scholarly journals NP-Hardness of the Problem of Optimal Box Positioning

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 711
Author(s):  
Alexei V. Galatenko ◽  
Stepan A. Nersisyan ◽  
Dmitriy N. Zhuk
Keyword(s):  
Reduction Technique ◽  
Satisfiability Problem ◽  
Np Hard ◽  
Polynomial Reduction ◽  
Finite Set ◽  
The Given ◽  
Np Hardness

We consider the problem of finding a position of a d-dimensional box with given edge lengths that maximizes the number of enclosed points of the given finite set P ⊂ R d , i.e., the problem of optimal box positioning. We prove that while this problem is polynomial for fixed values of d, it is NP-hard in the general case. The proof is based on a polynomial reduction technique applied to the considered problem and the 3-CNF satisfiability problem.

scholarly journals NP-hardness of quadratic Euclidean 1-Mean and 1-Median 2-Clustering problem with the constraints on the cluster sizes

2019 ◽  
Vol 489 (4) ◽  
pp. 339-343
Author(s):  
A. V. Kel’manov ◽  
A. V. Pyatkin ◽  
V. I. Khandeev
Keyword(s):  
Euclidean Space ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Clustering Problem ◽  
Geometric Center ◽  
Finite Set ◽  
The Given ◽  
Np Hardness

In the paper, we consider a problem of clustering a finite set of N points in d-dimensional Euclidean space into two clusters minimizing the sum over all clusters of the intracluster sums of the distances between clusters elements and their centers. The center of one cluster is defined as centroid (geometric center). The center of the other one is a sought point in the input set. We analyze the variant of the problem with the given clusters sizes. We have proved the strong NP-hardness of this problem.

Finding an Unknown Acyclic Orientation of a Given Graph

2009 ◽  
Vol 19 (1) ◽  
pp. 121-131 ◽  
Author(s):  
OLEG PIKHURKO
Keyword(s):  
Complete Graph ◽  
Discrete Mathematics ◽  
Np Hard ◽  
Worst Case ◽  
Acyclic Orientation ◽  
Multiplicative Factor ◽  
The Given

Let c(G) be the smallest number of edges we have to test in order to determine an unknown acyclic orientation of the given graph G in the worst case. For example, if G is the complete graph on n vertices, then c(G) is the smallest number of comparisons needed to sort n numbers.We prove that c(G) ≤ (1/4 + o(1))n2 for any graph G on n vertices, answering in the affirmative a question of Aigner, Triesch and Tuza [Discrete Mathematics144 (1995) 3–10]. Also, we show that, for every ϵ > 0, it is NP-hard to approximate the parameter c(G) within a multiplicative factor 74/73 − ϵ.

APPROXIMATING POLYGONS AND SUBDIVISIONS WITH MINIMUM-LINK PATHS

1993 ◽  
Vol 03 (04) ◽  
pp. 383-415 ◽  
Author(s):  
LEONIDAS J. GUIBAS ◽  
JOHN E. HERSHBERGER ◽  
JOSEPH S.B. MITCHELL ◽  
JACK SCOTT SNOEYINK
Keyword(s):  
Greedy Algorithms ◽  
Np Hard ◽  
Basic Approach ◽  
Line Segments ◽  
Minimum Number ◽  
Polygonal Chains ◽  
The Given ◽  
Plane Polygon

We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We give some variants that have linear or O(n log n) algorithms approximating polygonal chains of n segments. We also show that approximating subdivisions and approximating with chains with. no self-intersections are NP-hard.

