Combinatorial Optimization Problems with Testing

In stochastic versions of combinatorial optimization problems, the goal is to optimize an objective function under constraints that specify the feasible solutions, when some parameters involve uncertainty. We focus on maximizing a linear objective ∑N i=1Wixi, where x1, . . . , xN are the decision variables, W1, . . . ,WN are mutually independent random coefficients, and the constraints are deterministic. This formulation captures many real-world problems and has been extensively studied in Operations Research. In most prior work, the uncertainty in the objective is modeled by unknown distributions and various mechanisms are used to acquire information on these distributions. In contrast, we assume that the distributions are known and use a mechanism to acquire information on realizations. We consider problems where a decision-maker can test any desired coefficient Wi, which reveals its realization wi and incurs a fixed cost c > 0. Testing a coefficient reduces uncertainty, which can only improve the optimization, but it may or may not compensate for the testing cost. The decision-maker can test coefficients in a sequential and possibly adaptive manner, and then has to return a feasible solution based on the current information. A policy is a set of rules that determines the sequential decisions of the decision-maker in every possible situation. The goal is to find a policy that maximizes the expected profit, where profit is defined as the returned solution’s objective value minus all the testing costs. A policy that obtains optimal expected profit can usually be computed by dynamic programming. However, if the dimension of the problem is high, this method becomes intractable. As an alternative, we study policies that are myopic – a myopic policy decides whether to test coefficients based on the marginal profit from exactly one test, without considering possible future effects, i.e., it makes decisions based on a limited horizon of one test. Compared to dynamic programming, myopic policies are simpler and easier to compute, but more restrictive and thus possibly suboptimal. We show that myopic policies can actually achieve optimal expected profit for a number of interesting problems: 1. Selection with testing, where the feasible solutions represent selecting exactly one of the N coefficients and each decision variable xi indicates whether coefficient Wi is selected; 2. Maximum spanning tree with testing, where the feasible solutions represent spanning trees in a given graph G and each xi indicates whether the corresponding edge is included in the tree; 3. Linear optimization over a polymatroid with testing, where the feasible solutions are defined by a given submodular function. For more general optimization problems with testing, we derive weaker results in the form of sufficient conditions under which a myopic rule for deciding whether to stop testing is optimal.