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We consider the Bin Packing problem with a partition matroid constraint. The input is a set of items of sizes in $(0,1]$, and a partition matroid over the items. The goal is to pack all items in a minimum number of unit-size bins, such that…
Optimal packing of objects in containers is a critical problem in various real-life and industrial applications. This paper investigates the two-dimensional packing of convex polygons without rotations, where only translations are allowed.…
In the Bin Packing problem one is given $n$ items with weights $w_1,\ldots,w_n$ and $m$ bins with capacities $c_1,\ldots,c_m$. The goal is to find a partition of the items into sets $S_1,\ldots,S_m$ such that $w(S_j) \leq c_j$ for every bin…
Packing cost accounts for a large part of the e-commerce logistics cost. Mining the patterns of customer orders and designing suitable packing bins help to reduce operating cost. In the classical bin packing problem, a given set of…
This paper proves a bottom-left placement theorem for the rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given rectangles into a rectangular container without overlapping, then we can achieve a…
In this paper we deal with the Bin Packing Problem with Conflicts on interval graphs: given an interval graph, a nonnegative integer weight for each vertex, and a nonnegative integer B, find a partition of the vertex set of the graph into k…
We consider the online vector bin packing problem where $n$ items specified by $d$-dimensional vectors must be packed in the fewest number of identical $d$-dimensional bins. Azar et al. (STOC'13) showed that for any online algorithm $A$,…
We present an $n\Delta^{O(k^2)}$ time algorithm to obtain an optimal solution for $1$-dimensional cutting stock problem: the bin packing problem of packing $n$ items onto unit capacity bins under the restriction that the number of item…
In this paper, we study the following knapsack problem: Given a list of squares with profits, we are requested to pack a sublist of them into a rectangular bin (not a unit square bin) to make profits in the bin as large as possible. We…
We investigate computational problems involving large weights through the lens of kernelization, which is a framework of polynomial-time preprocessing aimed at compressing the instance size. Our main focus is the weighted Clique problem,…
We study the 2-dimensional vector packing problem, which is a generalization of the classical bin packing problem where each item has 2 distinct weights and each bin has 2 corresponding capacities. The goal is to group items into minimum…
This paper formally proposes a problem about the efficient utilization of the four dimensional space-time. Given a cuboid container, a finite number of rigid cuboid items, and the time length that each item should be continuous baked in the…
For bin packing, the input consists of $n$ items with sizes $s_1,...,s_n \in [0,1]$ which have to be assigned to a minimum number of bins of size 1. Recently, the second author gave an LP-based polynomial time algorithm that employed…
For bin packing, the input consists of n items with sizes s_1,...,s_n in [0,1] which have to be assigned to a minimum number of bins of size 1. The seminal Karmarkar-Karp algorithm from '82 produces a solution with at most OPT + O(log^2…
We solve the Bin Packing problem in $O^*(2^k)$ time, where $k$ is the number of items less or equal to one third of the bin capacity. This parameter measures the distance from the polynomially solvable case of only large (i.e., greater than…
Kernelization algorithms are polynomial-time reductions from a problem to itself that guarantee their output to have a size not exceeding some bound. For example, d-Set Matching for integers d>2 is the problem of finding a matching of size…
The Bin Packing Problem involves efficiently packing items into a limited number of bins without exceeding their capacity. In this paper, we try to answer a specific question in this field. Mathematically the combinatorial optimization…
We consider the {\em Shaped Partition Problem} of partitioning $n$ given vectors in real $k$-space into $p$ parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary…
Sets of $d\times d$ matrices sharing a common invariant cone enjoy special properties, which are widely used in applications. However, finding this cone or even proving its existence/non-existence is hard. This problem is known to be…
We study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly…