The Problem

问题属于dynamics programming大类。
Given $n$ items { 1 , 2 , . . . , n } \{1,2,...,n\} {1,2,...,n}, and each has a given nonnegative weight $w_i$ and value $v_i$. We are also given a bound $W$.
Goal: select a subset $S$ of items, which maximizes ${\sum }_{i\in S}{v}_i$ subject to ${\sum }_{i\in S}{w}_i\le W$.

Designing the Algorithm

对于动态规划问题，一般从最后一种情况入手。
在添加最后一个element的时候，不光需要知道 OPT(n-1)，也需要知道$W-w_n$，即添加最后一个element时的bound是多少。

Assume that $W$ is an integer, and all requests $i=1,...,n$ have integer weights $w_i$. We will have a subproblem for each$ i=0,1,…,n$ and each integer $0\le w\le W$. We will use $OPT(i,w)$ to denote the value of the optimal solution using a subset of the items { 1 , . . . , i } \{1, . . . , i\} {1,...,i} with maximum allowed weight $w$, that is,
$$OPT(i,w)=\underset{S}{\max }\sum _{j\in S}{v}_j$$
where S ⊆ { 1 , . . . , i } S\subseteq \{1,...,i\} S⊆{1,...,i}

定理（6.8）：if $w<w_i$, then $OPT(i,w)=OPT(i-1,w)$. Otherwise $$OPT(i,w)=\max (OPT(i-1,w),v_i+OPT(i-1,w-w_i))$$

Analyzing the Algorithm

定理（6.9）：The Subset-Sum($n,W$) Algorithm correctly computes the optimal value of the problem, and runs in $O(nW)$ time.

定理（6.10）： Given a table $M$ of the optimal values of the subproblems, the optimal set $S$ can be found in $O(n)$ time.

Reference

Algorithm Design (Jon Kleinberg)