Exact Algorithms and Time/Space Tradeoffs

Years and Authors of Summarized Original Work

• 2010; Lokshtanov, Nederlof

Problem Definition

In the subset sum problem, we are given integers a₁, …, a _n , t and are asked to find a subset $X \subseteq \{ 1,\ldots ,n\}$ such that $\sum _{i\in X}a_{i} = t$. In the Knapsack problem, we are given a₁, …, a _n , b₁, …, b _n , t, u and are asked to find a subset $X \subseteq \{ 1,\ldots ,n\}$ such that $\sum _{i\in X}a_{i} \leq t$ and $\sum _{i\in X}b_{i} \geq u$. It is well known that both problems can be solved in O(nt) time using dynamic programming. However, as is typical for dynamic programming, these algorithms require a lot of working memory and are relatively hard to execute in parallel on several processors: the above algorithms use O(t) space which may be exponential in the input size.

This raises the question: when can we avoid these disadvantages and still be (approximately) as fast as dynamic programming algorithms? It appears that by (slightly) loosening the time budget,...