On Guillotine Separable Packings for the Two-Dimensional Geometric Knapsack Problem

In two-dimensional geometric knapsack problem, we are given a set of n axis-aligned rectangular items and an axis-aligned square-shaped knapsack. Each item has integral width, integral height and an associated integral profit. The goal is to find a (non-overlapping axis-aligned) packing of a maximum profit subset of rectangles into the knapsack. A well-studied and frequently used constraint in practice is to allow only packings that are guillotine separable, i.e., every rectangle in the packing can be obtained by recursively applying a sequence of edge-to-edge axis-parallel cuts that do not intersect any item of the solution. In this paper we study approximation algorithms for the geometric knapsack problem under guillotine cut constraints. We present polynomial time (1+?)-approximation algorithms for the cases with and without allowing rotations by 90 degrees, assuming that all input numeric data are polynomially bounded in n. In comparison, the best-known approximation factor for this setting is 3+? [Jansen-Zhang, SODA 2004], even in the cardinality case where all items have the same profit. Our main technical contribution is a structural lemma which shows that any guillotine packing can be converted into another structured guillotine packing with almost the same profit. In this packing, each item is completely contained in one of a constant number of boxes and ?-shaped regions, inside which the items are placed by a simple greedy routine. In particular, we provide a clean sufficient condition when such a packing obeys the guillotine cut constraints which might be useful for other settings where these constraints are imposed

Tight Approximation Algorithms For Geometric Bin Packing with Skewed Items

In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan [SODA\u2714]. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko [FOCS\u2705] obtained an APTAS for this problem. Let ? be the smallest constant such that for every set I of items, the number of bins in the optimal solution to G2BP for I is upper bounded by ? opt(I) + c, where opt(I) is the number of bins in the optimal solution to 2BP for I and c is a constant. It is known that 4/3 ? ? ? 1.692. Bansal and Khan [SODA\u2714] conjectured that ? = 4/3. The conjecture, if true, will imply a (4/3+?)-approximation algorithm for 2BP. According to convention, for a given constant ? > 0, a rectangle is large if both its height and width are at least ?, and otherwise it is called skewed. We make progress towards the conjecture by showing ? = 4/3 for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on ? was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS

Fully Dynamic Algorithms for Knapsack Problems with Polylogarithmic Update Time

Knapsack problems are among the most fundamental problems in optimization. In the Multiple Knapsack problem, we are given multiple knapsacks with different capacities and items with values and sizes. The task is to find a subset of items of maximum total value that can be packed into the knapsacks without exceeding the capacities. We investigate this problem and special cases thereof in the context of dynamic algorithms and design data structures that efficiently maintain near-optimal knapsack solutions for dynamically changing input. More precisely, we handle the arrival and departure of individual items or knapsacks during the execution of the algorithm with worst-case update time polylogarithmic in the number of items. As the optimal and any approximate solution may change drastically, we maintain implicit solutions and support polylogarithmic time query operations that can return the computed solution value and the packing of any given item. While dynamic algorithms are well-studied in the context of graph problems, there is hardly any work on packing problems (and generally much less on non-graph problems). Motivated by the theoretical interest in knapsack problems and their practical relevance, our work bridges this gap