scholarly journals Exact simulation of Gaussian boson sampling in polynomial space and exponential time

2020 ◽  
Vol 2 (2) ◽  
Author(s):  
Nicolás Quesada ◽  
Juan Miguel Arrazola
Keyword(s):  
Polynomial Space ◽  
Exact Simulation ◽  
Exponential Time ◽  
Boson Sampling

APPROXIMATING ASYMMETRIC TSP IN EXPONENTIAL TIME

2014 ◽  
Vol 25 (01) ◽  
pp. 89-99 ◽  
Author(s):  
ALEXANDER GOLOVNEV
Keyword(s):  
Traveling Salesman Problem ◽  
Directed Graph ◽  
Short Note ◽  
Simple Algorithm ◽  
Computable Function ◽  
Traveling Salesman ◽  
Polynomial Space ◽  
Exponential Time ◽  
Running Time ◽  
Polynomial Factor

Let G be a complete directed graph with n vertices and integer edge weights in range [0,M]. It is well known that an optimal Traveling Salesman Problem (TSP) in G can be solved in 2n time and space (all bounds are given within a polynomial factor of the input length, i.e., poly(n, log M)) and this is still the fastest known algorithm. If we allow a polynomial space only, then the best known algorithm has running time 4nnlog n. For TSP with bounded weights there is an algorithm with 1.657n · M running time. It is a big challenge to develop an algorithm with 2n time and polynomial space. Also, it is well-known that TSP cannot be approximated within any polynomial time computable function unless P=NP. In this short note we propose a very simple algorithm that, for any 0 < ε < 1, finds (1+ε)-approximation to asymmetric TSP in 2nε−1 time and ε−1 · poly(n, log M) space. Thereby, for any fixed ε, the algorithm needs 2n steps and polynomial space to compute (1 + ε)-approximation.

scholarly journals Parameterized Single-Exponential Time Polynomial Space Algorithm for Steiner Tree

2015 ◽  
pp. 494-505 ◽  
Author(s):  
Fedor V. Fomin ◽  
Petteri Kaski ◽  
Daniel Lokshtanov ◽  
Fahad Panolan ◽  
Saket Saurabh
Keyword(s):  
Steiner Tree ◽  
Polynomial Space ◽  
Exponential Time ◽  
Single Exponential

scholarly journals Finding and Counting Permutations via CSPs

Algorithmica ◽  
2021 ◽  
Author(s):  
Benjamin Aram Berendsohn ◽  
László Kozma ◽  
Dániel Marx
Keyword(s):  
Constraint Satisfaction ◽  
Polynomial Space ◽  
Permutation Patterns ◽  
Counting Problem ◽  
Exponential Time ◽  
Running Time ◽  
Algorithmic Question ◽  
Stack Sorting ◽  
Exponential Time Algorithms ◽  
New Algorithms

AbstractPermutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k. In this work we give two new algorithms for this well-studied problem, one whose running time is $$n^{k/4 + o(k)}$$ n k / 4 + o ( k ) , and a polynomial-space algorithm whose running time is the better of $$O(1.6181^n)$$ O ( 1 . 6181 n ) and $$O(n^{k/2 + 1})$$ O ( n k / 2 + 1 ) . These results improve the earlier best bounds of $$n^{0.47k + o(k)}$$ n 0.47 k + o ( k ) and $$O(1.79^n)$$ O ( 1 . 79 n ) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when $$k \in \varOmega (\log {n})$$ k ∈ Ω ( log n ) . We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction. Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time $$f(k) \cdot n^{o(k/\log {k})}$$ f ( k ) · n o ( k / log k ) would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing (4321-avoiding) and 3-decreasing (1234-avoiding) permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that a sub-exponential running time is unlikely with the current techniques, even for patterns from these restricted classes.

scholarly journals New Algorithms for Mixed Dominating Set

2021 ◽  
Vol vol. 23 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Louis Dublois ◽  
Michael Lampis ◽  
Vangelis Th. Paschos
Keyword(s):  
Lower Bound ◽  
Dominating Set ◽  
Exact Algorithm ◽  
Polynomial Space ◽  
Parameterized Algorithms ◽  
Exponential Time ◽  
Open Questions ◽  
Fpt Algorithm ◽  
Exponential Space ◽  
New Algorithms

A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle the problem's complexity parameterized by treewidth and pathwidth by giving an algorithm running in time $O^*(5^{tw})$ (improving the current best $O^*(6^{tw})$), as well as a lower bound showing that our algorithm cannot be improved under the Strong Exponential Time Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound of $O^*((2 - \varepsilon)^{pw})$). Furthermore, by using a simple but so far overlooked observation on the structure of minimal solutions, we obtain branching algorithms which improve both the best known FPT algorithm for this problem, from $O^*(4.172^k)$ to $O^*(3.510^k)$, and the best known exponential-time exact algorithm, from $O^*(2^n)$ and exponential space, to $O^*(1.912^n)$ and polynomial space. Comment: This paper has been accepted to IPEC 2020

scholarly journals Parameterized Single-Exponential Time Polynomial Space Algorithm for Steiner Tree

2019 ◽  
Vol 33 (1) ◽  
pp. 327-345 ◽  
Author(s):  
Fedor V. Fomin ◽  
Petteri Kaski ◽  
Daniel Lokshtanov ◽  
Fahad Panolan ◽  
Saket Saurabh
Keyword(s):  
Steiner Tree ◽  
Polynomial Space ◽  
Exponential Time ◽  
Single Exponential

The Structure and Photochromism of 2,4,4,6-Tetraphenyl-4H-thiopyran

1992 ◽  
Vol 57 (6) ◽  
pp. 1326-1334 ◽  
Author(s):  
Jaroslav Vojtěchovský ◽  
Jindřich Hašek ◽  
Stanislav Nešpůrek ◽  
Mojmír Adamec
Keyword(s):  
Solid State ◽  
Uv Light ◽  
Order Reaction ◽  
Boat Conformation ◽  
X Rays ◽  
Exponential Time ◽  
First Order ◽  
Double Plane ◽  
The Mean ◽  
Independent Molecules

2,4,4,6-Tetraphenyl-4H-thiopyran, C29H22S, orthorhombic, Pna21, a = 17.980(4), b = 6.956(2), c = 34.562(11) Å, V = 4323(2) Å3, Z = 8, Dx = 1.237 g cm-3, F(000) = 1696, λ(CuKα) = 1.54184 A, μ = 1.372 mm-2, T = 294 K. The final R was 0.050 for the unique set of 3103 observed reflections. The central 4H-thiopyran ring forms a boat conformation for both symmetrically independent molecules with average boat angles 4.4(3) and 6.8(3)° at S and C(sp3), respectively. The mean planes of phenyls at the position 2 and 6 are turned from the double plane of 4H-thiopyran by 42.5(5) and 35.8(3)°, respectively. The investigated material undergoes a photochromic change in the solid state after irradiation with UV light or X-rays. The maximum of the new absorption band is situated at 564 nm. The non-exponential time dependence of photochromic bleaching is analysed in terms of a dispersive first-order reaction.

A Combinatorial Problem Which Is Complete in Polynomial Space

1976 ◽  
Vol 23 (4) ◽  
pp. 710-719 ◽  
Author(s):  
S. Even ◽  
R. E. Tarjan
Keyword(s):  
Combinatorial Problem ◽  
Polynomial Space
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