scholarly journals Application of incremental satisfiability problem solvers for non-deterministic polynomial-time hard problems as illustrated by minimal Boolean formula synthesis problem

2020 ◽  
Vol 20 (6) ◽  
pp. 841-847
Author(s):  
K.I. Chukharev
Keyword(s):  
Polynomial Time ◽  
Synthesis Problem ◽  
Satisfiability Problem ◽  
Boolean Formula ◽  
Hard Problems ◽  
Problem Solvers

scholarly journals Are Stable Instances Easy?

2012 ◽  
Vol 21 (5) ◽  
pp. 643-660 ◽  
Author(s):  
YONATAN BILU ◽  
NATHAN LINIAL
Keyword(s):  
Polynomial Time ◽  
Discrete Optimization ◽  
Optimization Problem ◽  
Discrete Optimization Problem ◽  
Hard Problem ◽  
Np Hard ◽  
Max Cut Problem ◽  
Np Hard Problem ◽  
Hard Problems ◽  
Np Hard Problems

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP-hard problems are easier to solve, and in particular, whether there exist algorithms that solve in polynomial time all sufficiently stable instances of some NP-hard problem. The paper focuses on the Max-Cut problem, for which we show that this is indeed the case.

TYPE INFERENCE FOR FIRST-CLASS MESSAGES WITH FEATURE CONSTRAINTS

2000 ◽  
Vol 11 (01) ◽  
pp. 29-63
Author(s):  
MARTIN MÜLLER ◽  
SUSUMU NISHIMURA
Keyword(s):  
Polynomial Time ◽  
Type System ◽  
Type Inference ◽  
Satisfiability Problem ◽  
Recent Proposal ◽  
Inference Problem ◽  
Np Complete ◽  
Feature Constraints ◽  
Feature Trees ◽  
Selection Constraint

We present a constraint system, OF, of feature trees that is appropriate to specify and implement type inference for first-class messages. OF extends traditional systems of feature constraints by a selection constraint x <y> z, "by first-class feature tree" y, which is in contrast to the standard selection constraint x[f]y, "by fixed feature" f. We investigate the satisfiability problem of OF and show that it can be solved in polynomial time, and even in quadratic time if the number of features is bounded. We compare OF with Treinen's system EF of feature constraints with first-class features, which has an NP-complete satisfiability problem. This comparison yields that the satisfiability problem for OF with negation is NP-hard. Further we obtain NP-completeness, for a specific subclass of OF with negation that is useful for a related type inference problem. Based on OF we give a simple account of type inference for first-class messages in the spirit of Nishimura's recent proposal, and we show that it has polynomial time complexity: We also highlight an immediate extension of this type system that appears to be desirable but makes type inference NP-complete.

FROM XSAT TO SAT BY EXHIBITING BOOLEAN FUNCTIONS

2009 ◽  
Vol 18 (05) ◽  
pp. 783-799
Author(s):  
RICHARD OSTROWSKI ◽  
LIONEL PARIS
Keyword(s):  
Graph Coloring ◽  
Boolean Functions ◽  
Structural Information ◽  
Conjunctive Normal Form ◽  
Satisfiability Problem ◽  
Boolean Formula ◽  
Truth Assignment ◽  
Coloring Problem ◽  
Graph Coloring Problem ◽  
Speed Up

Given a Boolean formula in conjunctive normal form (CNF), the Exact Satisfiability problem (XSAT), a variant of the Satisfiability problem (SAT), consists in finding an assignment to the variables such that each clause contains exactly one satisfied literal. Best algorithms to solve this problem run in [Formula: see text] ([Formula: see text] for X3SAT). Another possibility is to transform each clause in a set of equivalent clauses for the Satisfiability problem and to use modern and powerful solvers (zChaff, Berkmin, MiniSat, RSat etc.) to find such truth assignment. In this paper we introduce three new encodings from XSAT instances to SAT instances that lead to a lot of structural information (equivalency gates and and gates) which is naturally hidden in the pairwise transformation. Some solvers (lsat,march_dl,eqsatz) can take into account this kinds of structural information to make simplifications as pretreatment and speed-up the resolution. Then we show the interest of dealing with the XSAT formalism by introducing an encoding of binary CSP and graph coloring problem into XSAT instances. Preliminary results on real-world binary CSP and graph coloring problem show the importance of exhibiting equivalencies for the XSAT problem.

scholarly journals On A Quantum Proof Of The Equivalence of P And NP

2019 ◽  
Author(s):  
Adewale Oluwasanmi
Keyword(s):  
Polynomial Time ◽  
Solution Space ◽  
Quantum Systems ◽  
Satisfiability Problem ◽  
Quantum Mechanical ◽  
Correctness Proof ◽  
Classical Systems ◽  
Classical Architecture ◽  
General Quantum

