ON THE DIFFICULTY OF DECIDING THE CONVEXITY OF POLYNOMIALS OVER SIMPLEXES

1996 ◽  
Vol 06 (02) ◽  
pp. 227-229 ◽  
Author(s):  
BAINING GUO
Keyword(s):  
Decision Problem ◽  
Np Hard

We analyze the complexity of deciding the convexity of polynomials over simplexes. For polynomials of degree four or higher, the decision problem requires accomplishing an NP-hard task; when the degrees of the polynomials are less than four, the problem can be solved in NC time.

scholarly journals Computational Complexity of Topological Invariants

2014 ◽  
Vol 58 (1) ◽  
pp. 27-32
Author(s):  
Manuel Amann
Keyword(s):  
Computational Complexity ◽  
Decision Problem ◽  
Topological Invariants ◽  
Np Hard ◽  
Connected Space ◽  
Finite Dimensional ◽  
Lusternik Schnirelmann ◽  
Simply Connected

AbstractWe answer the following question posed by Lechuga: given a simply connected spaceXwith bothH*(X; ℚ) and π*(X) ⊗ ℚ being finite dimensional, what is the computational complexity of an algorithm computing the cup length and the rational Lusternik—Schnirelmann category ofX?Basically, by a reduction from the decision problem of whether a given graph isk-colourable fork≥ 3, we show that even stricter versions of the problems above are NP-hard.

scholarly journals ON THE ROMAN BONDAGE NUMBER OF A GRAPH

2013 ◽  
Vol 05 (01) ◽  
pp. 1350001 ◽  
Author(s):  
A. BAHREMANDPOUR ◽  
FU-TAO HU ◽  
S. M. SHEIKHOLESLAMI ◽  
JUN-MING XU
Keyword(s):  
Decision Problem ◽  
Maximum Degree ◽  
Minimum Weight ◽  
Domination Number ◽  
Np Hard ◽  
Roman Domination ◽  
Minimum Cardinality ◽  
Bondage Number ◽  
Roman Domination Number ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph G = (V, E) is a function f : V → {0, 1, 2} such that every vertex v ∈ V with f(v) = 0 has at least one neighbor u ∈ V with f(u) = 2. The weight of a RDF is the value f(V(G)) = Σu∈V(G) f(u). The minimum weight of a RDF on a graph G is called the Roman domination number, denoted by γR(G). The Roman bondage number bR(G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E′ ⊆ E(G) for which γR(G - E′) > γR(G). In this paper, we first show that the decision problem for determining bR(G) is NP-hard even for bipartite graphs and then we establish some sharp bounds for bR(G) and characterizes all graphs attaining some of these bounds.

scholarly journals Progress on Roman and Weakly Connected Roman Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1846
Author(s):  
Joanna Raczek ◽  
Rita Zuazua
Keyword(s):  
Graph Theory ◽  
Bipartite Graph ◽  
Decision Problem ◽  
Hard Problem ◽  
Np Hard ◽  
Np Hard Problem

A graph G for which γR(G)=2γ(G) is the Roman graph, and if γRwc(G)=2γwc(G), then G is the weakly connected Roman graph. In this paper, we show that the decision problem of whether a bipartite graph is Roman is a co-NP-hard problem. Next, we prove similar results for weakly connected Roman graphs. We also study Roman trees improving the result of M.A. Henning’s A characterization of Roman trees, Discuss. Math. Graph Theory 22 (2002). Moreover, we give a characterization of weakly connected Roman trees.

scholarly journals Optimizing Distributed Real-Time Embedded System Handling Dependence and Several Strict Periodicity Constraints

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Omar Kermia
Keyword(s):  
Embedded System ◽  
Real Time ◽  
Execution Time ◽  
Decision Problem ◽  
Schedulability Analysis ◽  
Multiprocessor Scheduling ◽  
Np Hard ◽  
Necessary And Sufficient

This paper focuses on real-time nonpreemptive multiprocessor scheduling with precedence and strict periodicity constraints. Since this problem is NP-hard, there exist several approaches to resolve it. In addition, because of periodicity constraints our problem stands for a decision problem which consists in determining if, a solution exists or not. Therefore, the first criterion on which the proposed heuristic is evaluated is its schedulability. Then, the second criterion on which the proposed heuristic is evaluated is its execution time. Hence, we performed a schedulability analysis which leads to a necessary and sufficient schedulability condition for determining whether a task satisfies its precedence and periodicity constraints on a processor where others tasks have already been scheduled. We also present two multiperiodic applications.

scholarly journals Deciding Relaxed Two-Colourability: A Hardness Jump

2009 ◽  
Vol 18 (1-2) ◽  
pp. 53-81 ◽  
Author(s):  
R. BERKE ◽  
T. SZABÓ
Keyword(s):  
Decision Problem ◽  
Maximum Degree ◽  
Time Algorithm ◽  
Np Hard ◽  
Running Time ◽  
Symmetric Version ◽  
Component Order ◽  
Colour Classes

