Algorithms the min-max regret 0-1 Integer Linear Programming Problem with Interval Data

On the Finite Optimal Convergence of Logic-Based Benders' Decomposition in Solving 0-1 Min-max Regret Optimization Problems with Interval Costs

This paper addresses a class of problems under interval data uncertainty composed of min-max regret versions of classical 0-1 optimization problems with interval costs. We refer to them as interval 0-1 min-max regret problems. The state-of-the-art exact algorithms for this class of problems work by solving a corresponding mixed integer linear programming formulation in a Benders' decomposition fashion. Each of the possibly exponentially many Benders' cuts is separated on the fly through the resolution of an instance of the classical 0-1 optimization problem counterpart. Since these separation …

A double oracle approach for minmax regret optimization problems with interval data

In this paper, we provide a generic anytime lower bounding procedure for minmax regret optimization problems. We show that the lower bound obtained is always at least as accurate as the lower bound recently proposed by Chassein and Goerigk (2015). This lower bound can be viewed as the optimal value of a linear programming relaxation of a mixed integer programming formulation of minmax regret optimization, but the contribution of the paper is to compute this lower bound via a double oracle algorithm (McMahan et al., 2003) that we specify. The double oracle algorithm is designed by relying on a …

Online MinCut: Competitive and Regret Analysis

In this paper we study the mincut problem in the online setting. We consider two distinct models: A) competitive analysis and B) regret analysis. In the competitive setting we consider the vertex arrival model; whenever a new vertex arrives it's neighborhood with respect to the set of known vertices is revealed. An online algorithm must make an irrevocable decision to determine the side of the cut that the vertex must belong to in order to minimize the size of the final cut. Various models are considered. 1) For classical and advice models we give tight bounds on the competitive ratio of deter…

Efficient Algorithms for k-Regret Minimizing Sets

A regret minimizing set Q is a small size representation of a much larger database P so that user queries executed on Q return answers whose scores are not much worse than those on the full dataset. In particular, a k-regret minimizing set has the property that the regret ratio between the score of the top-1 item in Q and the score of the top-k item in P is minimized, where the score of an item is the inner product of the item's attributes with a user's weight (preference) vector. The problem is challenging because we want to find a single representative set Q whose regret ratio is small with …

Competitive ratio versus regret minimization: achieving the best of both worlds

We consider online algorithms under both the competitive ratio criteria and the regret minimization one. Our main goal is to build a unified methodology that would be able to guarantee both criteria simultaneously. For a general class of online algorithms, namely any Metrical Task System (MTS), we show that one can simultaneously guarantee the best known competitive ratio and a natural regret bound. For the paging problem we further show an efficient online algorithm (polynomial in the number of pages) with this guarantee. To this end, we extend an existing regret minimization algorithm (speci…

A Polynomial Time Algorithm for Minimax-Regret Evacuation on a Dynamic Path

A dynamic path network is an undirected path with evacuees situated at each vertex. To evacuate the path, evacuees travel towards a designated sink (doorway) to exit. Each edge has a capacity, the number of evacuees that can enter the edge in unit time. Congestion occurs if an evacuee has to wait at a vertex for other evacuees to leave first. The basic problem is to place k sinks on the line, with an associated evacuation strategy, so as to minimize the total time needed to evacuate everyone. The minmax-regret version introduces uncertainty into the input, with the number of evacuees at vertic…

Complexity of interval minmax regret scheduling on parallel identical machines with total completion time criterion

In this paper, we consider the problem of scheduling jobs on parallel identical machines, where the processing times of jobs are uncertain: only interval bounds of processing times are known. The optimality criterion of a schedule is the total completion time. In order to cope with the uncertainty, we consider the maximum regret objective and we seek a schedule that performs well under all possible instantiations of processing times. Although the deterministic version of the considered problem is solvable in polynomial time, the minmax regret version is known to be weakly NP-hard even for a si…

无遗憾Mⁿ-凹函数最大化：随机老虎机算法与对抗性全信息设置的难度

No-Regret M${}^{\natural}$-Concave Function Maximization: Stochastic Bandit Algorithms and Hardness of Adversarial Full-Information Setting

针对Mⁿ-凹函数无法预先获取完整信息的问题，提出随机老虎机设置下的无遗憾算法并证明其鲁棒性，同时指出对抗性全信息设置中多项式时间算法无法实现低遗憾，为在线优化提供理论基础与新方法。

近似双次模 regret 最小化在 Billboard 和社交媒体广告中

Approximately Bisubmodular Regret Minimization in Billboard and Social Media Advertising

本文提出了一种近似双次模影响模型，用于解决多模式广告中非单调、非次模的 regret 最小化问题，并通过实验验证了其在减少总 regret 和满足广告需求方面的有效性。

Regret Minimization with Noisy Observations

In a typical optimization problem, the task is to pick one of a number of options with the lowest cost or the highest value. In practice, these cost/value quantities often come through processes such as measurement or machine learning, which are noisy, with quantifiable noise distributions. To take these noise distributions into account, one approach is to assume a prior for the values, use it to build a posterior, and then apply standard stochastic optimization to pick a solution. However, in many practical applications, such prior distributions may not be available. In this paper, we study s…