Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInteger Linear Programming Formulations for Double Roman Domination Problem
64
0
0.0
(
0
)
Added by
Yongtang Shi
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
For a graph $G= (V,E)$, a double Roman dominating function (DRDF) is a function $f : V to {0,1,2,3}$ having the property that if $f (v) = 0$, then vertex $v$ must have at least two neighbors assigned $2$ under $f$ or {at least} one neighbor $u$ with $f (u) = 3$, and if $f (v) = 1$, then vertex $v$ must have at least one neighbor $u$ with $f (u) ge 2$. In this paper, we consider the double Roman domination problem, which is an optimization problem of finding the DRDF $f$ such that $sum_{vin V} f (v)$ is minimum. We propose {five integer linear programming (ILP) formulations and one mixed integer linear programming formulation with polynomial number of constraints for this problem. Some additional valid inequalities and bounds are also proposed for some of these formulations.} Further, we prove that {the first four models indeed solve the double Roman domination problem, and the last two models} are equivalent to the others regardless of the variable relaxation or usage of a smaller number of constraints and variables. Additionally, we use one ILP formulation to give an $H(2(Delta+1))$-approximation algorithm. All proposed formulations and approximation algorithm are evaluated on randomly generated graphs to compare the performance.
rate research
Read More
For a graph $G,$ the set $D subseteq V(G)$ is a porous exponential dominating set if $1 le sum_{d in D} left( 2 right)^{1-dist(d,v)}$ for every $v in V(G),$ where $dist(d,v)$ denotes the length of the shortest $dv$ path. The porous exponential dominating number of $G,$ denoted $gamma_e^*(G),$ is the minimum cardinality of a porous exponential dominating set. For any graph $G,$ a technique is derived to determine a lower bound for $gamma_e^*(G).$ Specifically for a grid graph $H,$ linear programing is used to sharpen bound found through the lower bound technique. Lower and upper bounds are determined for the porous exponential domination number of the King Grid $mathcal{K_n},$ the Slant Grid $mathcal{S_n},$ and the $n$-dimensional hypercube $Q_n.$
A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with dual tree-depth d and largest entry D are solvable in time g(d,D)poly(n) for some function g. However, the dual tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth, and thus does not reflect its geometric structure. We prove that the minimum dual tree-depth of a row-equivalent matrix is equal to the branch-depth of the matroid defined by the columns of the matrix. We design a fixed parameter algorithm for computing branch-depth of matroids represented over a finite field and a fixed parameter algorithm for computing a row-equivalent matrix with minimum dual tree-depth. Finally, we use these results to obtain an algorithm for integer programming running in time g(d*,D)poly(n) where d* is the branch-depth of the constraint matrix; the branch-depth cannot be replaced by the more permissive notion of branch-width.
Property A is a form of weak amenability for groups and metric spaces introduced as an approach to the famous Novikov higher signature conjecture, one of the most important unsolved problems in topology. We show that property A can be reduced to a sequence of linear programming optimization problems on finite graphs. We explore the dual problems, which turn out to have interesting interpretations as combinatorial problems concerning the maximum total supply of flows on a network. Using isoperimetric inequalities, we relate the dual problems to the Cheeger constant of the graph and explore the role played by symmetry of a graph to obtain a striking characterization of the difference between an expander and a graph without property A. Property A turns out to be a new measure of connectivity of a graph that is relevant to graph theory. The dual linear problems can be solved using a variety of methods, which we demonstrate on several enlightening examples. As a demonstration of the power of this linear programming approach we give elegant proofs of theorems of Nowak and Willett about graphs without property A.
Ahlswede and Katona (1977) posed the following isodiametric problem in Hamming spaces: For every $n$ and $1le Mle2^{n}$, determine the minimum average Hamming distance of binary codes with length $n$ and size $M$. Fu, Wei, and Yeung (2001) used linear programming duality to derive a lower bound on the minimum average distance. However, their linear programming approach was not completely exploited. In this paper, we improve Fu-Wei-Yeungs bound by finding a better feasible solution to their dual program. For fixed $0<ale1/2$ and for $M=leftlceil a2^{n}rightrceil $, our feasible solution attains the asymptotically optimal value of Fu-Wei-Yeungs dual program as $ntoinfty$. Hence for $0<ale1/2$, all possible asymptotic bounds that can be derived by Fu-Wei-Yeungs linear program have been characterized. Furthermore, noting that the average distance of a code is closely related to weights of Fourier coefficients of a Boolean function, we also apply the linear programming technique to prove bounds on Fourier weights of a Boolean function of various degrees.
This paper discusses the odds problem, proposed by Bruss in 2000, and its variants. A recurrence relation called a dynamic programming (DP) equation is used to find an optimal stopping policy of the odds problem and its variants. In 2013, Buchbinder, Jain, and Singh proposed a linear programming (LP) formulation for finding an optimal stopping policy of the classical secretary problem, which is a special case of the odds problem. The proposed linear programming problem, which maximizes the probability of a win, differs from the DP equations known for long time periods. This paper shows that an ordinary DP equation is a modification of the dual problem of linear programming including the LP formulation proposed by Buchbinder, Jain, and Singh.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
