# Homomorphic Hashing for Sparse Coefficient Extraction

@article{Kaski2012HomomorphicHF, title={Homomorphic Hashing for Sparse Coefficient Extraction}, author={Petteri Kaski and Mikko Koivisto and Jesper Nederlof}, journal={ArXiv}, year={2012}, volume={abs/1203.4063} }

We study classes of Dynamic Programming (DP) algorithms which, due to their algebraic definitions, are closely related to coefficient extraction methods. DP algorithms can easily be modified to exploit sparseness in the DP table through memorization. Coefficient extraction techniques on the other hand are both space-efficient and parallelisable, but no tools have been available to exploit sparseness. We investigate the systematic use of homomorphic hash functions to combine the best of these…

## 12 Citations

### Space-Time Tradeoffs for Subset Sum: An Improved Worst Case Algorithm

- Computer Science, MathematicsICALP
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The strategy for dealing with arbitrary instances is to instead inject the randomness into the dissection process itself by working over a carefully selected but random composite modulus, and to introduce explicit space---time controls into the algorithm by means of a "bailout mechanism".

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We present randomized algorithms that solve Subset Sum and Knapsack instances with n items in O*(20.86n) time, where the O*(·) notation suppresses factors polynomial in the input size, and polynomial…

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The concept of kernelization, developed within the field of parameterized complexity theory, is used to give a mathematical analysis of the power of data reduction for dealing with fundamental NP-hard graph problems and it is proved that Treewidth and Pathwidth do not admit polynomial kernels parameterized by the vertex-deletion distance to a clique, unless thePolynomial hierarchy collapses.

### Faster Space-Efficient Algorithms for Subset Sum, k-Sum and Related Problems

- Computer Science, MathematicsSIAM J. Comput.
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We present randomized algorithms that solve subset sum and knapsack instances with $n$ items in $O^*(2^{0.86n})$ time, where the $O^*(\cdot)$ notation suppresses factors polynomial in the input size,…

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A Monte Carlo algorithm is given that determines the existence of a set cover of size $\sigma n$ in $O^*(2^{(1-\Omega(\sigma^4))n})$ time and outputs NO if $\chi(G) > s$ and YES with constant probability if $\ chi(G)\leq s-1$.

### Model Counting for CNF Formulas of Bounded Modular Treewidth

- Mathematics, Computer ScienceAlgorithmica
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It is shown that the number of satisfying assignments can be computed in polynomial time for CNF formulas whose incidence graphs have bounded modular treewidth, and the first one to harness this technique for #SAT.

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A characterization in terms of the popular density parameter $n/\log_2 t$: if all instances of density at least $1.003$ admit a truly faster algorithm, then so does every instance, which goes against the current intuition that instances ofdensity 1 are the hardest, and therefore is a step toward answering the open question in the affirmative.

### Solving #SAT and MAXSAT by Dynamic Programming

- Computer Science, MathematicsJ. Artif. Intell. Res.
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These algorithms extend all previous results for MaxSAT and #SAT achieved by dynamic programming along structural decompositions of the incidence graph of the input formula, as a proof of concept that warrants further research.

### Solving MaxSAT and #SAT on Structured CNF Formulas

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A structural parameter of CNF formulas is proposed and used to identify instances of weighted MaxSAT and #SAT that can be solved in polynomial time.

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A generic method to transform a ranged problem into an exact problem (i.e. A ranged problem for which l?=?u) is developed and has several intriguing applications in exact exponential algorithms and parameterized complexity.

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