Various forms of dependent rounding are useful when handling a mixture of “hard” (e.g., matroid) constraints and “soft” (packing) constraints. We consider a few classes of such problems that arise in facility location, where one aims for small additive violations of the packing constraints, and where we require substantial “near-independence” properties among the variables being rounded. While ...

This paper examines the performance of hill-climbing algorithms on standard test problems for combinatorial auctions (CAs). On single-unit CAs, deterministic hill-climbers are found to perform well, and their performance can be improved signiicantly by randomizing them and restarting them several times, or by using them collectively. For some problems this good performance is shown to be no bet...

We study constrained versions of the knapsack problem in which dependencies between items are given by a graph. In one version, an item can be selected only if one of its neighbours is also selected. In the other version, an item can be selected only when all its neighbours are also selected. These problems generalize and unify several problems including the prize collecting and budgeted maximu...

We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b ≤ Ax ≤ b x ∈ Zn with b ≤ (AU)y ≤ b y ∈ Zn, where U is a unimodular matrix computed via basis reduction, to make the columns of AU short (i.e. have small Euclidean norm), and nearly orthogonal (see e.g. [20], [17]). Our approach is termed column basis reduction, and the reformula...

We address the classical knapsack problem and a variant in which an upper bound is imposed on the number of items that can be selected. We show that appropriate combinations of rounding techniques yield novel and powerful ways of rounding. As an application of these techniques, we present a linear-storage Polynomial Time Approximation Scheme (PTAS) and a Fully Polynomial Time Approximation Sche...

The stochastic non-linear fractional knapsack problem is a challenging optimization problem with numerous applications, including resource allocation. The goal is to find the most valuable mix of materials that fits within a knapsack of fixed capacity. When the value functions of the involved materials are fully known and differentiable, the most valuable mixture can be found by direct applicat...

In this paper, we study the fault-tolerant matroid median and knapsack problems. These two problems generalize many fundamental clustering facility location problems, such as uniform k-median, location, median, etc. We present a versatile iterative rounding framework obtain unifying constant-factor approximation algorithm.

In this paper, we present a new methodology to adapt any kind of lattice reduction algorithms to deal with the modular knapsack problem. In general, the modular knapsack problem can be solved using a lattice reduction algorithm, when its density is low. The complexity of lattice reduction algorithms to solve those problems is upper-bounded in the function of the lattice dimension and the maximu...

We present algorithms for the following three-dimensional (3D) guillotine cutting problems: Unbounded Knapsack, Cutting Stock and Strip Packing. We consider the case where the items have fixed orientation and the case where orthogonal rotations around all axes are allowed. For the Unbounded 3D Knapsack problem, we extend the recurrence formula proposed by Beasley for the Rectangular Knapsack Pr...

We consider variants of the classic bin packing and multiple knapsack problems, in which sets of items of di erent classes (colours) need to be placed in bins; the items may have di<erent sizes and values. Each bin has a limited capacity, and a bound on the number of distinct classes of items it can accommodate. In the class-constrained multiple knapsack (CCMK) problem, our goal is to maximize ...