Abstract We propose a general solution approach for min-max-robust counterparts of combinatorial optimization problems with uncertain linear objectives. We focus on the discrete scenario case, but our approach can be extended to other types of uncertainty sets such as polytopes or ellipsoids. Concerning the underlying certain problem, the algorithm is entirely oracle-based, i.e., our approach only requires a (primal) algorithm for solving the certain problem. It is thus particularly useful in case the certain problem is well-studied but its combinatorial structure cannot be directly exploited in a tailored robust optimization approach, or in situations where the underlying problem is only defined implicitly by a given software. The idea of our algorithm is to solve the convex relaxation of the robust problem by a simplicial decomposition approach, the main challenge being the non-differentiability of the objective function in the case of discrete or polytopal uncertainty. The resulting dual bounds are then used within a tailored branch-and-bound framework for solving the robust problem to optimality. By a computational evaluation, we show that our method outperforms straightforward linearization approaches on the robust minimum spanning tree problem. Moreover, using the Concorde solver for the certain oracle, our approach computes much better dual bounds for the robust traveling salesman problem in the same amount of time.

Abstract We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower’s problem. More precisely, adopting the robust optimization approach and assuming that the follower’s profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower’s reaction from the leader’s perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader’s objective function.

Abstract We consider the bilevel minimum spanning tree (BMST) problem where the leader and the follower choose a spanning tree together, according to different objective functions. We show that this problem is NP‐hard, even in the special case where the follower only controls a matching. Moreover, we give some evidence that BMST might even remain hard in case the follower controls only few edges. On the positive side, we present a ‐approximation algorithm for BMST, where is the number of vertices. Moreover, we show that 2‐approximating BMST is fixed‐parameter tractable and that, in case of uniform costs on leader's edges, even solving BMST exactly is fixed‐parameter tractable. We finally consider bottleneck variants of BMST and settle the complexity landscape of all combinations of sum or bottleneck objective functions for the leader and follower, for the optimistic as well as the pessimistic setting.

Binary programs with a quadratic objective function are NP-hard in general, even if the linear optimization problem over the same feasible set is tractable. In this paper, we address such problems by computing quadratic global underestimators of the objective function that are separable but not necessarily convex. Exploiting the binary constraint on the variables, a minimizer of the separable underestimator over the feasible set can be computed by solving an appropriate linear minimization problem over the same feasible set. Embedding the resulting lower bounds into a branch-and-bound framework, we obtain an exact algorithm for the original quadratic binary program. The main practical challenge is the fast computation of an appropriate underestimator, which in our approach reduces to solving a series of semidefinite programs. We exploit the special structure of the resulting problems to obtain a tailored coordinate-descent method for their solution. Our extensive experimental results on various quadratic combinatorial optimization problems show that our approach outperforms both CPLEX and the related QCR method as well as the SDP-based software BiqCrunch on instances of the quadratic shortest path problem and the quadratic assignment problem.

The critical node selection problem (CNP) has important applications in telecommunication, supply chain design, and disease propagation prevention. In practice, the weights on the connections are often uncertain or hard to estimate. For this reason, robust optimization approaches have been considered recently for CNP. In this article, we address very general uncertainty sets, only requiring a linear optimization oracle for the set of potential scenarios. In particular, we can deal with discrete scenario based uncertainty, gamma uncertainty, and ellipsoidal uncertainty. For this general class of robust critical node selection problems, we propose an exact solution method based on Benders decomposition. The Benders subproblem, which in our approach is a robust optimization problem, is efficiently solved by applying the Floyd-Warshall algorithm. The presented approach is tested on 384 instances based on Forest-Fire, Barabási-Albert, Erdős-Rényi, and Watts-Strogatz graphs with different number of nodes and edges, where running times are compared to CPLEX being directly applied to the robust problem formulation. The computational results show the advantage of the proposed approach in handling the uncertainty thus outperforming CPLEX most notably for the ellipsoidal uncertainty cases.

The quadratic linear ordering problem naturally generalizes various optimization problems such as bipartite crossing minimization or the betweenness problem, which includes linear arrangement. These problems have important applications, e.g., in automatic graph drawing and computational biology. We present a new polyhedral approach to the quadratic linear ordering problem that is based on a linearization of the quadratic objective function. Our main result is a reformulation of the 3-dicycle inequalities using quadratic terms. After linearization, the resulting constraints are shown to be face-inducing for the polytope corresponding to the unconstrained quadratic problem. We use this result both within a branch-and-cut algorithm and within a branch-and-bound algorithm based on semidefinite programming. Experimental results for bipartite crossing minimization show that this approach clearly outperforms other methods.

The aim of automatic graph drawing is the development of algorithms for creating nice and easily readable layouts of abstractly given graphs. For the special case of rooted trees of unbounded degree, John Q. Walker II presented a drawing algorithm in this journal in 1990. This algorithm is an extension of the Reingold–Tilford algorithm. It yields very good results and is therefore widely used. Furthermore, it is widely assumed to run in linear time, as the author claims in his article. However, the algorithm in its presented form clearly needs quadratic runtime. We explain the reasons for that and state a revised algorithm that creates the same layouts in linear time. Copyright © 2006 John Wiley & Sons, Ltd.