An exact tensor network for the 3SAT problem

We construct a tensor network that delivers an unnormalized quantum state whose coefficients are the solutions to a given instance of 3SAT, an NP-complete problem. The tensor network contraction that corresponds to the norm of the state counts the number of solutions to the instance. It follows that exact contractions of this tensor network are in the \#P-complete computational complexity class, thus believed to be a hard task. Furthermore, we show that for a 3SAT instance with $n$ bits, it is enough to perform a polynomial number of contractions of the tensor network structure associated to the computation of local observables to obtain one of the explicit solutions to the problem, if any. Physical realization of a state described by a generic tensor network is equivalent to finding the satisfying assignment of a 3SAT instance and, consequently, this experimental task is expected to be hard.

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AN EFFICIENT METHOD TO DETERMINE THE INTENDED SOLUTION FOR A SYSTEM OF GEOMETRIC CONSTRAINTS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195905001701 ◽

2005 ◽

Vol 15 (03) ◽

pp. 279-298 ◽

Cited By ~ 12

Author(s):

HILDERICK A. VAN DER MEIDEN ◽

WILLEM F. BRONSVOORT

Keyword(s):

Geometric Constraints ◽

Number Of Solutions ◽

Selection Scheme ◽

Selection Rules ◽

Backtracking Search ◽

Complete Problem ◽

Single Solution ◽

Constraint Problem ◽

Np Complete ◽

Geometric Elements

The number of solutions of a geometric constraint problem is generally exponential to the number of geometric elements in the problem. Finding a single intended solution, satisfying additional criteria, typically results in an NP-complete problem. A prototype-based selection scheme is presented here that avoids this problem. First, a resemblance relation between configurations is formally defined. This relation should be satisfied between the intended solution and a prototype configuration. The resemblance relation is in our approach satisfied by applying selection rules to subproblems in a bottom-up solving approach. The resulting solving algorithm is polynomial, because the selection rules are not used as search heuristic, but to unambiguously select a single solution such that no backtracking search is needed. For many applications, in particular CAD, this solution is both meaningful and intuitive.

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A Multilevel Memetic Algorithm for Large SAT-Encoded Problems

Evolutionary Computation ◽

10.1162/evco_a_00078 ◽

2012 ◽

Vol 20 (4) ◽

pp. 641-664 ◽

Cited By ~ 11

Author(s):

Noureddine Bouhmala

Keyword(s):

Original Problem ◽

Memetic Algorithm ◽

Satisfying Assignment ◽

Boolean Expression ◽

Complete Problem ◽

Starting Solution ◽

Previous Level ◽

Np Complete ◽

Problem Instances ◽

Hardware Designs

Many researchers have focused on the satisfiability problem and on many of its variants due to its applicability in many areas of artificial intelligence. This NP-complete problem refers to the task of finding a satisfying assignment that makes a Boolean expression evaluate to True. In this work, we introduce a memetic algorithm that makes use of the multilevel paradigm. The multilevel paradigm refers to the process of dividing large and difficult problems into smaller ones, which are hopefully much easier to solve, and then work backward toward the solution of the original problem, using a solution from a previous level as a starting solution at the next level. Results comparing the memetic with and without the multilevel paradigm are presented using problem instances drawn from real industrial hardware designs.

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On the Computational Complexity of Gossip Protocols

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2017/106 ◽

2017 ◽

Cited By ~ 4

Author(s):

Krzysztof R. Apt ◽

Eryk Kopczyński ◽

Dominik Wojtczak

Keyword(s):

Computational Complexity ◽

Private Information ◽

Epistemic Logic ◽

Distributed Programs ◽

Distributed Protocols ◽

Partial Correctness ◽

Complete Problem ◽

Gossip Protocol ◽

Np Complete ◽

Gossip Protocols

Gossip protocols deal with a group of communicating agents, each holding a private information, and aim at arriving at a situation in which all the agents know each other secrets. Distributed epistemic gossip protocols are particularly simple distributed programs that use formulas from an epistemic logic. Recently, the implementability of these distributed protocols was established (which means that the evaluation of these formulas is decidable), and the problems of their partial correctness and termination were shown to be decidable, but their exact computational complexity was left open. We show that for any monotonic type of calls the implementability of a distributed epistemic gossip protocol is a P^{NP}_{||}-complete problem, while the problems of its partial correctness and termination are in coNP^{NP}.

