On the Complexities of Selected Satisfiability and Equivalence Queries over Boolean Formulas and Inclusion Queries over Hulls

This paper is concerned with the computational complexities of three types of queries, namely, satisfiability, equivalence, and hull inclusion. The first two queries are analyzed over the domain of CNF formulas, while hull inclusion queries are analyzed over continuous and discrete sets defined by rational polyhedra. Although CNF formulas can be represented by polyhedra over discrete sets, we analyze them separately on account of their distinct structure. In particular, we consider the NAESAT and XSAT versions of satisfiability over HornCNF, 2CNF, and Horn⊕2CNF formulas. These restricted families find applications in a number of practical domains. From the hull inclusion perspective, we are primarily concerned with the question of checking whether two succinct descriptions of a set of points are equivalent. In particular, we analyze the complexities of integer hull inclusion over 2SAT and Horn polyhedra. Hull inclusion problems are important from the perspective of deriving minimal descriptions of point sets. One of the surprising consequences of our work is the stark difference in complexities between equivalence problems in the clausal and polyhedral domains for the same polyhedral structure.

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MAINTAINING VISIBILITY INFORMATION OF PLANAR POINT SETS WITH A MOVING VIEWPOINT

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195907002343 ◽

2007 ◽

Vol 17 (04) ◽

pp. 297-304 ◽

Cited By ~ 2

Author(s):

OLIVIER DEVILLERS ◽

VIDA DUJMOVIĆ ◽

HAZEL EVERETT ◽

SAMUEL HORNUS ◽

SUE WHITESIDES ◽

...

Keyword(s):

Linear Space ◽

Point Sets ◽

Worst Case ◽

Planar Point ◽

Set Of Points

Given a set of n points in the plane, we consider the problem of computing the circular ordering of the points about a viewpoint q and efficiently maintaining this ordering information as q moves. In linear space, and after O(n log n) preprocessing time, our solution maintains the view at a cost of O( log n) amortized time (resp.O( log 2 n) worst case time) for each change. Our algorithm can also be used to maintain the set of points sorted according to their distance to q .

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NAE-resolution: A new resolution refutation technique to prove not-all-equal unsatisfiability

Mathematical Structures in Computer Science ◽

10.1017/s096012952000016x ◽

2020 ◽

Vol 30 (7) ◽

pp. 736-751

Author(s):

Hans Kleine Büning ◽

P. Wojciechowski ◽

K. Subramani

Keyword(s):

Normal Form ◽

Conjunctive Normal Form ◽

Principal Result ◽

Satisfying Assignment ◽

Boolean Formulas ◽

Complete Procedure ◽

Np Complete ◽

Resolution Refutation ◽

Original Formula ◽

Cnf Formulas

AbstractIn this paper, we analyze Boolean formulas in conjunctive normal form (CNF) from the perspective of read-once resolution (ROR) refutation schemes. A read-once (resolution) refutation is one in which each clause is used at most once. Derived clauses can be used as many times as they are deduced. However, clauses in the original formula can only be used as part of one derivation. It is well known that ROR is not complete; that is, there exist unsatisfiable formulas for which no ROR exists. Likewise, the problem of checking if a 3CNF formula has a read-once refutation is NP-complete. This paper is concerned with a variant of satisfiability called not-all-equal satisfiability (NAE-satisfiability). A CNF formula is NAE-satisfiable if it has a satisfying assignment in which at least one literal in each clause is set to false. It is well known that the problem of checking NAE-satisfiability is NP-complete. Clearly, the class of CNF formulas which are NAE-satisfiable is a proper subset of satisfiable CNF formulas. It follows that traditional resolution cannot always find a proof of NAE-unsatisfiability. Thus, traditional resolution is not a sound procedure for checking NAE-satisfiability. In this paper, we introduce a variant of resolution called NAE-resolution which is a sound and complete procedure for checking NAE-satisfiability in CNF formulas. The focus of this paper is on a variant of NAE-resolution called read-once NAE-resolution in which each clause (input or derived) can be part of at most one NAE-resolution step. Our principal result is that read-once NAE-resolution is a sound and complete procedure for 2CNF formulas. Furthermore, we provide an algorithm to determine the smallest such NAE-resolution in polynomial time. This is in stark contrast to the corresponding problem concerning 2CNF formulas and ROR refutations. We also show that the problem of checking whether a 3CNF formula has a read-once NAE-resolution is NP-complete.

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Testing satisfiability of CNF formulas by computing a stable set of points

Annals of Mathematics and Artificial Intelligence ◽

10.1007/s10472-005-0420-x ◽

2004 ◽

Vol 43 (1-4) ◽

pp. 65-89 ◽

Cited By ~ 2

Author(s):

Eugene Goldberg

Keyword(s):

Stable Set ◽

Set Of Points ◽

Cnf Formulas

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Chebychev Approximations of Finite Point Sets with Application to Planar Kinematic Synthesis

Journal of Mechanical Design ◽

10.1115/1.3454021 ◽

1979 ◽

Vol 101 (1) ◽

pp. 32-40 ◽

Cited By ~ 7

Author(s):

Y. L. Sarkisyan ◽

K. C. Gupta ◽

B. Roth

Keyword(s):

Point Sets ◽

Finite Point ◽

Kinematic Synthesis ◽

Radial Deviation ◽

Point Set ◽

Planar Linkages ◽

Straight Lines ◽

Set Of Points

In the first part of this paper we consider the problem of determining circles which best approximate a given set of points. The approximation is one which minimizes the maximum radial deviation of the points from the approximating circle. Then a similar procedure is developed for determining straight lines which best approximate a given point set. The final parts of the paper illustrate the application of these results to synthesizing planar linkages.

