arXiv subjects

Ariel Kulik

Publications and source records attributed to Ariel Kulik.

29 records · Page 2Linked to original sources

arXiv2023

An EPTAS for Budgeted Matching and Budgeted Matroid Intersection

We study the budgeted versions of the well known matching and matroid intersection problems. While both problems admit a polynomial-time approximation scheme (PTAS) [Berger et al. (Math. Programming, 2011), Chekuri, Vondrak and Zenklusen (SODA 2011)], it has been an intriguing open question whether these problems admit a fully PTAS (FPTAS), or even an efficient PTAS (EPTAS). In this paper we answer the second part of this question affirmatively, by presenting an EPTAS for budgeted matching and budgeted matroid intersection. A main component of our scheme is a novel construction of representative sets for desired solutions, whose cardinality depends only on $\varepsilon$, the accuracy parameter. Thus, enumerating over solutions within a representative set leads to an EPTAS. This crucially distinguishes our algorithms from previous approaches, which rely on exhaustive enumeration over the solution set. Our ideas for constructing representative sets may find use in tackling other budgeted optimization problems, and are thus of independent interest.

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arXiv2023

Computing Generalized Convolutions Faster Than Brute Force

In this paper, we consider a general notion of convolution. Let $D$ be a finite domain and let $D^n$ be the set of $n$-length vectors (tuples) of $D$. Let $f : D \times D \to D$ be a function and let $\oplus_f$ be a coordinate-wise application of $f$. The $f$-Convolution of two functions $g,h : D^n \to \{-M,\ldots,M\}$ is $$(g \otimes_f h)(\textbf{v}) := \sum_{\substack{\textbf{v}_g,\textbf{v}_h \in D^n\\ \text{s.t. } \textbf{v}_g \oplus_f \textbf{v}_h}} g(\textbf{v}_g) \cdot h(\textbf{v}_h)$$ for every $\textbf{v} \in D^n$. This problem generalizes many fundamental convolutions such as Subset Convolution, XOR Product, Covering Product or Packing Product, etc. For arbitrary function $f$ and domain $D$ we can compute $f$-Convolution via brute-force enumeration in $\widetilde{O}(|D|^{2n}\mathrm{polylog}(M))$ time. Our main result is an improvement over this naive algorithm. We show that $f$-Convolution can be computed exactly in $\widetilde{O}((c \cdot |D|^2)^{n}\mathrm{polylog}(M))$ for constant $c := 3/4$ when $D$ has even cardinality. Our main observation is that a \emph{cyclic partition} of a function $f : D \times D \to D$ can be used to speed up the computation of $f$-Convolution, and we show that an appropriate cyclic partition exists for every $f$. Furthermore, we demonstrate that a single entry of the $f$-Convolution can be computed more efficiently. In this variant, we are given two functions $g,h : D^n \to \{-M,\ldots,M\}$ alongside with a vector $\textbf{v} \in D^n$ and the task of the $f$-Query problem is to compute integer $(g \otimes_f h)(\textbf{v})$. This is a generalization of the well-known Orthogonal Vectors problem. We show that $f$-Query can be computed in $\widetilde{O}(|D|^{\fracω{2} n}\mathrm{polylog}(M))$ time, where $ω\in [2,2.372)$ is the exponent of currently fastest matrix multiplication algorithm.

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arXiv2022

An EPTAS for Budgeted Matroid Independent Set

We consider the budgeted matroid independent set problem. The input is a ground set, where each element has a cost and a non-negative profit, along with a matroid over the elements and a budget. The goal is to select a subset of elements which maximizes the total profit subject to the matroid and budget constraints. Several well known special cases, where we have, e.g., a uniform matroid and a budget, or no matroid constraint (i.e., the classic knapsack problem), admit a fully polynomial-time approximation scheme (FPTAS). In contrast, already a slight generalization to the multi-budgeted matroid independent set problem has a PTAS but does not admit an efficient polynomial-time approximation scheme (EPTAS). This implies a PTAS for our problem, which is the best known result prior to this work. Our main contribution is an EPTAS for the budgeted matroid independent set problem. A key idea of the scheme is to find a representative set for the instance, whose cardinality depends solely on $1/\varepsilon$, where $\varepsilon > 0$ is the accuracy parameter of the scheme. The representative set is identified via matroid basis minimization, which can be solved by a simple greedy algorithm. Our scheme enumerates over subsets of the representative set and extends each subset using a linear program. The notion of representative sets may be useful in solving other variants of the budgeted matroid independent set problem.

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arXiv2021

General Knapsack Problems in a Dynamic Setting

The world is dynamic and changes over time, thus any optimization problem used to model real life problems must address this dynamic nature, taking into account the cost of changes to a solution over time. The multistage model was introduced with this goal in mind. In this model we are given a series of instances of an optimization problem, corresponding to different times, and a solution is provided for each instance. The strive for obtaining near-optimal solutions for each instance on one hand, while maintaining similar solutions for consecutive time units on the other hand, is quantified and integrated into the objective function. In this paper we consider the Generalized Multistage $d$-Knapsack problem, a generalization of the multistage variants of the Multiple Knapsack problem, as well as the $d$-Dimensional Knapsack problem. We present a PTAS for Generalized Multistage $d$-Knapsack.

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arXiv2021

Modular and Submodular Optimization with Multiple Knapsack Constraints via Fractional Grouping

A multiple knapsack constraint over a set of items is defined by a set of bins of arbitrary capacities, and a weight for each of the items. An assignment for the constraint is an allocation of subsets of items to the bins which adheres to bin capacities. In this paper we present a unified algorithm that yields efficient approximations for a wide class of submodular and modular optimization problems involving multiple knapsack constraints. One notable example is a polynomial time approximation scheme for Multiple-Choice Multiple Knapsack, improving upon the best known ratio of $2$. Another example is Non-monotone Submodular Multiple Knapsack, for which we obtain a $(0.385-\varepsilon)$-approximation, matching the best known ratio for a single knapsack constraint. The robustness of our algorithm is achieved by applying a novel fractional variant of the classical linear grouping technique, which is of independent interest.

