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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. 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