Randomized Interpolation and Approximationof Sparse Polynomials
SIAM Journal on Computing
Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems
Journal of the ACM (JACM)
Random knapsack in expected polynomial time
Journal of Computer and System Sciences - Special issue: STOC 2003
Open problems around exact algorithms
Discrete Applied Mathematics
Smoothed analysis: an attempt to explain the behavior of algorithms in practice
Communications of the ACM - A View of Parallel Computing
On problems without polynomial kernels
Journal of Computer and System Sciences
Kernelization: New Upper and Lower Bound Techniques
Parameterized and Exact Computation
Saving space by algebraization
Proceedings of the forty-second ACM symposium on Theory of computing
On the Compressibility of $\mathcal{NP}$ Instances and Cryptographic Applications
SIAM Journal on Computing
Determinant Sums for Undirected Hamiltonicity
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Exact Exponential Algorithms
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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∈Xω(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.