We study relaxations of proper two-colourings, such that the order of the induced monochromatic components in one (or both) of the colour classes is bounded by a constant. A colouring of a graph G is called (C1, C2)-relaxed if every monochromatic component induced by vertices of the first (second) colour is of order at most C1 (C2, resp.). We prove that the decision problem ‘Is there a (1, C)-relaxed colouring of a given graph G of maximum degree 3?’ exhibits a hardness jump in the component order C. In other words, there exists an integer f(3) such that the decision problem is NP-hard for every 2 ≤ C < f(3), while every graph of maximum degree 3 is (1, f(3))-relaxed colourable. We also show f(3) ≤ 22 by way of a quasilinear time algorithm, which finds a (1, 22)-relaxed colouring of any graph of maximum degree 3. Both the bound on f(3) and the running time greatly improve earlier results. We also study the symmetric version, that is, when C1 = C2, of the relaxed colouring problem and make the first steps towards establishing a similar hardness jump.

SORT ORDER PROBLEMS IN RELATIONAL DATABASES

1998 ◽  
Vol 09 (04) ◽  
pp. 399-429
Author(s):  
KIM S. LARSEN
Keyword(s):  
Decision Problem ◽  
Relational Databases ◽  
Original Problem ◽  
Linear Time ◽  
Efficient Algorithms ◽  
Relational Algebra ◽  
Np Hard ◽  
Total Work ◽  
Compact Representations ◽  
Np Complete

A relation of degree k can be sorted lexicographically in k! different ways, i.e., according to any one of the possible permutations of the schema of the relation. Such permutations are referred to as sort orders. When evaluating unary and binary relational algebra operators using sort-merge algorithms, sort orders fulfilling the constraints enforced by the operators are chosen for the operand relations. The relations are then sorted according to their assigned sort orders, and the result is obtained by merging. Should the operands already be sorted according to one of the permissible sort orders, then only a merging is required. The sort order of the result will depend on the sort orders of the operands. When evaluating whole relational algebra expressions, the result of one operation will be used as an operand to the next. It is desirable to choose sort orders in such a way that the result of one operation will automatically fulfill the requirements of the next. In general, one would like to find a minimal number of operators in the expression for which this cannot be obtained, bearing in mind the overall goal of minimizing the total work. We show that this problem is NP-hard, and that the corresponding decision problem is NP-complete. However, most simplifications of the original problem give rise to efficient algorithms. In fact, most frequently occurring queries can be analyzed in linear time in the size of the query. This is due to the fact that only a very limited number of subsets of all permutations of schemas can be encountered in the algorithms, which means that compact representations for these subsets can be found.

scholarly journals Symbolic Top-k Planning

2020 ◽  
Vol 34 (06) ◽  
pp. 9967-9974
Author(s):  
David Speck ◽  
Robert Mattmüller ◽  
Bernhard Nebel
Keyword(s):  
Empirical Analysis ◽  
Decision Problem ◽  
State Of The Art ◽  
Np Hard ◽  
Current State ◽  
Novel Approach ◽  
Planning Task ◽  
Ordinary Case

The objective of top-k planning is to determine a set of k different plans with lowest cost for a given planning task. In practice, such a set of best plans can be preferred to a single best plan generated by ordinary optimal planners, as it allows the user to choose between different alternatives and thus take into account preferences that may be difficult to model. In this paper we show that, in general, the decision problem version of top-k planning is PSPACE-complete, as is the decision problem version of ordinary classical planning. This does not hold for polynomially bounded plans for which the decision problem turns out to be PP-hard, while the ordinary case is NP-hard. We present a novel approach to top-k planning, called sym-k, which is based on symbolic search, and prove that sym-k is sound and complete. Our empirical analysis shows that sym-k exceeds the current state of the art for both small and large k.

LABELING POINTS WITH CIRCLES

2001 ◽  
Vol 11 (02) ◽  
pp. 181-195 ◽  
Author(s):  
TYCHO STRIJK ◽  
ALEXANDER WOLFF
Keyword(s):  
Decision Problem ◽  
Constant Factor ◽  
Equal Size ◽  
Np Hard ◽  
Approximation Factor

We present a new algorithm for labeling points with circles of equal size. Our algorithm tries to maximize the label size. It improves the approximation factor of the only known algorithm for this problem by more than 50% to about 1/20. At the same time, our algorithm keeps the O(n log n) time bound of its predecessor. In addition, we show that the decision problem is NP-hard and that it is NP-hard to approximate the maximum label size beyond a certain constant factor.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

Special Cases of the Decision Problem

1951 ◽  
Vol 49 (22) ◽  
pp. 203-221 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Decision Problem ◽  
Special Cases
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