A New Branch and Bound Algorithm for the Cyclic Bandwidth Problem

Abstract

The Cyclic Bandwidth problem (CB) for graphs consists in labeling the vertices of a guest graph G by distinct vertices of a host cycle C (both of order n) in such a way that the maximum distance in the cycle between adjacent vertices in G is minimized. The CB problem arises in application areas like VLSI designs, data structure representations and interconnection networks for parallel computer systems. In this paper a new Branch and Bound (B&B) algorithm for the CB problem is introduced. Its key components were carefully devised after an in-depth analysis of the given problem. The practical effectiveness of this algorithm is shown through extensive experimentation over 20 standard graphs. The results show that the proposed exact algorithm attains the lower bounds for these graphs (of order n ≤ 40) expending a reasonable computational time.

An Iterated Three-Phase Search Approach for Solving the Cyclic Bandwidth Problem

“…To our knowledge, there are only three published algorithms on solving the CBP. In [14], a branch and bound algorithm was proposed that can solve some standard instances (like path, mesh and cycle) of small sizes limited to 40 vertices. To handle larger instances, a heuristic algorithm based on the tabu search metaheuristic (named TScb) was presented in [15].…”

Tabu search for the cyclic bandwidth problem

“…The CB problem has been exactly solved for twenty small instances (n < 40) using a branch & bound algorithm (B&B) recently proposed by Romero-Monsivais et al [12]. However, this algorithm becomes impractical when the number of vertices n in the studied graph increases, since the size of the search space suffers a combinatorial explosion.…”

“…Most of the graphs in our second set (24 of them) were previously used by Duarte et al [13] and Lozano et al [14] as benchmarks instances for the antibandwidth problem [1]. The rest of the instances (4 graphs) were collected by us to complement this set with small graphs that can be solved exactly using the B&B algorithm proposed in [12].…”

“…The first one is composed of 85 standard graphs with known optimal solutions from seven different families including paths, cycles, meshes, trees, hypercubes and caterpillars. Certain of them were previously used to evaluate a branch & bound (B&B) algorithm for the CB problem [12]. The second set contains 28 graphs produced from real-world scientific and engineering applications, some of which were recently used as benchmark instances for other graph labeling problem [13,14].…”

Branch and bound algorithm with applications to robust stability

“…Related branch and bound approaches in robust stability are Bemporad et al [7], Romero-Monsivais et al [24], or Sakizlis et al [26], see also [22]. Here we discuss a novel branch and bound algorithm, which has three key ingredients leading to a significantly better performance.…”