A strongly polynomial minimum cost circulation algorithm
Combinatorica
SIAM Journal on Computing
A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
A faster strongly polynomial minimum cost flow algorithm
Operations Research
Polynomial-time approximation algorithms for the Ising model
SIAM Journal on Computing
The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
A combinatorial algorithm minimizing submodular functions in strongly polynomial time
Journal of Combinatorial Theory Series B
Markov Chain Algorithms for Planar Lattice Structures
SIAM Journal on Computing
Approximate counting by dynamic programming
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
Journal of the ACM (JACM)
Random Walks on Truncated Cubes and Sampling 0-1 Knapsack Solutions
SIAM Journal on Computing
Counting independent sets up to the tree threshold
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Simulated annealing in convex bodies and an O*(n4) volume algorithm
Journal of Computer and System Sciences - Special issue on FOCS 2003
Simple deterministic approximation algorithms for counting matchings
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Correlation decay and deterministic FPTAS for counting list-colorings of a graph
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Adaptive simulated annealing: A near-optimal connection between sampling and counting
Journal of the ACM (JACM)
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
Pseudorandom generators for polynomial threshold functions
Proceedings of the forty-second ACM symposium on Theory of computing
Computational Transition at the Uniqueness Threshold
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
A very simple algorithm for estimating the number of k‐colorings of a low‐degree graph
Random Structures & Algorithms
Improved inapproximability results for counting independent sets in the hard-core model
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
An FPTAS for #Knapsack and Related Counting Problems
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
A Deterministic Polynomial-Time Approximation Scheme for Counting Knapsack Solutions
SIAM Journal on Computing
A Deterministic Polynomial-Time Approximation Scheme for Counting Knapsack Solutions
SIAM Journal on Computing
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Given $n$ elements with nonnegative integer weights $w_1, \ldots, w_n$ and an integer capacity $C$, we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most the given capacity. We give a deterministic algorithm that estimates the number of solutions to within relative error $1\pm\varepsilon$ in time polynomial in $n$ and $1/\varepsilon$ (fully polynomial approximation scheme). More precisely, our algorithm takes time $O(n^3\varepsilon^{-1}\log(n/\varepsilon))$. Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes were known first by Morris and Sinclair via Markov chain Monte Carlo techniques and subsequently by Dyer via dynamic programming and rejection sampling.