arXiv subjects

Waldo Gálvez

Publications and source records attributed to Waldo Gálvez.

11 recordsLinked to original sources

arXiv2026

A Framework for the Design of Efficient Diversification Algorithms for Packing and Covering Problems

There has been considerable recent interest in computing a diverse collection of solutions to a given optimization problem. Given a classical optimization problem $Π$ (e.g., minimum spanning tree, maximum independent set) with input size $n$ and an integer $k\geq 1$, the goal is to generate a collection of $k$ maximally diverse solutions to $Π$. This diverse-X paradigm not only allows the user to generate very different solutions, but also helps make systems more secure and robust by handling uncertainty and achieving energy efficiency. For several problems $Π$ solvable in polynomial time, there are efficient $\text{poly}(n,k)$-time exact or approximation algorithms available for their diverse-solution variants. In contrast, only FPT algorithms are known for NP-complete problems such as vertex covers and independent sets. In the worst case, these algorithms run in time $\exp((kn)^a)$ for some $a>0$. In this work, we address this gap and give a general framework for $\text{poly}(n,k)$-time \emph{bi-criteria} PTAS for the diverse variants of several NP-hard problems such as \textsc{Knapsack}, \textsc{Maximum Weight Independent Set} (MWIS) and \textsc{Minimum Weight Vertex Cover} (MWVC) in planar graphs. More precisely, given a maximization (or, minimization) problem $Π$, an integer $k \geq 1$, user-specified quality threshold $ c \in (0, 1) $, and user-controlled error parameters $δ,\varepsilon \in (0,1)$, our bi-approximation algorithm returns a collection of $k$ solutions $S_1,S_2,\ldots,S_k$ that are $(1-δ)c$-optimal and whose diversity is at least a $(1-\varepsilon)$-fraction of the maximum diversity achievable by any collection of $k$ $c$-optimal solutions. To complement our positive results, we prove that the diverse-solution variants of \KP and MWIS mentioned above do not admit standard single-criterion PTASs.

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arXiv2025

On the 2D Demand Bin Packing Problem: Hardness and Approximation Algorithms

We study a two-dimensional generalization of the classical Bin Packing problem, denoted as 2D Demand Bin Packing. In this context, each bin is a horizontal timeline, and rectangular tasks (representing electric appliances or computational requirements) must be allocated into the minimum number of bins so that the sum of the heights of tasks at any point in time is at most a given constant capacity. We prove that simple variants of the problem are NP-hard to approximate within a factor better than $2$, namely when tasks have short height and when they are squares, and provide best-possible approximation algorithms for them; we also present a simple $3$-approximation for the general case. All our algorithms are based on a general framework that computes structured solutions for relatively large tasks, while including relatively small tasks on top via a generalization of the well-known First-Fit algorithm for Bin Packing.

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arXiv2025

Computing Diverse and Nice Triangulations

We initiate the study of computing diverse triangulations to a given polygon. Given a simple $n$-gon $P$, an integer $ k \geq 2 $, a quality measure $σ$ on the set of triangulations of $P$ and a factor $ α\geq 1 $, we formulate the Diverse and Nice Triangulations (DNT) problem that asks to compute $k$ \emph{distinct} triangulations $T_1,\dots,T_k$ of $P$ such that a) their diversity, $\sum_{i < j} d(T_i,T_j) $, is as large as possible \emph{and} b) they are nice, i.e., $σ(T_i) \leq ασ^* $ for all $1\leq i \leq k$. Here, $d$ denotes the symmetric difference of edge sets of two triangulations, and $σ^*$ denotes the best quality of triangulations of $P$, e.g., the minimum Euclidean length. As our main result, we provide a $\mathrm{poly}(n,k)$-time approximation algorithm for the DNT problem that returns a collection of $k$ distinct triangulations whose diversity is at least $1 - Θ(1/k)$ of the optimal, and each triangulation satisfies the quality constraint. This is accomplished by studying \emph{bi-criteria triangulations} (BCT), which are triangulations that simultaneously optimize two criteria, a topic of independent interest. We complement our approximation algorithms by showing that the DNT problem and the BCT problem are NP-hard. Finally, for the version where diversity is defined as $\min_{i < j} d(T_i,T_j) $, we show a reduction from the problem of computing optimal Hamming codes, and provide an $n^{O(k)}$-time $\tfrac12$-approximation algorithm. This improves over the naive ${C_{n-2} \choose k} \approx 2^{O(nk)}$ time bound for enumerating all $k$-tuples among the triangulations of a simple $n$-gon, where $C_n$ denotes the $n$-th Catalan number.

