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The k-th Roman domination problem is polynomial on interval graphs

JOURNAL OF COMBINATORIAL OPTIMIZATION 2024年第3期48卷 1-14页

作者： Li, Peng Chongqing Univ Technol 69 Hongguang Rd Chongqing Peoples R China

Let G be some simple graph and k be any positive integer. Take h : V ( G )- {0,1, , ... ,k+1} and v is an element of V (G), let AN(h)(v) denote the set of vertices w is an element of N-G (v) with h (w) >= 1. Let AN(h)[v] h [v] = AN(h)(v) boolean OR {v}. The function h is a [k]-Roman dominating function of G if h ( AN(h)[v]) >= |AN(h)(v)|+ k holds for any v is an element of V(G). The minimum weight of such a function is called the k-th Roman Domination number of G, which is denoted by gamma(kR) (G). In 2020, Banerjee et al. presented linear time algorithms to compute the double Roman domination number on proper interval graphs and block graphs. They posed the open question that whether there is some polynomial time algorithm to solve the double Roman domination problem on interval graphs. It is argued that the interval graph is a nontrivial graph class. In this article, we design a simple dynamic polynomial time algorithm to solve the k-th Roman domination problem on interval graphs for each fixed integer k > 1.

关键词： Domination Roman domination k-th Roman domination polynomial time algorithm Interval graphs

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Energy-aware lot sizing problem: Complexity analysis and exact algorithms

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INTERNATIONAL JOURNAL OF PRODUCTION ECONOMICS 2018年 203卷 254-263页

作者： Rapine, Christophe Goisque, Guillaume Akbalik, Ayse Univ Lorraine LCOMS Technopole Metz Metz France Univ Lorraine Metz France

The single-item lot sizing problem under a periodic energy limitation is considered in this paper. Identical and parallel capacitated machines constitute the production system, each one consuming a certain amount of energy when being switched on, when reserved, and when producing. We consider a cost for starting-up the machines, a reservation cost to keep the machines ready for production, in addition to classical lot sizing costs, such as, joint setup cost, unit production cost and unit holding cost, all being time-dependent. Besides the classical lot sizing decisions of how much and in which periods to produce, we have to decide the number of machines to switch on and to switch off in each period. We show that this problem is NP-hard even under restricted conditions. In contrast, assuming stationary energy parameters, we propose two polynomial time dynamic programming algorithms to solve the problem to optimality. The first algorithm is proposed for the case with null setup cost, null reservation cost and null reservation energy consumption, and runs in O ((MT4)-T-5) time, with M being the number of machines and T the number of periods. We show that we can extend this algorithm to solve the generalized version of the problem in time complexity O ((MT6)-T-6).

关键词： Lot sizing problem Energy Capacitated parallel machines polynomial time algorithm Complexity

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Burning a graph is hard

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DISCRETE APPLIED MATHEMATICS 2017年 232卷 73-87页

作者： Bessy, Stephane Bonato, Anthony Janssen, Jeannette Rautenbach, Dieter Roshanbin, Elham LIRMM Montpellier France Ryerson Univ Dept Math Toronto ON M5B 2K3 Canada Dalhousie Univ Dept Math & Stat Halifax NS B3H 3J5 Canada Ulm Univ Inst Optimizat & Operat Res Ulm Germany

Graph burning is a model for the spread of social contagion. The burning number is a graph parameter associated with graph burning that measures the speed of the spread of contagion in a graph;the lower the burning number, the faster the contagion spreads. We prove that the corresponding graph decision problem is NP-complete when restricted to acyclic graphs with maximum degree three, spider graphs and path-forests. We provide polynomial time algorithms for finding the burning number of spider graphs and path forests if the number of arms and components, respectively, are fixed. Finally, we describe a polynomial time approximation algorithm with approximation factor 3 for general graphs. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Graph burning Computational complexity NP-complete 3-partition polynomial time algorithm Approximation algorithm

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A well-solvable special case of the bounded knapsack problem

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OPERATIONS RESEARCH LETTERS 2011年第2期39卷 118-120页

作者： Deineko, Vladimir G. Woeginger, Gerhard J. Tech Univ Eindhoven Dept Math & Comp Sci NL-5600 MB Eindhoven Netherlands Univ Warwick Warwick Business Sch Coventry CV4 7AL W Midlands England

We identify a polynomially solvable special case of the bounded knapsack problem that is characterized by a set of simple inequalities relating item weight ratios to item profit ratios. Our result generalizes and extends a corresponding result of Zukerman, et al. [M. Zukerman, L. Jia, T. Neame, G.J. Woeginger, A polynomially solvable special case of the unbounded knapsack problem, Operations Research Letters 29 (2001) 13-16] for the unbounded knapsack problem. (c) 2011 Elsevier B.V. All rights reserved.

关键词： Combinatorial optimization Computational complexity polynomial time algorithm Cross ratio ordered

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Some scheduling problems with deteriorating jobs and learning effects

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COMPUTERS & INDUSTRIAL ENGINEERING 2010年第1期58卷 25-28页

作者： Yang, Dar-Li Kuo, Wen-Hung Natl Formosa Univ Dept Informat Management Yunlin 632 Taiwan

This paper considers some scheduling problems with deteriorating jobs and learning effects. The following objective functions are considered: the makespan, the total completion times, and the total absolute differences in completion times. Several polynomial time algorithms are proposed to optimally solve the single-machine scheduling problems. Finally, we show that several special cases of the flowshop scheduling problems remain polynomially solvable under the proposed model. (C) 2009 Published by Elsevier Ltd.

