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

ω

:

U

 → {1,…,

N

}, and integers 0 ≤ 

l

 ≤ 

u

 ≤ ∞. Then the problem is to decide whether there is an

$X \in \mathcal{F}$

such that

l

 ≤ ∑ 

e

 ∈ 

X

ω

(

e

) ≤ 

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

and

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.