Adding integer constraints converts a convex continuous feasible region into a non-convex set of isolated points, which eliminates the smooth geometric properties needed to find solutions in polynomial time. The Branch and Bound algorithm manages this complexity by dividing the original search space into a tree of continuous subproblems and calculating objective bounds at each node. By comparing these continuous bounds against the best known integer solution, the solver systematically prunes vast subtrees of unpromising choices without having to explore every potential solution.

Integer constraints turn easy continuous problems into NP-hard challenges by breaking smooth geometric shapes into isolated points, causing a combinatorial explosion. Branch and Bound navigates this difficulty by relaxing those integer rules to find optimistic bounds, allowing the algorithm to systematically prune unpromising sub-trees without checking every combination.

Adding integer constraints makes the problem harder because the solution must satisfy discrete choices, such as 0 or 1, instead of allowing any fractional value. This creates many possible combinations, making ILP problems generally NP-hard. Branch and Bound handles this by dividing the problem into smaller subproblems, using bounds from their LP relaxations to eliminate branches that cannot lead to a better integer solution.

Adding integer constraints transforms an easy continuous Linear Programming problem into an NP-hard problem because it destroys the continuous, convex search space, turning it into a non-convex set of isolated points that causes the number of possible solutions to scale exponentially with each added discrete variable. Branch and Bound navigates this exponential complexity through a systematic divide-and-conquer strategy: it branches by splitting fractional variables into sub-nodes, computes bounds at each node by solving the continuous LP relaxation to determine the best possible outcome for that sub-tree, and prunes (discards) any sub-tree whose bound is worse than the best known valid integer solution, allowing the solver to skip vast regions of the search space without evaluating every individual combination.