Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets

Consider the following problem: given a set system (U,I) and an edge-weighted graph G = (U, E) on the same universe U, find the set A in I such that the Steiner tree cost with terminals A is as large as possible: "which set in I is the most difficult to connect up?" This is an example of a max-min problem: find the set A in I such that the value of some minimization (covering) problem is as large as possible. In this paper, we show that for certain covering problems which admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system I is given by a p-system or q-knapsacks or both. This result is similar to results for constrained maximization of submodular functions. Although many natural covering problems are not even approximately submodular, we show that one can use properties of the online algorithm as a surrogate for submodularity. Moreover, we give stronger connections between max-min optimization and two-stage robust optimization, and hence give improved algorithms for robust versions of various covering problems, for cases where the uncertainty sets are given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and http://arxiv.org/abs/0912.1045 appeared in ICALP 201

A 3/2-approximation algorithm for some minimum-cost graph problems

International audienceWe consider a class of graph problems introduced in a paper of Goemans and Williamson that involve finding forests of minimum edge cost. This class includes a number of location/routing problems; it also includes a problem in which we are given as input a parameter k, and want to find a forest such that each component has at least k vertices. Goemans and Williamson gave a 2-approximation algorithm for this class of problems. We give an improved 3/2-approximation algorithm

Prophet Inequalities with Limited Information

In the classical prophet inequality, a gambler observes a sequence of stochastic rewards $V_1,...,V_n$ and must decide, for each reward $V_i$, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value $V_i$. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose $k$ items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards $V_1,...,V_n$ are drawn. The assumption that the gambler knows the distribution from which $V_1,...,V_n$ are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, \emph{even with full knowledge of the distribution}. Specifically, we provide a novel single-sample algorithm when the gambler can choose any $k$ elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related \emph{secretary problem} to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. We apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders

Robust Assignment Using Redundant Robots on Transport Networks with Uncertain Travel Time

This paper considers the problem of assigning mo- bile robots to goals on transport networks with uncertain and potentially correlated information about travel times. Our aim is to produce optimal assignments, such that the average waiting time at destinations is minimized. Since noisy travel time estimates result in sub-optimal assignments, we propose a method that offers robustness to uncertainty by making use of redundant robot assignments. However, solving the redundant assignment problem optimally is strongly NP-hard. Hence, we exploit structural properties of our mathematical problem formulation to propose a polynomial-time, near-optimal solution. We demonstrate that our problem can be reduced to minimizing a supermodular cost function subject to a matroid constraint. This allows us to develop a greedy assignment algorithm, for which we derive sub-optimality bounds. We demonstrate the effectiveness of our approach with simulations on transport networks with correlated uncertain edge costs and uncertain node positions that lead to noisy travel time estimates. Comparisons to benchmark algorithms show that our method performs near-optimally and significantly better than non-redundant assignment. Finally, our findings include results on the benefit of diversity and complementarity in redundant robot coalitions; these insights contribute towards providing resilience to uncertainty through targeted composition of robot coalitions.This work was supported by ARL DCIST CRA W911NF- 17-2-0181, by the Centre for Digital Built Britain, under InnovateUK grant number RG96233, for the research project “Co-Evolving Built Environments and Mobile Autonomy for Future Transport and Mobility”, and by the Engineering and Physical Sciences Research Council (grant EP/S015493/1)

Cross-monotonic Cost-Sharing Methods for Network Design Games

In this thesis we consider some network design games that arise from common network design problems. A network design game involves multiple players who control nodes in a network, each of which has a personal interest in seeing their nodes connected in some manner. To this end, the players will submit bids to a mechanism whose task will be to select which of the players to connect, how to connect their nodes, and how much to charge each player for the connection. We rely on many fundamental results from mechanism design (from [8], [9] & [5]) in this thesis and focus our efforts on designing and analyzing cost-sharing methods. That is, for a given set of players and their connection requirements, our algorithms compute a solution that satisfies all the players’ requirements and calculates ’fair’ prices to charge each of them for the connection. Our cost-sharing methods use a primal-dual framework developed by Agrawal, Klein and Ravi in [1] and generalized by Goemans &Williamson in [3]. We modify the algorithms by using the concept of death-time introduced by K¨onemann, Leonardi & Sch¨afer in [6]. Our main result is a 2-budget balanced and cross-monotonic cost sharing method for the downwards monotone set cover game, which arises naturally from any downwards monotone 0, 1-function. We have also designed a 2-budget balanced and cross-monotonic cost sharing method for two versions of the edge cover game arising from the edge cover problem. These games are special cases of the downwards monotone set cover game. By a result by Immorlica, Mahdian & Mirrokni in [4] our result is best possible for the edge cover game. We also designed a cross-monotonic cost sharing method for a network design game we call the Even Parity Connection game arising from the T-Join problem that generalizes proper cut requirement functions. We can show our algorithm returns cost shares that recover at least half the cost of the solution. We conjecture that our cost sharing method for the even parity connection game is competitive and thus 2-budget balance