In the stochastic knapsack problem, we are given a knapsack of size B, and a set of jobs whose sizes and rewards are drawn from a known probability distribution. However, the only way to know the actual size and reward is to schedule the job-when it completes, we get to know these values. How should we schedule jobs to maximize the expected total reward? We know constant-factor approximations for this problem when we assume that rewards and sizes are independent random variables, and that we cannot prematurely cancel jobs after we schedule them. What can we say when either or both of these assumptions are changed?The stochastic knapsack problem is of interest in its own right, but techniques developed for it are applicable to other stochastic packing problems. Indeed, ideas for this problem have been useful for budgeted learning problems, where one is given several arms which evolve in a specified stochastic fashion with each pull, and the goal is to pull the arms a total of B times to maximize the reward obtained. Much recent work on this problem focus on the case when the evolution of the arms follows a martingale, i.e., when the expected reward from the future is the same as the reward at the current state. What can we say when the rewards do not form a martingale?In this paper, we give constant-factor approximation algorithms for the stochastic knapsack problem with correlations and/or cancellations, and also for budgeted learning problems where the martingale condition is not satisfied, using similar ideas. Indeed, we can show that previously proposed linear programming relaxations for these problems have large integrality gaps. We propose new time-indexed LP relaxations; using a decomposition and "gap-filling" approach, we convert these fractional solutions to distributions over strategies, and then use the LP values and the time ordering information from these strategies to devise a randomized adaptive scheduling algorithm. We hope our LP formulation and decomposition methods may provide a new way to address other correlated bandit problems with more general contexts. *
Mixed-integer quadratic programming (MIQP) is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP [11] and integer linear programming is in NP [1,12,4,9].
Abstract. We consider packing LP's with m rows where all constraint coefficients are normalized to be in the unit interval. The n columns arrive in random order and the goal is to set the corresponding decision variables irrevocably when they arrive to obtain a feasible solution maximizing the expected reward. Previous (1 − )-competitive algorithms require the right-hand side of the LP to be Ω( m 2 log n ), a bound that worsens with the number of columns and rows. However, the dependence on the number of columns is not required in the single-row case and known lower bounds for the general case are also independent of n. Our goal is to understand whether the dependence on n is required in the multirow case, making it fundamentally harder than the single-row version. We refute this by exhibiting an algorithm which is (1 − )-competitive as long as the righthand sides are Ω( m 2 2 log m ). Our techniques refine previous PAC-learning based approaches which interpret the online decisions as linear classifications of the columns based on sampled dual prices. The key ingredient of our improvement comes from a non-standard covering argument together with the realization that only when the columns of the LP belong to few 1-d subspaces we can obtain small such covers; bounding the size of the cover constructed also relies on the geometry of linear classifiers. General packing LP's are handled by perturbing the input columns, which can be seen as making the learning problem more robust.
We prove that any minimal valid function for the k-dimensional infinite group relaxation that is piecewise linear with at most k + 1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k = 1, and Cornuéjols and Molinaro for k = 2.
In this paper, we study the strength of Chvátal-Gomory (CG) cuts and more generally aggregation cuts for packing and covering integer programs (IPs). Aggregation cuts are obtained as follows: Given an IP formulation, we first generate a single implied inequality using aggregation of the original constraints, then obtain the integer hull of the set defined by this single inequality with variable bounds, and finally use the inequalities describing the integer hull as cutting-planes. Our first main result is to show that for packing and covering IPs, the CG and aggregation closures can be 2-approximated by simply generating the respective closures for each of the original formulation constraints, without using any aggregations. On the other hand, we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general IPs. The proof of the above stated results for the case of covering IPs with bounds require the development of some new structural results, which may be of independent interest. Finally, we examine the strength of cuts based on k different aggregation inequalities simultaneously, the so-called multi-row cuts, and show that every packing or covering IP with a large integrality gap also has a large k-aggregation closure rank. In particular, this rank is always at least of the order of the logarithm of the integrality gap.
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