HKZOs ½¨½+É כs“¦, “mš¸”“ ëH ]

jH 3lq&h†<Êú þj™è 2“ K\¦ ½¨½+É כs.

2. (1)éߖ> 1.

Using Approximate binary search with=O(1)

can find L6OP T 6U s.t U \L=O(1) inO(log (U₀\L₀)× poly(n, < d_max>)) time.

é ߖ> 2.

Once U\L=O(1) then -approximation in poly(n, < d_max>,¹) time :

Delete x_j with c_j > U, scale ˆc_j = b_L\nc^cj , or ˇc_j = d_L\ne^cj , and apply the poly(n, c_max, d_max) algorithm.

Then,poly(n,_L\n,d^cj

max) ”=” poly(n, d_max,¹)

For converse, suffices to set= _O(nc¹

max).

(2) Consider gknap(max,6), max{c^Tx : x ∈ Q, d^Tx 6 B, x ∈ {0,1}ⁿ.}, equivalent to find maximum γ 6 nc_max s.t. B > min{d^Tx : x ∈ Q, c^Tx > γ, x ∈ {0,1}ⁿ.} which is solvable in poly(n, < dmax >, cmax, γ) ”=” poly(n, cmax, < dmax >). so gknap, fully polynomially approx- imable.

The converse proved similarly.

3. (1) K\"f ]X c_ ×æu °úכ`¦ Wc“¦ €, c_ o'XO_ >hú\¦ k“¦ € E[Wc] = (1− ¹

2^k)w_cH†d`¦ ~1> ·ú˜Ãº e”. (1− ¹

2^k)> ¹₂sÙ¼–Ð „^‰ ×æu_ l@/°úכ“Ér E[W] =X

c∈C

E[W_c]> 1 2

X

c∈C

w_c > 1 2OPT.

—

¸ŽH ]X\"f k>2s€ E[W]> ³₄OPT H†d`¦ ·ú˜ ú e”.

(2)

E[W|x₁ =a1, . . . , xi =ai]

=E[W|x₁ =a₁, . . . , x_i=a_i, x_i+1 = True]·¹₂ +E[W|x₁=a₁, . . . , x_i =a_i, x_i+1= False]·¹₂

1

"f,E[W|x₁=a1, . . . , xi =ai, xi+1= True],¢¸H E[W|x₁ =a1, . . . , xi =ai, xi+1= False]H E[W|x₁ =a₁, . . . , x_i =a_i]˜Ð &ôÇ. "f sQôÇ ›'a>\¦ x_i[þt_ e”_–Ð &ñôÇ íH"f\

('‘ íH"f“¦ &ñ) &h6 x€ —¸ŽH xi[þts ”o°úכ`¦ °ú> ÷&H íHçߖ,E[W]ü< °ú  H 3lq&h†<Êú\¦ ” KìøÍ×¼r"f &ñ&hܼ–Ð ½¨Kt> )a.

—

¸ŽH i\ @/K l@/°úכ E[W|x₁ = a₁, . . . , x_i = a_i]\¦ †½Órçߖ\ >íߖ½+É Ãº e”€, sH †½Ó r

çߖ ·ú˜“¦o1pus)a.”o°úכs1lqwn&hܼ–Ð ÅÒ#QtHâĺ sH†½Órçߖ\ >íߖ½+Éúe”.

4. (1) e”__ e ∈ U\¦ ÒqtyŒ•. e\¦ Ÿí†<ʍH |9½+˓Ér ´ú§ f>h, "f þj™è ôÇ>h_ |9

½

+Ë SH fractional cover_ K\"f x_S > _f¹ ÷&ôÇ. "f, 0A_ ·ú˜“¦o1pus ‚×þ˜ôÇ |9½+Ë S_ |9½+Ë CH0pxK ÷&#Q ôÇ.

Õ

ªo“¦ x_S > _f¹“ âĺ\ëߖ 1s ÷&Ù¼–Ð, fractional solution[þt“Ér fC\¦ œíõ#Œ 7£xt ·ú§H

. "f, ئ§4)a &ñúK_ 3lq&h†<Êú °úכ“Ér fractional cover_ 3lq&h†<Êú °úכ OPT_p_ fC\¦ Å t

·ú§H.

(2) ·ú˜“¦o1pus =åQ€ &!Q÷&t ·ú§€Œ¤ ×æ4Ÿ¤&!Q)a "鶙è \O. ÕªQÙ¼–Ð "é¶, Š©œ@/ ëH]j _

KH —¸¿º 0pxKs.α= 1õ β =f{9M:_ ¢-ao)a©œ˜Ð#ŒÄ»›¸|`¦ëߖ7á¤l M:ëH\ 6£§

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