scholarly journals Incidence dimension and 2-packing number in graphs

2022 ◽  
Author(s):  
Dragana Božović ◽  
Aleksander Kelenc ◽  
Iztok Peterin ◽  
Ismael G. Yero
Keyword(s):  
Np Hard ◽  
Packing Number ◽  
Close Relationship ◽  
The Given

Abstract. Let G = (V,E) be a graph. A set of vertices A is an incidence generator for G if for any two distinct edges e,f ∈ E(G) there exists a vertex from A which is an endpoint of either e or f. The smallest cardinality of an incidence generator for G is called the incidence dimension and is denoted by dimI(G). A set of vertices P ⊆ V (G) is a 2-packing of G if the distance in G between any pair of distinct vertices from P is larger than two. The largest cardinality of a 2-packing of G is the packing number of G and is denoted by ρ(G). In this article, the incidence dimension is introduced and studied. The given results show a close relationship between dimI(G) and ρ(G). We rst note that the complement of any 2-packing in graph G is an incidence generator for G, and further show that either dimI(G) = ρ(G) or dimI(G) = ρ(G)−1 for any graph G. In addition, we present some bounds for dimI(G) and prove that the problem of determining the incidence dimension of a graph is NP-hard.

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.

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.

Line-Constrained k-Median, k-Means, and k-Center Problems in the Plane

2016 ◽  
Vol 26 (03n04) ◽  
pp. 185-210
Author(s):  
Haitao Wang ◽  
Jingru Zhang
Keyword(s):  
Time Algorithm ◽  
Efficient Algorithms ◽  
Additional Constraint ◽  
Distance Measures ◽  
Np Hard ◽  
Center Problem ◽  
Median Problem ◽  
One Dimensional ◽  
The Given

The (weighted) [Formula: see text]-median, [Formula: see text]-means, and [Formula: see text]-center problems in the plane are known to be NP-hard. In this paper, we study these problems with an additional constraint that requires the sought [Formula: see text] facilities to be on a given line. We present efficient algorithms for various distance measures such as [Formula: see text]. We assume that all [Formula: see text] weighted points are given sorted by their projections on the given line. For [Formula: see text]-median, our algorithms for [Formula: see text] and [Formula: see text] metrics run in [Formula: see text] time and [Formula: see text] time, respectively. For [Formula: see text]-means, which is defined only on the squared [Formula: see text] distance, we give an [Formula: see text] time algorithm. For [Formula: see text]-center, our algorithms run in [Formula: see text] time for all three metrics, and in [Formula: see text] time for the unweighted version under [Formula: see text] and [Formula: see text] metrics. While our results for the [Formula: see text]-center problem are optimal, the results for the [Formula: see text]-median problem almost match the best algorithms for the corresponding one-dimensional problems.

FINDING TYPE SETS IS NP-HARD

1991 ◽  
Vol 01 (04) ◽  
pp. 437-444 ◽  
Author(s):  
DAVID HOBBY
Keyword(s):  
Finite Algebra ◽  
Hard Problem ◽  
Np Hard ◽  
Complete Problem ◽  
Type 3 ◽  
The Given ◽  
Type Sets

It is shown that determining the type set of the variety generated by a finite algebra is a P-Space-hard problem. This is done by interpreting into it the P-Space-complete problem of determining if a given function is a composition of a set of unary functions on a set. Specifically, the given function is a composition of the others just when the type 3 is not in the type set of the variety generated by the algebra that is constructed. A lemma that restricts the types that are possible in the variety generated by a given algebra is also given.

scholarly journals Total Weak Roman Domination in Graphs

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 831 ◽  
Author(s):  
Abel Cabrera Martínez ◽  
Luis P. Montejano ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Open Neighbourhood ◽  
Np Hard ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Close Relationship ◽  
Roman Dominating Function ◽  
Domination In Graphs ◽  
Dominating Function

Given a graph G = ( V , E ) , a function f : V → { 0 , 1 , 2 , ⋯ } is said to be a total dominating function if ∑ u ∈ N ( v ) f ( u ) > 0 for every v ∈ V , where N ( v ) denotes the open neighbourhood of v. Let V i = { x ∈ V : f ( x ) = i } . We say that a function f : V → { 0 , 1 , 2 } is a total weak Roman dominating function if f is a total dominating function and for every vertex v ∈ V 0 there exists u ∈ N ( v ) ∩ ( V 1 ∪ V 2 ) such that the function f ′ , defined by f ′ ( v ) = 1 , f ′ ( u ) = f ( u ) - 1 and f ′ ( x ) = f ( x ) whenever x ∈ V ∖ { u , v } , is a total dominating function as well. The weight of a function f is defined to be w ( f ) = ∑ v ∈ V f ( v ) . In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by γ t r ( G ) , which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on γ t r ( G ) and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard.

Study on the Spatial Policy of Coordinated Development in Beijing-Tianjin-Hebei Region

2021 ◽  
Vol 27 (7) ◽  
pp. 65-72
Author(s):  
Liu Jingjuan ◽  
Keyword(s):  
Industrial Structure ◽  
Integration Process ◽  
Coordinated Development ◽  
Systematic Analysis ◽  
Spatial Policy ◽  
Close Relationship ◽  
Preliminary Preparation ◽  
Relationship Of ◽  
The Given ◽  
New Situation

The article focuses on the spatial policy of the coordinated development of the Beijing-Tianjin-Hebei region under the new situation in China. Through a systematic analysis of the political integration process of the given region, the validity of the integration implementation and its pathways for improving are investigated. Based on the real situation of regional development, it is necessary to adopt a regional spatial policy to further advance the integration process, including integrated planning for the development of the region, promoting the integration of infrastructure construction, strengthening the integration of the industrial structure and resource allocation, etc. It was revealed that the coordinated development of the Beijing-Tianjin-Hebei region gradually moved to the period of full implementation of the strategy after a period of preliminary preparation and a period of reaching consensus. The coordinated development of the region is, in fact, a spatial strategy with the aim of achieving a spatial balance of regional economic development by the optimal distribution of production elements. So, in accordance with the current requirements for the formation of additional benefits and the qualitative development of the regional economy, the spatial policy should be based on the close relationship of “prosperity of humanity” and “prosperity of the region”

On Covering Points with Minimum Turns

2015 ◽  
Vol 25 (01) ◽  
pp. 1-9 ◽  
Author(s):  
Minghui Jiang
Keyword(s):  
Np Hard ◽  
Line Segments ◽  
Polygonal Chain ◽  
Axis Parallel ◽  
A Chain ◽  
Set Of Points ◽  
The Given

We study the problem of finding a polygonal chain of line segments to cover a set of points in ℝd, d≥2, with the goal of minimizing the number of links or turns in the chain. A chain of line segments that covers all points in the given set is called a covering tour if the chain is closed, and is called a covering path if the chain is open. A covering tour or a covering path is rectilinear if all segments in the chain are axis-parallel. We prove that the two problems Minimum-Link Rectilinear Covering Tour and Minimum-Link Rectilinear Covering Path are both NP-hard in ℝ10.

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.

Minding the Close Relationship

1999 ◽  
Author(s):  
John H. Harvey ◽  
Julia Omarzu
Keyword(s):  
Close Relationship
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