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arXiv2021

Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

Given a $k$-vertex-connected graph $G$ and a set $S$ of extra edges (links), the goal of the $k$-vertex-connectivity augmentation problem is to find a set $S' \subseteq S$ of minimum size such that adding $S'$ to $G$ makes it $(k+1)$-vertex-connected. Unlike the edge-connectivity augmentation problem, research for the vertex-connectivity version has been sparse. In this work we present the first polynomial time approximation algorithm that improves the known ratio of 2 for $2$-vertex-connectivity augmentation, for the case in which $G$ is a cycle. This is the first step for attacking the more general problem of augmenting a $2$-connected graph. Our algorithm is based on local search and attains an approximation ratio of $1.8704$. To derive it, we prove novel results on the structure of minimal solutions.

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arXiv2021

Machine Covering in the Random-Order Model

In the Online Machine Covering problem jobs, defined by their sizes, arrive one by one and have to be assigned to $m$ parallel and identical machines, with the goal of maximizing the load of the least-loaded machine. In this work, we study the Machine Covering problem in the recently popular random-order model. Here no extra resources are present, but instead the adversary is weakened in that it can only decide upon the input set while jobs are revealed uniformly at random. It is particularly relevant to Machine Covering where lower bounds are usually associated to highly structured input sequences. We first analyze Graham's Greedy-strategy in this context and establish that its competitive ratio decreases slightly to $Θ\left(\frac{m}{\log(m)}\right)$ which is asymptotically tight. Then, as our main result, we present an improved $\tilde{O}(\sqrt[4]{m})$-competitive algorithm for the problem. This result is achieved by exploiting the extra information coming from the random order of the jobs, using sampling techniques to devise an improved mechanism to distinguish jobs that are relatively large from small ones. We complement this result with a first lower bound showing that no algorithm can have a competitive ratio of $O\left(\frac{\log(m)}{\log\log(m)}\right)$ in the random-order model. This lower bound is achieved by studying a novel variant of the Secretary problem, which could be of independent interest.

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arXiv2021

A (2+ε)-Approximation Algorithm for Maximum Independent Set of Rectangles

We study the Maximum Independent Set of Rectangles (MISR) problem, where we are given a set of axis-parallel rectangles in the plane and the goal is to select a subset of non-overlapping rectangles of maximum cardinality. In a recent breakthrough, Mitchell [2021] obtained the first constant-factor approximation algorithm for MISR. His algorithm achieves an approximation ratio of 10 and it is based on a dynamic program that intuitively recursively partitions the input plane into special polygons called corner-clipped rectangles (CCRs), without intersecting certain special horizontal line segments called fences. In this paper, we present a $(2+ε)$-approximation algorithm for MISR which is also based on a recursive partitioning scheme. First, we use a partition into a class of axis-parallel polygons with constant complexity each that are more general than CCRs. This allows us to provide an arguably simpler analysis and at the same time already improves the approximation ratio to 6. Then, using a more elaborate charging scheme and a recursive partitioning into general axis-parallel polygons with constant complexity, we improve our approximation ratio to $2+ε$. In particular, we construct a recursive partitioning based on more general fences which can be sequences of up to $O(1/ε)$ line segments each. This partitioning routine and our other new ideas may be useful for future work towards a PTAS for MISR.

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arXiv2021

Approximation Algorithms for Demand Strip Packing

In the Demand Strip Packing problem (DSP), we are given a time interval and a collection of tasks, each characterized by a processing time and a demand for a given resource (such as electricity, computational power, etc.). A feasible solution consists of a schedule of the tasks within the mentioned time interval. Our goal is to minimize the peak resource consumption, i.e. the maximum total demand of tasks executed at any point in time. It is known that DSP is NP-hard to approximate below a factor 3/2, and standard techniques for related problems imply a (polynomial-time) 2-approximation. Our main result is a (5/3+eps)-approximation algorithm for any constant eps>0. We also achieve best-possible approximation factors for some relevant special cases.

