A data-science team is building a cost-bounded nationwide fiber-optic backbone. For every candidate link (i,j) it defines a binary variable x_ij that equals 1 if the link is installed, 0 otherwise. The network must (1) connect all K data-center nodes in a single component, (2) contain no cycles, and (3) respect a given budget. Which additional modeling device BEST guarantees requirements (1) and (2) in a mixed-integer linear program that can be handled directly by a branch-and-bound MILP solver?
Dualize the budget and connectivity constraints with Lagrange multipliers and solve the resulting unconstrained objective using gradient descent.
Replace the connectivity constraint with an objective that minimizes the average shortest-path length between all node pairs, dropping the cycle restriction.
Introduce single-commodity flow conservation constraints from a chosen root node: supply K-1 units at the root, demand one unit at every other node, and allow flow only on edges where x_ij = 1.
Relax x_ij to continuous values in and add a quadratic penalty term for every three-edge cycle to discourage loops.
Adding single-commodity flow constraints rooted at an arbitrarily chosen source node forces exactly K−1 units of flow to leave the root and exactly one unit to reach every other node; flow is allowed only on edges where x_ij = 1. These balance constraints simultaneously ensure connectivity (a path must exist from the root to every node) and eliminate cycles (a connected graph with K nodes and K−1 edges is a tree). Because all constraints remain linear and x_ij stay binary, the resulting model is a standard MILP that commercial branch-and-bound solvers can optimize. Continuous relaxations with penalty terms may still yield fractional or disconnected solutions, minimizing average path length does not prohibit cycles, and dualizing the constraints for gradient descent removes integrality, so none of those alternatives reliably produce a feasible spanning-tree topology.
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What are single-commodity flow constraints and how do they ensure connectivity and no cycles?
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Why are the other modeling approaches insufficient for meeting the network requirements?