We present an algorithm, along with a correctness proof, for solving the 3 Satisfiability problem that is inspired by quantum mechanical principles and that runs in polynomial time with respect to the size of the input problem. Even though we term both our algorithm and its associated proof as quantum (for reasons which we will demonstrate), it is intended to be run on standard classical architecture. In the article, we posit that the 3 Satisfiability problem has an intrinsic complex quantum form that can be programmed in order to build a model of the solution space for satisfiable instances or show that such a model cannot be constructed. This yields surprising results on the ability for classical systems to abstractly simulate general quantum systems.

scholarly journals Rotation Based MSS/MCS Enumeration

10.29007/8btb ◽  
2020 ◽  
Author(s):  
Jaroslav Bendík ◽  
Ivana Cerna
Keyword(s):  
Time Limit ◽  
Satisfiability Problem ◽  
Experimental Comparison ◽  
Boolean Formula ◽  
Enumeration Algorithm ◽  
Complete Enumeration ◽  
Enumeration Algorithms

Given an unsatisfiable Boolean Formula F in CNF, i.e., a set of clauses, one is often interested in identifying Maximal Satisfiable Subsets (MSSes) of F or, equivalently, the complements of MSSes called Minimal Correction Subsets (MCSes). Since MSSes (MC- Ses) find applications in many domains, e.g. diagnosis, ontologies debugging, or axiom pinpointing, several MSS enumeration algorithms have been proposed. Unfortunately, finding even a single MSS is often very hard since it naturally subsumes repeatedly solving the satisfiability problem. Moreover, there can be up to exponentially many MSSes, thus their complete enumeration is often practically intractable. Therefore, the algorithms tend to identify as many MSSes as possible within a given time limit. In this work, we present a novel MSS enumeration algorithm called RIME. Compared to existing algorithms, RIME is much more frugal in the number of performed satisfiability checks which we witness via an experimental comparison. Moreover, RIME is several times faster than existing tools.

scholarly journals Characterization and computation of ancestors in reaction systems

2020 ◽  
Author(s):  
Roberto Barbuti ◽  
Anna Bernasconi ◽  
Roberta Gori ◽  
Paolo Milazzo
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Boolean Formula ◽  
Minimal Size ◽  
Minimum Cardinality ◽  
Reaction Systems ◽  
Target Set ◽  
The Cost

Abstract In reaction systems, preimages and nth ancestors are sets of reactants leading to the production of a target set of products in either 1 or n steps, respectively. Many computational problems on preimages and ancestors, such as finding all minimum-cardinality nth ancestors, computing their size or counting them, are intractable. In this paper, we characterize all nth ancestors using a Boolean formula that can be computed in polynomial time. Once simplified, this formula can be exploited to easily solve all preimage and ancestor problems. This allows us to directly relate the difficulty of ancestor problems to the cost of the simplification so that new insights into computational complexity investigations can be achieved. In particular, we focus on two problems: (i) deciding whether a preimage/nth ancestor exists and (ii) finding a preimage/nth ancestor of minimal size. Our approach is constructive, it aims at finding classes of reactions systems for which the ancestor problems can be solved in polynomial time, in exact or approximate way.

A Polynomial Algorithm for Constructing a Large Bipartite Subgraph, with an Application to a Satisfiability Problem

1982 ◽  
Vol 34 (3) ◽  
pp. 519-524 ◽  
Author(s):  
Svatopluk Poljak ◽  
Daniel Turzík
Keyword(s):  
Connected Graph ◽  
Polynomial Algorithm ◽  
Short Proof ◽  
Satisfiability Problem ◽  
Boolean Formula ◽  
Complete Problem ◽  
Bipartite Subgraph ◽  
Np Complete ◽  
Image Position

Let G be a symmetric connected graph without loops. Denote by b(G) the maximum number of edges in a bipartite subgraph of G. Determination of b(G) is polynomial for planar graphs ([6], [8]); in general it is an NP-complete problem ([5]). Edwards in [1], [2] found some estimates of b(G) which give, in particular,for a connected graph G of n vertices and m edges, whereand ﹛x﹜ denotes the smallest integer ≧ x.We give an 0(V3) algorithm which for a given graph constructs a bipartite subgraph B with at least f(m, n) edges, yielding a short proof of Edwards’ result.Further, we consider similar methods for obtaining some estimates for a particular case of the satisfiability problem. Let Φ be a Boolean formula of variables x1, …, xn.

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