MULTILINEAR EQUATIONS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

2011 ◽  
Vol 21 (01n02) ◽  
pp. 35-59 ◽  
Author(s):  
A. CHERUBINI ◽  
C. NUCCIO ◽  
E. RODARO
Keyword(s):  
Normal Form ◽  
Inverse Semigroups ◽  
Satisfiability Problem ◽  
Finite Set ◽  
Context Free ◽  
Schützenberger Automaton ◽  
Amalgamated Product

Let S = S1 *U S2 = Inv〈X; R〉 be the free amalgamated product of the finite inverse semigroups S1, S2 and let Ξ be a finite set of unknowns. We consider the satisfiability problem for multilinear equations over S, i.e. equations wL ≡ wR with wL, wR ∈ (X ∪ X-1 ∪ Ξ ∪ Ξ-1)+ such that each x ∈ Ξ labels at most one edge in the Schützenberger automaton of either wL or wR relative to the presentation 〈X ∪ Ξ|R〉. We prove that the satisfiability problem for such equations is decidable using a normal form of the words wL, wR and the fact that the language recognized by the Schützenberger automaton of any word in (X ∪ X-1)+) relative to the presentation 〈X|R〉 is context-free.

scholarly journals ON SPANNERS OF GEOMETRIC GRAPHS

2009 ◽  
Vol 20 (01) ◽  
pp. 135-149 ◽  
Author(s):  
JOACHIM GUDMUNDSSON ◽  
MICHIEL SMID
Keyword(s):  
Real Number ◽  
Upper Bound ◽  
Weighted Graph ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Np Hard ◽  
Minimum Number ◽  
Np Hardness

Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every real number t with [Formula: see text], there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω(n1+1/t) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(n1+2/(t-1)) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.

scholarly journals Llull and Copeland Voting Computationally Resist Bribery and Constructive Control

2009 ◽  
Vol 35 ◽  
pp. 275-341 ◽  
Author(s):  
P. Faliszewski ◽  
E. Hemaspaandra ◽  
L. A. Hemaspaandra ◽  
J. Rothe
Keyword(s):  
Polynomial Time ◽  
Network Flow ◽  
Np Hard ◽  
Voter Preferences ◽  
External Agent ◽  
Fixed Parameter ◽  
Election System ◽  
Election Systems ◽  
The Given ◽  
Flexible Models

Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Constructive control of elections refers to attempts by an agent to, via such actions as addition/deletion/partition of candidates or voters, ensure that a given candidate wins. Destructive control refers to attempts by an agent to, via the same actions, preclude a given candidate's victory. An election system in which an agent can sometimes affect the result and it can be determined in polynomial time on which inputs the agent can succeed is said to be vulnerable to the given type of control. An election system in which an agent can sometimes affect the result, yet in which it is NP-hard to recognize the inputs on which the agent can succeed, is said to be resistant to the given type of control. Aside from election systems with an NP-hard winner problem, the only systems previously known to be resistant to all the standard control types were highly artificial election systems created by hybridization. This paper studies a parameterized version of Copeland voting, denoted by Copeland^\alpha, where the parameter \alpha is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates. In every previously studied constructive or destructive control scenario, we determine which of resistance or vulnerability holds for Copeland^\alpha for each rational \alpha, 0 \leq \alpha \leq 1. In particular, we prove that Copeland^{0.5}, the system commonly referred to as ``Copeland voting,'' provides full resistance to constructive control, and we prove the same for Copeland^\alpha, for all rational \alpha, 0 < \alpha < 1. Among systems with a polynomial-time winner problem, Copeland voting is the first natural election system proven to have full resistance to constructive control. In addition, we prove that both Copeland^0 and Copeland^1 (interestingly, Copeland^1 is an election system developed by the thirteenth-century mystic Llull) are resistant to all standard types of constructive control other than one variant of addition of candidates. Moreover, we show that for each rational \alpha, 0 \leq \alpha \leq 1, Copeland^\alpha voting is fully resistant to bribery attacks, and we establish fixed-parameter tractability of bounded-case control for Copeland^\alpha. We also study Copeland^\alpha elections under more flexible models such as microbribery and extended control, we integrate the potential irrationality of voter preferences into many of our results, and we prove our results in both the unique-winner model and the nonunique-winner model. Our vulnerability results for microbribery are proven via a novel technique involving min-cost network flow.

scholarly journals Convex Quadrangulations of Bichromatic Point Sets

2019 ◽  
Vol 29 (04) ◽  
pp. 289-299
Author(s):  
Alexander Pilz ◽  
Carlos Seara
Keyword(s):  
Open Problem ◽  
Np Hard ◽  
Point Sets ◽  
Finite Set

We consider quadrangulations of red and blue points in the plane where each face is convex and no edge connects two points of the same color. In particular, we show that the following problem is NP-hard: Given a finite set [Formula: see text] of points with each point either red or blue, does there exist a convex quadrangulation of [Formula: see text] in such a way that the predefined colors give a valid vertex 2-coloring of the quadrangulation? We consider this as a step towards solving the corresponding long-standing open problem on monochromatic point sets.