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Images of Linear Conditions on a Manhattan Plane

Geometry & Graphics ◽

10.12737/2308-4898-2020-3-14 ◽

2020 ◽

Vol 8 (1) ◽

pp. 3-14

Author(s):

V. Yurkov

Keyword(s):

Plane Partition ◽

Point Sets ◽

Geometric Problem ◽

Line Segments ◽

Finite Set ◽

Isolated Points ◽

Finite Sum ◽

Lattice Construction ◽

Set Of Points ◽

The Given

In this paper are considered planar point sets generated by linear conditions, which are realized in rectangular or Manhattan metric. Linear conditions are those expressed by the finite sum of the products of distances by numerical coefficients. Finite sets of points and lines are considered as figures defining linear conditions. It has been shown that linear conditions can be defined relative to other planar figures: lines, polygons, etc. The design solutions of the following general geometric problem are considered: for a finite set of figures (points, line segments, polygons...) specified on a plane with a rectangular metric, which are in a common position, it is necessary to construct sets that satisfy any linear condition. The problems in which the given sets are point and segment ones have been considered in detail, and linear conditions are represented as a sum or as relations of distances. It is proved that solution result can be isolated points, broken lines, and areas on the plane. Sets of broken lines satisfying the given conditions form families of isolines for the given condition. An algorithm for building isoline families is presented. The algorithm is based on the Hanan lattice construction and the isolines behavior in each node and each sub-region of the lattice. For isoline families defined by conditions for relation of distances, some of their properties allowing accelerate their construction process are proved. As an example for application of the described theory, the problem of plane partition into regions corresponding to a given set of points, lines and other figures is considered. The problem is generalized problem of Voronoi diagram construction, and considered in general formulation. It means the next: 1) the problem is considered in rectangular metric; 2) all given points may be integrated in various figures – separate points, line segments, triangles, quadrangles etc.; 3) the Voronoi diagram’s property of proximity is changed for property of proportionality. Have been represented examples for plane partition into regions, determined by two-point sets.

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A Bound on a Convexity Measure for Point Sets

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195919500110 ◽

2019 ◽

Vol 29 (04) ◽

pp. 301-306

Author(s):

Danny Rorabaugh

Keyword(s):

Upper Bound ◽

Interior Angle ◽

Point Sets ◽

Planar Point ◽

Angle Measure ◽

Convex Position ◽

Point Set ◽

Set Of Points ◽

Convexity Measure

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most [Formula: see text]. We can thus talk about the convexity of a set of points in terms of its min-max interior angle measure. The main result presented here is a nontrivial upper bound of the min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.

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A Randomized Approach to Volume Constrained Polyhedronization Problem

Journal of Computing and Information Science in Engineering ◽

10.1115/1.4029559 ◽

2015 ◽

Vol 15 (1) ◽

Cited By ~ 1

Author(s):

Jiju Peethambaran ◽

Amal Dev Parakkat ◽

Ramanathan Muthuganapathy

Keyword(s):

Three Dimensional ◽

Greedy Heuristic ◽

Point Sets ◽

Maximal Volume ◽

Simple Polyhedron ◽

Practical Algorithms ◽

Finite Set ◽

Potential Applications ◽

Brute Force Method ◽

Set Of Points

Given a finite set of points in R3, polyhedronization deals with constructing a simple polyhedron such that the vertices of the polyhedron are precisely the given points. In this paper, we present randomized approximation algorithms for minimal volume polyhedronization (MINVP) and maximal volume polyhedronization (MAXVP) of three dimensional point sets in general position. Both, MINVP and MAXVP, problems have been shown to be NP-hard and to the best of our knowledge, no practical algorithms exist to solve these problems. It has been shown that for any point set S in R3, there always exists a tetrahedralizable polyhedronization of S. We exploit this fact to develop a greedy heuristic for MINVP and MAXVP constructions. Further, we present an empirical analysis on the quality of the approximation results of some well defined point sets. The algorithms have been validated by comparing the results with the optimal results generated by an exhaustive searching (brute force) method for MINVP and MAXVP for some well chosen point sets of smaller sizes. Finally, potential applications of minimum and maximum volume polyhedra in 4D printing and surface lofting, respectively, have been discussed.

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A Note on Convexly Independent Subsets of an Infinite Set of Points

Georgian Mathematical Journal ◽

10.1515/gmj.2002.303 ◽

2002 ◽

Vol 9 (2) ◽

pp. 303-307

Author(s):

A. Kharazishvili

Keyword(s):

Euclidean Space ◽

Point Sets ◽

Set Of Points ◽

Infinite Set

Abstract We consider convexly independent subsets of a given infinite set of points in the plane (Euclidean space) and evaluate the cardinality of such subsets. It is demonstrated, in particular, that situations are essentially different for countable and uncountable point sets.

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