Britta Peis

We propose a model for recoverable robust optimization with commitment . Given a combinatorial optimization problem and uncertainty about elements that may fail, we ask for a robust solution that, after the failing elements are revealed, can be augmented in a limited way. However, we commit to preserve the non-failing elements of the initial solution. We settle the computational complexity of such a robust counterpart of various classical polynomial-time solvable combinatorial optimization problems. We show, for the weighted matroid independent set problem, that an optimal solution to the nominal problem is also optimal for its robust counterpart. Indeed, matroids are provably the only structures with this strong property. Robust counterparts of other problems are -hard such as the matching problem and the stable set problem, even in bipartite graphs. However, we establish polynomial-time algorithms for the robust counterparts of the unweighted stable set problem in bipartite graphs and the weighted stable set problem in interval graphs, also known as the interval scheduling problem.

In train routing, the headway is the minimum distance that must be maintained between successive trains for safety and robustness. We introduce a model for train routing that requires a fixed headway to be maintained between trains, and study the problem of minimizing the makespan, i.e., the arrival time of the last train, in a single-source single-sink network. For this problem, we first show that there exists an optimal solution where trains move in convoys, that is, the optimal paths for any two trains are either the same or are arc-disjoint. Via this insight, we are able to reduce the approximability of our train routing problem to that of the min-max disjoint paths problem, which asks for a collection of disjoint paths where the maximum length of any path in the collection is as small as possible. While min-max disjoint paths inherits a strong inapproximability result on directed acyclic graphs from the multi-level bottleneck assignment problem, we show that a natural greedy composition approach yields a logarithmic approximation in the number of disjoint paths for series-parallel graphs. We also present an alternative analysis of this approach that yields a guarantee depending on how often the decomposition tree of the series-parallel graph alternates between series and parallel compositions on any root-leaf path.

We consider dynamic auctions for finding Walrasian equilibria in markets with indivisible items and strong gross substitutes valuation functions. Each price adjustment step in these auction algorithms requires finding an inclusion-wise minimal maximally overdemanded set or an inclusion-wise minimal maximally underdemanded set at the current prices. Both can be formulated as a submodular function minimization problem. We observe that minimizing this submodular function corresponds to a polymatroid sum problem, and using this viewpoint, we give a fast and simple push-relabel algorithm for finding the required sets. This improves on the previously best running time of Murota, Shioura and Yang (ISAAC 2013). Our algorithm is an adaptation of the push-relabel framework by Frank and Miklós (JJIAM 2012) to the particular setting. We obtain a further improvement for the special case of unit-supplies. We further show the following monotonicity properties of Walrasian prices: both the minimal and maximal Walrasian prices can only increase if supply of goods decreases, or if the demand of buyers increases. This is derived from a fine-grained analysis of market prices. We call packing prices a price vector such that there is a feasible allocation where each buyer obtains a utility maximizing set. Conversely, by covering prices we mean a price vector such that there exists a collection of utility maximizing sets of the buyers that include all available goods. We show that for strong gross substitutes valuations, the component-wise minimal packing prices coincide with the minimal Walrasian prices and the component-wise maximal covering prices coincide with the maximal Walrasian prices. These properties in turn lead to the price monotonicity results.

We propose a model for recoverable robust optimization with commitment. Given a combinatorial optimization problem and uncertainty about elements that may fail, we ask for a robust solution that, after the failing elements are revealed, can be augmented in a limited way. However, we commit to preserve the remaining elements of the initial solution. We consider different polynomial-time solvable combinatorial optimization problems and settle the computational complexity of their robust counterparts with commitment. We show for the weighted matroid basis problem that an optimal solution to the nominal problem is also optimal for its robust counterpart. Indeed, matroids are provably the only structures with this strong property. Robust counterparts of other problems are NP-hard such as the matching and the stable set problem, even in bipartite graphs. However, we establish polynomial-time algorithms for the robust counterparts of the unweighted stable set problem in bipartite graphs and the weighted stable set problem in interval graphs, also known as the interval scheduling problem.

In a Stackelberg max closure game, we are given a digraph whose vertices correspond to projects from which firms can choose and whose arcs represent precedence constraints. Some projects are under the control of a leader who sets prices in the first stage of the game, while in the second stage, the firms choose a feasible subset of projects of maximum value. For a single follower, the leader’s problem of finding revenue-maximizing prices can be solved in strongly polynomial time. In this paper, we focus on the setting with multiple followers and distinguish two situations. In the case in which only one copy of each project is available (limited supply), we show that the two-follower problem is solvable in strongly polynomial time, whereas the problem with three or more followers is NP-hard. In the case of unlimited supply, that is, when sufficient copies of each project are available, we show that the two-follower problem is already APX-hard. As a side result, we prove that Stackelberg min vertex cover on bipartite graphs with a single follower is APX-hard.

The famous Braess paradox describes the counterintuitive phenomenon in which, in certain settings, an increase of resources, such as a new road built within a congested network, may in fact lead to larger costs for the players in an equilibrium. In this paper, we consider general nonatomic congestion games and give a characterization of the combinatorial property of strategy spaces for which the Braess paradox does not occur. In short, matroid bases are precisely the required structure. We prove this characterization by two novel sensitivity results for convex separable optimization problems over polymatroid base polyhedra, which may be of independent interest.