arXiv subjects

Tian Liu

Publications and source records attributed to Tian Liu.

96 records · Page 6Linked to original sources

arXiv2008

Exact phase transition of backtrack-free search with implications on the power of greedy algorithms

Backtracking is a basic strategy to solve constraint satisfaction problems (CSPs). A satisfiable CSP instance is backtrack-free if a solution can be found without encountering any dead-end during a backtracking search, implying that the instance is easy to solve. We prove an exact phase transition of backtrack-free search in some random CSPs, namely in Model RB and in Model RD. This is the first time an exact phase transition of backtrack-free search can be identified on some random CSPs. Our technical results also have interesting implications on the power of greedy algorithms, on the width of random hypergraphs and on the exact satisfiability threshold of random CSPs.

cs.AI↗

arXiv2008

Quantum random number generator based on the photon number decision of weak laser pulses

We propose an approach to realize a quantum random number generator (QRNG) based on the photon number decision of weak laser pulses. This type of QRNG can generate true random numbers at a high speed and can be adjusted to zero bias conveniently, thus is suitable for the applications in quantum cryptography.

quant-ph↗

arXiv2008

From k-SAT to k-CSP: Two Generalized Algorithms

Constraint satisfaction problems (CSPs) models many important intractable NP-hard problems such as propositional satisfiability problem (SAT). Algorithms with non-trivial upper bounds on running time for restricted SAT with bounded clause length k (k-SAT) can be classified into three styles: DPLL-like, PPSZ-like and Local Search, with local search algorithms having already been generalized to CSP with bounded constraint arity k (k-CSP). We generalize a DPLL-like algorithm in its simplest form and a PPSZ-like algorithm from k-SAT to k-CSP. As far as we know, this is the first attempt to use PPSZ-like strategy to solve k-CSP, and before little work has been focused on the DPLL-like or PPSZ-like strategies for k-CSP.

cs.DS↗

arXiv2007

On Exponential Time Lower Bound of Knapsack under Backtracking

M.Aleknovich et al. have recently proposed a model of algorithms, called BT model, which generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model by Woeginger. BT model can be further divided into three kinds of fixed, adaptive and fully adaptive ones. They have proved exponential time lower bounds of exact and approximation algorithms under adaptive BT model for Knapsack problem. Their exact lower bound is $Ω(2^{0.5n}/\sqrt{n})$, in this paper, we slightly improve the exact lower bound to about $Ω(2^{0.69n}/\sqrt{n})$, by the same technique, with related parameters optimized.

cs.CC↗

arXiv2006

Improved Exponential Time Lower Bound of Knapsack Problem under BT model

M.Alekhnovich et al. recently have proposed a model of algorithms, called BT model, which covers Greedy, Backtrack and Simple Dynamic Programming methods and can be further divided into fixed, adaptive and fully adaptive three kinds, and have proved exponential time lower bounds of exact and approximation algorithms under adaptive BT model for Knapsack problem which are $Ω(2^{n/2}/\sqrt n)=Ω(2^{0.5n}/\sqrt n)$ and $Ω((1/ε)^{1/3.17})\approxΩ((1/ε)^{0.315})$(for approximation ratio $1-ε$) respectively (M. Alekhovich, A. Borodin, J. Buresh-Oppenheim, R. Impagliazzo, A. Magen, and T. Pitassi, Toward a Model for Backtracking and Dynamic Programming, \emph{Proceedings of Twentieth Annual IEEE Conference on Computational Complexity}, pp308-322, 2005). In this note, we slightly improved their lower bounds to $Ω(2^{(2-ε)n/3}/\sqrt{n})\approx Ω(2^{0.66n}/\sqrt{n})$ and $Ω((1/ε)^{1/2.38})\approxΩ((1/ε)^{0.420})$, and proposed as an open question what is the best achievable lower bounds for knapsack under adaptive BT models.

cs.CC↗

arXiv2006

On Lower Bound of Worst Case Error Probability for Quantum Fingerprinting with Shared Entanglement

This paper discusses properties of quantum fingerprinting with shared entanglement. Under certain restriction of final measurement, a relation is given between unitary operations of two parties. Then, by reducing to spherical coding problem, this paper gives a lower bound of worst case error probability for quantum fingerprinting with shared entanglement, showing a relation between worst case error probability and the amount of entanglement(measured by Schmidt number).

quant-ph↗