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arXiv2021

Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More

In the \textsc{2-Dimensional Knapsack} problem (2DK) we are given a square knapsack and a collection of $n$ rectangular items with integer sizes and profits. Our goal is to find the most profitable subset of items that can be packed non-overlappingly into the knapsack. The currently best known polynomial-time approximation factor for 2DK is $17/9+\varepsilon<1.89$ and there is a $(3/2+\varepsilon)$-approximation algorithm if we are allowed to rotate items by 90 degrees~{[}Gálvez et al., FOCS 2017{]}. In this paper, we give $(4/3+\varepsilon)$-approximation algorithms in polynomial time for both cases, assuming that all input data are {integers polynomially bounded in $n$}. Gálvez et al.'s algorithm for 2DK partitions the knapsack into a constant number of rectangular regions plus \emph{one} L-shaped region and packs items into those {in a structured way}. We generalize this approach by allowing up to a \emph{constant} number of {\emph{more general}} regions that can have the shape of an L, a U, a Z, a spiral, and more, and therefore obtain an improved approximation ratio. {In particular, we present an algorithm that computes the essentially optimal structured packing into these regions. }

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arXiv2018

Symmetry exploitation for Online Machine Covering with Bounded Migration

Online models that allow recourse are highly effective in situations where classical models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this setting, jobs arrive one by one and must be assigned to machines with the objective of maximizing the minimum machine load. When a job arrives, we are allowed to reassign some jobs as long as their total size is (at most) proportional to the processing time of the arriving job. The proportionality constant is called the migration factor of the algorithm. Using a rounding procedure with useful structural properties for online packing and covering problems, we design first a simple $(1.7 + \varepsilon)$-competitive algorithm using a migration factor of $O(1/\varepsilon)$ which maintains at every arrival a locally optimal solution with respect to the Jump neighborhood. After that, we present as our main contribution a more involved $(4/3+\varepsilon)$-competitive algorithm using a migration factor of $\tilde{O}(1/\varepsilon^3)$. At every arrival, we run an adaptation of the Largest Processing Time first (LPT) algorithm. Since the new job can cause a complete change of the assignment of smaller jobs in both cases, a low migration factor is achieved by carefully exploiting the highly symmetric structure obtained by the rounding procedure.

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arXiv2018

Improved Pseudo-Polynomial-Time Approximation for Strip Packing

We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of a fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be rotated. A reduction from the partition problem shows that no approximation better than 3/2 is possible for strip packing in polynomial time (assuming P$\neq$NP). Nadiradze and Wiese [SODA16] overcame this barrier by presenting a $(\frac{7}{5}+ε)$-approximation algorithm in pseudo-polynomial-time (PPT). As the problem is strongly NP-hard, it does not admit an exact PPT algorithm. In this paper, we make further progress on the PPT approximability of strip packing, by presenting a $(\frac43+ε)$-approximation algorithm. Our result is based on a non-trivial repacking of some rectangles in the \emph{empty space} left by the construction by Nadiradze and Wiese, and in some sense pushes their approach to its limit. Our PPT algorithm can be adapted to the case where we are allowed to rotate the rectangles by $90^\circ$, achieving the same approximation factor and breaking the polynomial-time approximation barrier of 3/2 for the case with rotations as well.

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arXiv2017

Approximating Geometric Knapsack via L-packings

We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is (2 + ε) [Jansen and Zhang, SODA 2004]. In this paper, we break the 2 approximation barrier, achieving a polynomial-time (17/9 + ε) < 1.89 approximation, which improves to (558/325 + ε) < 1.72 in the cardinality case. Essentially all prior work on 2DK approximation packs items inside a constant number of rectangular containers, where items inside each container are packed using a simple greedy strategy. We deviate for the first time from this setting: we show that there exists a large profit solution where items are packed inside a constant number of containers plus one L-shaped region at the boundary of the knapsack which contains items that are high and narrow and items that are wide and thin. As a second major and the main algorithmic contribution of this paper, we present a PTAS for this case. We believe that this will turn out to be useful in future work in geometric packing problems. We also consider the variant of the problem with rotations (2DKR), where items can be rotated by 90 degrees. Also, in this case, the best-known polynomial-time approximation factor (even for the cardinality case) is (2 + ε) [Jansen and Zhang, SODA 2004]. Exploiting part of the machinery developed for 2DK plus a few additional ideas, we obtain a polynomial-time (3/2 + ε)-approximation for 2DKR, which improves to (4/3 + ε) in the cardinality case.

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