关键词： Scheduling Single machine Deteriorating jobs Learning effects Flowshop polynomial time algorithm

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The Effects of Multitasking on Operations Scheduling

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PRODUCTION AND OPERATIONS MANAGEMENT 2015年第8期24卷 1248-1265页

作者： Hall, Nicholas G. Leung, Joseph Y. -T. Li, Chung-Lun Ohio State Univ Fisher Coll Business Columbus OH 43210 USA New Jersey Inst Technol Dept Comp Sci Newark NJ 07102 USA Hong Kong Polytech Univ Dept Logist & Maritime Studies Kowloon Hong Kong Peoples R China

This study considers a typical scheduling environment that is influenced by the behavioral phenomenon of multitasking. Under multitasking, the processing of a selected job suffers from interruption by other jobs that are available but unfinished. This situation arises in a wide variety of applications;for example, administration, manufacturing, and process and project management. Several classical solution methods for scheduling problems no longer apply in the presence of multitasking. The solvability of any scheduling problem under multitasking is no easier than that of the corresponding classical problem. We develop optimal algorithms for some fundamental and practical single machine scheduling problems with multitasking. For other problems, we show that they are computationally intractable, even though in some cases the corresponding problem in classical scheduling is efficiently solvable. We also study the cost increase and value gained due to multitasking. This analysis informs companies about how much it would be worthwhile to invest in measures to reduce or encourage multitasking.

关键词： scheduling multitasking polynomial time algorithm cost and value of multitasking

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GENERATING BOOLEAN MU-EXPRESSIONS

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ACTA INFORMATICA 1995年第2期32卷 171-187页

作者： EITER, T Information Systems Department Technical University of ViennaPaniglgasse 16 A-1040 Vienna Austria AT

In this paper, we consider the class of Boolean mu-functions, which are the Boolean functions definable by mu-expressions (Boolean expressions in which no variable occurs more than once). We present an algorithm which transforms a Boolean formula E into an equivalent mu-expression-if possible-in time linear in parallel to E parallel to times 2(nm), where parallel to E parallel to is the size of E and n(m) is the number of variables that occur more than once in E. As an application, we obtain a polynomial time algorithm for Mundici's problem of recognizing mu-functions from k-formulas [17]. Furthermore, we show that recognizing Boolean mu-functions is co-NP-complete for functions essentially dependent on all variables and we give a bound close to co-NP for the general case.

关键词： polynomial time Boolean Function time Linear time algorithm polynomial time algorithm

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Reconstructing a Phylogenetic Level-1 Network from Quartets

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BULLETIN OF MATHEMATICAL BIOLOGY 2014年第10期76卷 2517-2541页

作者： Keijsper, J. C. M. Pendavingh, R. A. Eindhoven Univ Technol NL-5600 MB Eindhoven Netherlands

We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on taxa from the set of all quartets containing a certain fixed taxon, in time. We also present a more general method which can handle more diverse quartet data, but which takes time. Both methods proceed by solving a certain system of linear equations over the two-element field . For a general dense quartet set, i.e. a set containing at least one quartet on every four taxa, our algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an -sized certificate of inconsistency otherwise. This answers a question raised by Gambette, Berry and Paul regarding the complexity of reconstructing a level-1 network from a dense quartet set, and more particularly regarding the complexity of constructing a cyclic ordering of taxa consistent with a dense quartet set.

关键词： Phylogenetic networks Cyclic ordering Quartets Level-1 networks Supernetwork method polynomial time algorithm

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Local structure when all maximal independent sets have equal weight

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SIAM JOURNAL ON DISCRETE MATHEMATICS 1998年第4期11卷 644-654页

作者： Caro, Y Ellingham, MN Ramey, JE Univ Haifa Oranim Sch Educ Dept Math IL-36006 Tivon Israel Vanderbilt Univ Dept Math Nashville TN 37240 USA Cumberland Coll Dept Math Williamsburg KY 40769 USA

In many combinatorial situations there is a notion of independence of a set of points. Maximal independent sets can be easily constructed by a greedy algorithm, and it is of interest to determine, for example, if they all have the same size or the same parity. Both of these questions may be formulated by weighting the points with elements of an abelian group, and asking whether all maximal independent sets have equal weight. If a set is independent precisely when its elements are pairwise independent, a graph can be used as a model. The question then becomes whether a graph, with its vertices weighted by elements of an abelian group, is well-covered, i.e., has all maximal independent sets of vertices with equal weight. This problem is known to be co-NP-complete in general. We show that whether a graph is well-covered or not depends on its local structure. Based on this, we develop an algorithm to recognize well-covered graphs. For graphs with n vertices and maximum degree Delta, it runs in linear time if Delta is bounded by a constant, and in polynomial time if Delta = O((3)root log n). We mention various applications to areas including hypergraph matchings and radius k independent sets. We extend our results to the problem of determining whether a graph has a weighting which makes it well-covered.

关键词： well-covered graph maximal independent set local structure polynomial time algorithm recognition algorithm hypergraph matching independence system

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The complexity of the equivalence and equation solvability problems over meta-Abelian groups

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JOURNAL OF ALGEBRA 2015年 433卷 208-230页

作者： Horvath, Gabor Univ Debrecen Inst Math H-4010 Debrecen Hungary

We provide polynomial time algorithms for deciding equation solvability and identity checking over groups that are semidirect products of two finite Abelian groups. Our main method is to reduce these problems to the sigma equation solvability and sigma equivalence problems over modules for commutative unital rings. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Finite meta-Abelian groups Equivalence Equation solvability Computational complexity polynomial time algorithm Module equivalence Module equation solvability

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