Reducing a Target Interval to a Few Exact Queries

Many combinatorial problems involving weights can be formulated as a so-called ranged problem. That is, their input consists of a universe $U$, a (succinctly-represented) set family $\mathcal{F} \subseteq 2^{U}$, a weight function $\omega:U \rightarrow {1,…,N}$, and integers $0 \leq l \leq u \leq \infty$. Then the problem is to decide whether there is an $X \in \mathcal{F}$ such that $l \leq \sum_{e \in X}\omega(e) \leq u$. Well-known examples of such problems include Knapsack, Subset Sum, Maximum Matching, and Traveling Salesman. In this paper, we develop a generic method to transform a ranged problem into an exact problem (i.e. a ranged problem for which $l=u$). We show that our method has several intriguing applications in exact exponential algorithms and parameterized complexity, namely: - In exact exponential algorithms, we present new insight into whether Subset Sum and Knapsack have efficient algorithms in both time and space. In particular, we show that the time and space complexity of Subset Sum and Knapsack are equivalent up to a small polynomial factor in the input size. We also give an algorithm that solves sparse instances of Knapsack efficiently in terms of space and time. - In parameterized complexity, we present the first kernelization results on weighted variants of several well-known problems. In particular, we show that weighted variants of Vertex Cover, Dominating Set, Traveling Salesman and Knapsack all admit polynomial randomized Turing kernels when parameterized by $|U|$. Curiously, our method relies on a technique more commonly found in approximation algorithms.

💡 Research Summary

The paper addresses a broad class of combinatorial problems that can be expressed as “ranged” weight‑constrained decision problems: given a universe U, a succinctly described family 𝔽 ⊆ 2^U, a weight function ω:U→{1,…,N}, and integers 0 ≤ l ≤ u ≤ ∞, decide whether there exists X∈𝔽 with l ≤ ∑_{e∈X}ω(e) ≤ u. Classic examples include Subset Sum, Knapsack, Maximum Matching, and the Traveling Salesman Problem (TSP). The authors develop a generic reduction that transforms any such ranged problem into a collection of “exact” problems, i.e., instances where the lower and upper bounds coincide (l = u).

The reduction works by repeatedly scaling and rounding the weight function and by partitioning the interval