A Branch and Bound Algorithm for Counting Independent Sets on Grid Graphs

Abstract

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The Branch-and-Bound Algorithm in Optimizing Mathematical Programming Models to Achieve Power Grid Observability

“…All optimization problems are written in YALMIP using symbolic syntax, with a strict declaration of the binary nature of the decision variables [26][27][28]. We initially optimize the 0/1 ILP model with a BBA [64][65][66][67][68] implemented by the bmibnb built-in solver in the YALMIP program in collaboration with external optimizer routines to find optimality [69][70][71][72][73][74][75][76][77][78][79].…”

“…6.1. Termination Criteria of BBA for Convergence of the Optimization Run [64][65][66][67] BBA is an in-front algorithm that is utilized to solve the proposed methods due to a good enough stopping criterion to fully accomplish the optimality criterion. As a result, the integer solver successfully returns a globally optimal point, avoiding being trapped in a local minimum or a suboptimal point [74].…”

Counting Rules for Computing the Number of Independent Sets of a Grid Graph

“…Meanwhile, for Figure 2, i(G) = 42 + 21 = 63. (42,30) De Ita et al [10] show how calculating the number of independent sets of a grid is possible by taking the Hamiltonian path as a guide while the two counting rules, the Fibonacci rule and the subtracted rule, are applied. Nevertheless, the time complexity of the aforementioned process is exponentially proportional to the maximum number of frond edges in any row of the grid.…”

“…Different exponential algorithms have been designed to compute the M-S index on square grids G m,n , starting with the super-exponential algorithm based on the transfer matrix method [5,6]. More recently, De Ita et al [10] propose an exponential-time algorithm based on the use of computing threads for computing i(G m,n ). Meanwhile, in this work, a branch-and-bound algorithm is introduced, whose time complexity is of order O((1.1939) (m−1) * (n−1) * poly(m, n)), where poly(m, n) represents a function with a polynomial-time complexity, and m and n are the dimensions of the grid graph.…”

“…Notice that all the above algorithms are of exponential order, and the last two algorithms have a different way of measuring the hard exponential complexity of counting independent sets on square grids, since the algorithm in [10] has an exponential growth based on the number of squares by row (or by columns) in the grid. However, for the algorithm presented here, the exponential growth depends on the number of applications of the ramification vertex rule until achieving basic case subgrids.…”