On bijunctive predicates over a finite set

2019 ◽  
Vol 29 (1) ◽  
pp. 49-58 ◽  
Author(s):  
Svetlana N. Selezneva
Keyword(s):  
Normal Forms ◽  
Satisfiability Problem ◽  
Polynomial Algorithms ◽  
Majority Function ◽  
Finite Set

Abstract The paper is concerned with representations of predicates over a finite set in the form of generalized conjunctive normal forms (GCNF). Properties of predicates GCNF are found which are preserved by some majority function. Such predicates are called generalized bijunctive predicates. These properties are used to construct new faster polynomial algorithms for the generalized satisfiability problem in the case when some majority function preserves all the original predicates.

Secrecy-Preserving Reasoning and Query Answering in Probabilistic Description Logic Knowledge Bases

10.29007/npd4 ◽  
2018 ◽  
Author(s):  
Gopalakrishnan Krishnasamy Sivaprakasam ◽  
Adrienne Raglin ◽  
Douglas Summers-Stay ◽  
Giora Slutzki
Keyword(s):  
Probabilistic Model ◽  
Description Logic ◽  
Knowledge Bases ◽  
Query Answering ◽  
Decomposition Algorithm ◽  
Probabilistic Description ◽  
Recursive Query ◽  
Finite Set ◽  
The Given

In this paper we study Secrecy-Preserving Query Answering problem underthe OpenWorld Assumption (OWA) for Prob-EL&gt;0;=1 Knowledge Bases(KBs). We have designed a tableau procedure to compute a semi model Mover the given KB which eventually is equivalent to a probabilistic modelto KB. Given a secrecy set S, which is a finite set of assertions, wecompute a function E, called an envelope of S, which assigns a set E() ofassertions to each world in the semi modal M. E provides logical protection to the secrecy set S against the reasoning of a querying agent. Once the semi model M and an envelope E are computed, we define the secrecy-preserving semi model ME.Based on the information available in ME, assertional queries with probabilisticoperators can be answered eciently while preserving secrecy. Tothe best of our knowledge, this work is first one studying secrecy-preservingreasoning in description logic augmented with probabilistic operators. Whenthe querying agent asks a query q, the reasoner answers “Yes” if informationabout q is available in ME; otherwise, the reasoner answers “Unknown”. Beingable to answer “Unknown” plays a key role in protecting secrecy underOWA. Since we are not computing all the consequences of the knowledgebase, answers to the queries based on just secrecy-preserving semi modelME could be erroneous. To fix this problem, we further augment our algorithmsby providing recursive query decomposition algorithm to make thequery answering procedure foolproof.1

Some polynomially solvable cases of the inverse ordered 1-median problem on trees

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3651-3664 ◽  
Author(s):  
Kien Nguyen
Keyword(s):  
Minimum Cost ◽  
Np Hard ◽  
Median Problem ◽  
Star Graphs ◽  
Polynomially Solvable ◽  
Np Hardness

We consider the problem of modifying the edge lengths of a tree at minimum cost such that a prespecified vertex become an ordered 1-median of the perturbed tree. We call this problem the inverse ordered 1-median problem on trees. Gassner showed in 2012 that the inverse ordered 1-median problem on trees is, in general, NP-hard. We, however, address some situations, where the corresponding inverse 1-median problem is polynomially solvable. For the problem on paths with n vertices, we develop an O(n3) algorithm based on a greedy technique. Furthermore, we prove the NP-hardness of the inverse ordered 1- median problem on star graphs and propose a quadratic algorithm that solves the inverse ordered 1-median problem on unweighted stars.

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