Branch and bound algorithm is used to solve mathematical and discrete optimization problems. This is an important technique because it provides a very flexible design for both combinatorial and mathematical optimization. This algorithm has been widely used by researchers for a number of years and has proven to be a good solution to a large number of problem.  So, what is branch and bound?

Variations

Using the branch and bound algorithm is a very effective method of solving combinatorial optimization problems. It is one of the most common methodologies used for NP-hard optimization problems. This algorithm consists of three algorithmic components: a search strategy, a pruning rule, and a global upper bound on the cost of the optimal solution. The search strategy relies on efficient estimation of the lower and upper bounds. When the admissible weight functions cannot improve on the stored solution, the search tree is pruned. The pruning rule is an important part of the branch and bound framework procedure. It is a set of rules that can be applied to the subproblems to weed out regions of the search space. The branching strategy spawns subproblems and forces them to take certain constraints. A common branching strategy is the standard integer branching rule. This strategy creates two new branches. However, this strategy does not consider all variables. A second branching strategy is the wide branching strategy. This technique explores all possible paths for each variable. This strategy leads to subproblems that are not optimal and require a lot of work to find a good solution. The branching strategy can be fixed by a lexicographic ordering rule. The global control value on the quality of the solution increases over time. The algorithm completeness has no effect on this. The higher the number of iterations, the more RAM the algorithm will use. Another common branching strategy is the generalized assignment problem strategy. This strategy compares all possible subproblems within a particular level of a tree. The advantage of this strategy is that it does not disrupt the structure of the pricing problem.

Search phase

Using branch and bound, one can solve combinatorial optimization problems quickly. The algorithm works by enumerating candidate solutions and then exploring subproblems to find the minimal path to an optimal solution. The method is often used in settings with adequate memory. The algorithm has two phases, which are the search phase and the verification phase. In the search phase, the algorithm is searching for a candidate solution that is at least as good as the current incumbent. It will also be checking the branches against upper and lower bounds for the optimal solution. If the branch cannot produce a better solution, it will be deleted. In the verification phase, the branching strategy is used to guide the algorithm to a feasible solution. The underlying branching rules are a set of constraints that limit the decision-making process. During this phase, the measure-of-best function is computed for each subproblem and used to select the next subproblem to explore. The first research question concerns the development of new search strategies. These strategies will be tested through numerical experiments. The second research question involves the study of branching strategies. These strategies will be used to determine whether they are effective in solving NP-hard problems. The third research question focuses on the unified theory of pruning rules. This theory is expected to provide more accurate theoretical bounds on the performance of branch and bound algorithms. The search phase is impacted by the branching strategy and pruning rules. The best-first search picks the node that has the highest potential improvement over the current objective value. It uses the measure-of-best function to calculate the value of each unexplored subproblem. This is done to ensure that the measure-of-best function is not overestimating the best solution.

Verification phase

During the verification phase, the branch and bound algorithm does the deed. It enumerates the best possible solutions for a given problem, and discards any branch that doesn’t yield a better solution. The enumerated solutions are stored in a tree structure, where the tree itself is considered the central node. The shortest path from any node to the next one is known as a branching rule, which is a key component of the B&B algorithm. The B&B algorithm has two phases, the search phase and the verification phase. The search phase involves the use of state-space searches, a node in the tree of feasible solutions. The best solution is selected, and the best candidate solution is then checked against the lower and upper bounds to determine the validity of the algorithm. The verification phase involves the identification of optimal solutions using pruning rules. The main goal of this phase is to identify the smallest number of unhelpful subproblems. These subproblems are the ones that don’t lead to optimal solutions, and require extensive work to discover. The B&B algorithm is a jack of all trades that is used in a variety of combinatorial optimization problems. It has the ability to efficiently find exact solutions to a wide variety of problems. The algorithm is most commonly employed to solve NP-hard optimization problems. It is a simple but effective methodology that has been employed for decades. It is often the backbone of a decision support system that aims to aid humans in the analysis of commercial data. It is also used in the production of zero-point predictions, which are used to detect phase-controlled faults.

Limitations

Among the numerous optimization algorithms, the Branch and Bound method is one of the most used. It’s applied to solve a variety of optimization problems, from combinatorial to discrete problems.  It can be known as the branch and bound problem. The Branch and Bound algorithm finds the smallest path to an optimal solution. It does so by systematically enumerating all possible solutions. The objective is to find a subset of solutions that will minimize the cost of the computed solution. When an optimal solution is found, the algorithm improves it by finding a minimum successor.  This is called least cost branch and bound. The Branch and Bound method is particularly useful for a large number of combinatorial optimization problems. It can locate the global optimum for a large percentage of problems. In particular, it can locate the global optimum for 85% of problems in the “glass-box” case. The algorithm uses a rooted decision tree, which represents the set of candidate solutions. The search tree is then pruned if it is not possible to improve on the stored solution. It’s also important to note that the problem’s domain should be discrete. In the worst case scenario, the number of nodes in the decision tree may be too large. This leads to the risk that partial solutions will be left out. In this scenario, the algorithm may explore all possible permutations, which can take an exponential amount of time. Another limitation of the Branch and Bound algorithm is that it is not parallelized. For example, if a problem has many subproblems, it can be challenging to parallelize the algorithm. In addition, the exploration of even a small portion of nodes can be resource-intensive. In the case of black-box constraints, the number of feasible solutions grows exponentially with the size of the network. This is problematic for sampling.

Research questions

Generally speaking, the branch and bound algorithm is a good fit for NP-hard optimization problems. It is often used to solve combinatorial optimization problems quickly and efficiently. However, its time-consuming nature makes it difficult to implement it in a parallelized fashion. Aside from the branch and bound algorithm itself, there are a number of other research questions that are associated with it. The first one is about the smallest number of useful subproblems that are possible in a combinatorial problem. The second one relates to the study of the most efficient ways to prune the tree. The third question concerns the most effective way to find a minimal path to an optimal solution. The branch and bound algorithm is just one of several techniques that are employed to solve these problems. Another method is the decomposition technique, which is used to break down a continuous problem into a dual of the original. This is a common practice for solving linear programming problems. The branch and bound framework is an umbrella term for a family of algorithms that have been used to solve a wide range of combinatorial optimization problems. It includes a tree search strategy, an implicitly enumerative branch scheme, and a precedence-based lower bound. It also incorporates devices for checking dynamic programming dominance in the search tree. The branch and bound method was first proposed in 1960 by Alison Doig and Ailsa Land. The simplest example of this is the use of a tree search to find the minimum feasible route to an optimal solution. The same algorithm can also be used to solve a more general combinatorial optimization problem. The branch and bound method is not without its limitations, but it has a reputation for being fast. In the worst case scenario, there may be too many branches for the algorithm to explore. This can be remedied by incorporating a learning phase in the process. If you like what you read, check out our other algorithm articles here.