Description of 2-integer continuous knapsack polyhedra

AbstractIn this paper we discuss the polyhedral structure of several mixed integer sets involving two integer variables. We show that the number of the corresponding facet-defining inequalities is polynomial on the size of the input data and their coefficients can also be computed in polynomial time using a known algorithm [D. Hirschberg, C. Wong, A polynomial-time algorithm for the knapsack problem with two variables, Journal of the Association for Computing Machinery 23 (1) (1976) 147–154] for the two integer knapsack problem. These mixed integer sets may arise as substructures of more complex mixed integer sets that model the feasible solutions of real application problems

Improvement of the branch and bound algorithm for solving the knapsack linear integer problem

The paper presents a new reformulation approach to reduce the complexity of a branch and bound algorithm for solving the knapsack linear integer problem. The branch and bound algorithm in general relies on the usual strategy of first relaxing the integer problem into a linear programing (LP) model. If the linear programming optimal solution is integer then, the optimal solution to the integer problem is available. If the linear programming optimal solution is not integer, then a variable with a fractional value is selected to create two sub-problems such that part of the feasible region is discarded without eliminating any of the feasible integer solutions. The process is repeated on all variables with fractional values until an integer solution is found. In this approach variable sum and additional constraints are generated and added to the original problem before solving. In order to do this the objective bound of knapsack problem is quickly determined. The bound is then used to generate a set of variable sum limits and four additional constraints. From the variable sum limits, initial sub-problems are constructed and solved. The optimal solution is then obtained as the best solution from all the sub-problems in terms of the objective value. The proposed procedure results in sub-problems that have reduced complexity and easier to solve than the original problem in terms of numbers of branch and bound iterations or sub-problems.The knapsack problem is a special form of the general linear integer problem. There are so many types of knapsack problems. These include the zero-one, multiple, multiple-choice, bounded, unbounded, quadratic, multi-objective, multi-dimensional, collapsing zero-one and set union knapsack problems. The zero-one knapsack problem is one in which the variables assume 0 s and 1 s only. The reason is that an item can be chosen or not chosen. In other words there is no way it is possible to have fractional amounts or items. This is the easiest class of the knapsack problems and is the only one that can be solved in polynomial by interior point algorithms and in pseudo-polynomial time by dynamic programming approaches. The multiple-choice knapsack problem is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The zero-one choice of taking an item is replaced by the selection of exactly one item out of each class of item

Faster space-efficient algorithms for Subset Sum, k -Sum, and related problems

We present randomized algorithms that solve subset sum and knapsack instances with n items in O∗ (20.86n) time, where the O∗ (∙ ) notation suppresses factors polynomial in the input size, and polynomial space, assuming random read-only access to exponentially many random bits. These results can be extended to solve binary integer programming on n variables with few constraints in a similar running time. We also show that for any constant k ≥ 2, random instances of k-Sum can be solved using O(nk -0.5polylog(n)) time and O(log n) space, without the assumption of random access to random bits.Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O(log n) space significantly faster than the trivial O(n2) time algorithm if no value occurs too often in the same list.</p

Improvement of the branch and bound algorithm for solving the knapsack linear integer problem

The paper presents a new reformulation approach to reduce the complexity of a branch and bound algorithm for solving the knapsack linear integer problem. The branch and bound algorithm in general relies on the usual strategy of first relaxing the integer problem into a linear programing (LP) model. If the linear programming optimal solution is integer then, the optimal solution to the integer problem is available. If the linear programming optimal solution is not integer, then a variable with a fractional value is selected to create two sub-problems such that part of the feasible region is discarded without eliminating any of the feasible integer solutions. The process is repeated on all variables with fractional values until an integer solution is found. In this approach variable sum and additional constraints are generated and added to the original problem before solving. In order to do this the objective bound of knapsack problem is quickly determined. The bound is then used to generate a set of variable sum limits and four additional constraints. From the variable sum limits, initial sub-problems are constructed and solved. The optimal solution is then obtained as the best solution from all the sub-problems in terms of the objective value. The proposed procedure results in sub-problems that have reduced complexity and easier to solve than the original problem in terms of numbers of branch and bound iterations or sub-problems.The knapsack problem is a special form of the general linear integer problem. There are so many types of knapsack problems. These include the zero-one, multiple, multiple-choice, bounded, unbounded, quadratic, multi-objective, multi-dimensional, collapsing zero-one and set union knapsack problems. The zero-one knapsack problem is one in which the variables assume 0 s and 1 s only. The reason is that an item can be chosen or not chosen. In other words there is no way it is possible to have fractional amounts or items. This is the easiest class of the knapsack problems and is the only one that can be solved in polynomial by interior point algorithms and in pseudo-polynomial time by dynamic programming approaches. The multiple-choice knapsack problem is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The zero-one choice of taking an item is replaced by the selection of exactly one item out of each class of item

Stochastic Combinatorial Optimization via Poisson Approximation

We study several stochastic combinatorial problems, including the expected utility maximization problem, the stochastic knapsack problem and the stochastic bin packing problem. A common technical challenge in these problems is to optimize some function of the sum of a set of random variables. The difficulty is mainly due to the fact that the probability distribution of the sum is the convolution of a set of distributions, which is not an easy objective function to work with. To tackle this difficulty, we introduce the Poisson approximation technique. The technique is based on the Poisson approximation theorem discovered by Le Cam, which enables us to approximate the distribution of the sum of a set of random variables using a compound Poisson distribution. We first study the expected utility maximization problem introduced recently [Li and Despande, FOCS11]. For monotone and Lipschitz utility functions, we obtain an additive PTAS if there is a multidimensional PTAS for the multi-objective version of the problem, strictly generalizing the previous result. For the stochastic bin packing problem (introduced in [Kleinberg, Rabani and Tardos, STOC97]), we show there is a polynomial time algorithm which uses at most the optimal number of bins, if we relax the size of each bin and the overflow probability by eps. For stochastic knapsack, we show a 1+eps-approximation using eps extra capacity, even when the size and reward of each item may be correlated and cancelations of items are allowed. This generalizes the previous work [Balghat, Goel and Khanna, SODA11] for the case without correlation and cancelation. Our algorithm is also simpler. We also present a factor 2+eps approximation algorithm for stochastic knapsack with cancelations. the current known approximation factor of 8 [Gupta, Krishnaswamy, Molinaro and Ravi, FOCS11].Comment: 42 pages, 1 figure, Preliminary version appears in the Proceeding of the 45th ACM Symposium on the Theory of Computing (STOC13

Polynomial Kernels for Weighted Problems

Kernelization is a formalization of efficient preprocessing for NP-hard problems using the framework of parameterized complexity. Among open problems in kernelization it has been asked many times whether there are deterministic polynomial kernelizations for Subset Sum and Knapsack when parameterized by the number $n$ of items. We answer both questions affirmatively by using an algorithm for compressing numbers due to Frank and Tardos (Combinatorica 1987). This result had been first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We further illustrate its applicability by giving polynomial kernels also for weighted versions of several well-studied parameterized problems. Furthermore, when parameterized by the different item sizes we obtain a polynomial kernelization for Subset Sum and an exponential kernelization for Knapsack. Finally, we also obtain kernelization results for polynomial integer programs