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P Systems with Evolutional Communication and Division Rules

Axioms ◽

10.3390/axioms10040327 ◽

2021 ◽

Vol 10 (4) ◽

pp. 327

Author(s):

David Orellana-Martín ◽

Luis Valencia-Cabrera ◽

Mario J. Pérez-Jiménez

Keyword(s):

Computational Complexity ◽

Complexity Theory ◽

Membrane Computing ◽

P Systems ◽

Membrane Systems ◽

Sat Problem ◽

Complete Problem ◽

Intractable Problems ◽

Np Complete ◽

The One

A widely studied field in the framework of membrane computing is computational complexity theory. While some types of P systems are only capable of efficiently solving problems from the class P, adding one or more syntactic or semantic ingredients to these membrane systems can give them the ability to efficiently solve presumably intractable problems. These ingredients are called to form a frontier of efficiency, in the sense that passing from the first type of P systems to the second type leads to passing from non-efficiency to the presumed efficiency. In this work, a solution to the SAT problem, a well-known NP-complete problem, is obtained by means of a family of recognizer P systems with evolutional symport/antiport rules of length at most (2,1) and division rules where the environment plays a passive role; that is, P systems from CDEC^(2,1). This result is comparable to the one obtained in the tissue-like counterpart, and gives a glance of a parallelism and the non-evolutionary membrane systems with symport/antiport rules.

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Atomic distributions in crystal structures solved by Boolean satisfiability techniques

Zeitschrift für Kristallographie - Crystalline Materials ◽

10.1515/zkri-2015-1887 ◽

2016 ◽

Vol 231 (2) ◽

Cited By ~ 1

Author(s):

Mathias Soeken ◽

Rolf Drechsler ◽

Reinhard X. Fischer

Keyword(s):

Crystal Structures ◽

Unsolved Problem ◽

Boolean Satisfiability ◽

Satisfying Assignment ◽

Interatomic Distances ◽

Complete Problem ◽

Atomic Distribution ◽

Sat Solver ◽

Np Complete ◽

Close Contacts

AbstractThe atomic distribution in crystal structures becomes very complex if atoms are disordered and randomly distributed over positions not being fully occupied. Interatomic distances between neighboring atoms might be too close for simultaneous occupancies and thus are mutually exclusive. The distribution of atoms over crystallographic positions avoiding close contacts with neighboring atoms represents an NP-complete problem that is believed to have no efficient solution. Here, we use Boolean satisfiability (SAT) techniques to find a valid atomic distribution pattern in the crystal structure. Distance constraints are encoded as conjunctions of logical disjunctions over Boolean variables and handed to a SAT solver. If a solution exists, the solver supplies a satisfying assignment to the Boolean variables yielding a valid distribution after decoding. That way the hitherto unsolved problem of distributing

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Relating Vertex and Global Graph Entropy in Randomly Generated Graphs

10.20944/preprints201805.0302.v1 ◽

2018 ◽

Author(s):

Philip Tee ◽

George Parisis ◽

Luc Berthouze ◽

Ian Wakeman

Keyword(s):

Computational Complexity ◽

Recent Work ◽

Random Graphs ◽

Real World ◽

Close Correlation ◽

Complete Problem ◽

Entropy Measures ◽

Vertex Set ◽

Np Complete ◽

The Greedy Algorithm

Combinatoric measures of entropy capture the complexity of a graph, but rely upon the calculation of its independent sets, or collections of non-adjacent vertices. This decomposition of the vertex set is a known NP-Complete problem and for most real world graphs is an inaccessible calculation. Recent work by Dehmer et al. and Tee et al. identified a number of alternative vertex level measures of entropy that do not suffer from this pathological computational complexity. It can be demonstrated that they are still effective at quantifying graph complexity. It is intriguing to consider whether there is a fundamental link between local and global entropy measures. In this paper, we investigate the existence of correlation between vertex level and global measures of entropy, for a narrow subset of random graphs. We use the greedy algorithm approximation for calculating the chromatic information and therefore Körner entropy. We are able to demonstrate close correlation for this subset of graphs and outline how this may arise theoretically.

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Popular Matching in Roommates Setting Is NP-hard

ACM Transactions on Computation Theory ◽

10.1145/3442354 ◽

2021 ◽

Vol 13 (2) ◽

pp. 1-20

Author(s):

Sushmita Gupta ◽

Pranabendu Misra ◽

Saket Saurabh ◽

Meirav Zehavi

Keyword(s):

Computational Complexity ◽

Np Hard ◽

Np Complete ◽

Open Question ◽

Popular Matching

An input to the P OPULAR M ATCHING problem, in the roommates setting (as opposed to the marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors in strict order, known as its preference. In the P OPULAR M ATCHING problem the objective is to test whether there exists a matching M * such that there is no matching M where more vertices prefer their matched status in M (in terms of their preferences) over their matched status in M *. In this article, we settle the computational complexity of the P OPULAR M ATCHING problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.

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