Hh/hw1

hw1/HuanhaiZhou/hw1.typ

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+#show smallcaps: set text(font: "linux libertine")
+
+#let bA = $bold(A)$
+#let bb = $bold(b)$
+#let bc = $bold(c)$
+#let bs = $bold(s)$
+#let bx = $bold(x)$
+
+#let setmin = $#smallcaps[Setmin]$
+#let knapsack = $#smallcaps[Knapsack]$
+#let approximateknapsack = $#smallcaps[Approximate-Knapsack]$
+
+== Problem 9.6
+Let $setmin(A,B)$ be a function with two input sets $A$ and $B$ and output $
+setmin(A,B) = cases(
+  A quad "if" abs(A) <= abs(B) "and" A != emptyset,
+  B quad "otherwise"
+)
+$
+where $abs(dot)$ denotes the cardinality of a set.
+
+We could write down the following state transition equations:
+$
+S_(i, v) = cases(S_(i-1, v) &quad "if" v < v_i,
+{i} &quad "if" v = v_i,
+setmin(S_(i-1, v), S_(i-1, v-v_i) union {i}) &quad "if" v > v_i
+) 
+$
+with initial conditions $
+S_(1, v) = cases({1} &quad "if" v = v_1,
+emptyset &quad "otherwise"
+)
+$
+
+There are $l times l v_max = l^2 v_max$ such $S_(i,v)$ in all and for each one we need constant overhead to calulcate. So the overall time complexity is $O(l^2 v_max)$.
+
+One need $log(v_max)$ bits to record the input data, hence the complexity is exponential in the input scale.
+
+== Problem 9.7
+Let $v_i^prime = floor(v_i\/K)$ be the scaled and truncated value. Note that $K omega_i <= v_i < (K+1) omega_i$, which means that picking up an object with value $v_i$ introduces error $epsilon < K$ in the overall value.
+
+Let $bold(sigma)^star in{0,1}^l$ be the optimal solution to the original $knapsack$ and $v_"opt"=sum sigma_i v_i$ be the associated max value.
+
+We here construct an solution to $approximateknapsack$ and require it to be an $(1-epsilon)$-approximation to $v_"opt"$.
+
+Let $abs(bold(sigma)^star)$ be the number of $1$'s in $bold(sigma)$ and $v_"trunc" equiv max_(w_i <= W) v_i. $
+
+$
+"total error" <= abs(sigma^star)K <= l K <= epsilon v_min <= epsilon v_"opt"
+$
+
+For the third inequality to hold, one must have $
+  K <= (epsilon v_"trunc")/l
+$ and we choose $
+  K = (epsilon v_"trunc")/l
+$ where $v_"trunc" equiv max_(w_i <= W) v_i$.
+
+== Problem 9.16
+
+To the linear program $
+max_bx (bc^T bx) quad "subject to" quad bA bx=bb "and" bx >= bold(0),
+$
+we associate an auxillary one $
+min_(bx,bs) sum s_i quad "subject to" quad bA bx+bs=bb, bx>=bold(0) "and" bs>= bold(0)
+$
+
+Choose $bx=bold(0), bs=bb$ as the initial solution and try solving the auxillary program.
+
+If the original program has a feasible solution that $bA bx=bb$ and $bx >= bold(0)$, the auxillary one should reach an optimal solution where $bs=bold(0), bA bx=bb$ and $bx >= bold(0)$. That is exactly an feasible solution of the original program.