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arXiv2021

An APTAS for Bin Packing with Clique-graph Conflicts

We study the following variant of the classic {\em bin packing} problem. Given a set of items of various sizes, partitioned into groups, find a packing of the items in a minimum number of identical (unit-size) bins, such that no two items of the same group are assigned to the same bin. This problem, known as {\em bin packing with clique-graph conflicts}, has natural applications in storing file replicas, security in cloud computing and signal distribution. Our main result is an {\em asymptotic polynomial time approximation scheme (APTAS)} for the problem, improving upon the best known ratio of $2$. %In particular, for any instance $I$ and a fixed $\eps \in (0,1)$, the items are packed in at most $(1+\eps)OPT(I) +1$ bins, where $OPT(I)$ is the minimum number of bins required for packing the instance. As a key tool, we apply a novel {\em Shift \& Swap} technique which generalizes the classic linear shifting technique to scenarios allowing conflicts between items. The major challenge of packing {\em small} items using only a small number of extra bins is tackled through an intricate combination of enumeration and a greedy-based approach that utilizes the rounded solution of a {\em linear program}.

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arXiv2021

An Almost Optimal Approximation Algorithm for Monotone Submodular Multiple Knapsack

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint. The input is a set $I$ of items, each has a non-negative weight, and a set of bins of arbitrary capacities. Also, we are given a submodular, monotone and non-negative function $f$ over subsets of the items. The objective is to find a packing of a subset of items $A \subseteq I$ in the bins such that $f(A)$ is maximized. Our main result is an almost optimal polynomial time $(1-e^{-1}-\varepsilon)$-approximation algorithm for the problem, for any $\varepsilon>0$. The algorithm relies on a structuring technique which converts a general multiple knapsack constraint to a constraint in which the bins are partitioned into groups of exponentially increasing cardinalities, each consisting of bins of uniform capacity. We derive the result by combining structuring with a refined analysis of techniques for submodular optimization subject to knapsack constraints.

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arXiv2021

A Refined Analysis of Submodular Greedy

Many algorithms for maximizing a monotone submodular function subject to a knapsack constraint rely on the natural greedy heuristic. We present a novel refined analysis of this greedy heuristic which enables us to: $(1)$ reduce the enumeration in the tight $(1-e^{-1})$-approximation of [Sviridenko 04] from subsets of size three to two; $(2)$ present an improved upper bound of $0.42945$ for the classic algorithm which returns the better between a single element and the output of the greedy heuristic.

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arXiv2019

Generalized Assignment via Submodular Optimization with Reserved Capacity

We study a variant of the \emph{generalized assignment problem} ({\sf GAP}) with group constraints. An instance of {\sf Group GAP} is a set $I$ of items, partitioned into $L$ groups, and a set of $m$ uniform (unit-sized) bins. Each item $i \in I$ has a size $s_i >0$, and a profit $p_{i,j} \geq 0$ if packed in bin $j$. A group of items is \emph{satisfied} if all of its items are packed. The goal is to find a feasible packing of a subset of the items in the bins such that the total profit from satisfied groups is maximized. We point to central applications of {\sf Group GAP} in Video-on-Demand services, mobile Device-to-Device network caching and base station cooperation in 5G networks. Our main result is a $\frac{1}{6}$-approximation algorithm for {\sf Group GAP} instances where the total size of each group is at most $\frac{m}{2}$. At the heart of our algorithm lies an interesting derivation of a submodular function from the classic LP formulation of {\sf GAP}, which facilitates the construction of a high profit solution utilizing at most half the total bin capacity, while the other half is \emph{reserved} for later use. In particular, we give an algorithm for submodular maximization subject to a knapsack constraint, which finds a solution of profit at least $\frac{1}{3}$ of the optimum, using at most half the knapsack capacity, under mild restrictions on element sizes. Our novel approach of submodular optimization subject to a knapsack \emph{with reserved capacity} constraint may find applications in solving other group assignment problems.

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arXiv2015

On Lagrangian Relaxation and Reoptimization Problems

We prove a general result demonstrating the power of Lagrangian relaxation in solving constrained maximization problems with arbitrary objective functions. This yields a unified approach for solving a wide class of {\em subset selection} problems with linear constraints. Given a problem in this class and some small $\eps \in (0,1)$, we show that if there exists an $r$-approximation algorithm for the Lagrangian relaxation of the problem, for some $r \in (0,1)$, then our technique achieves a ratio of $\frac{r}{r+1} -\! \eps$ to the optimal, and this ratio is tight. The number of calls to the $r$-approximation algorithm, used by our algorithms, is {\em linear} in the input size and in $\log (1 / \eps)$ for inputs with cardinality constraint, and polynomial in the input size and in $\log (1 / \eps)$ for inputs with arbitrary linear constraint. Using the technique we obtain (re)approximation algorithms for natural (reoptimization) variants of classic subset selection problems, including real-time scheduling, the {\em maximum generalized assignment problem (GAP)} and maximum weight independent set.

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arXiv2011

Approximations for Monotone and Non-monotone Submodular Maximization with Knapsack Constraints

Submodular maximization generalizes many fundamental problems in discrete optimization, including Max-Cut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks. In this paper we consider the problem of maximizing any submodular function subject to $d$ knapsack constraints, where $d$ is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through {\em extension by expectation} of the submodular function. Formally, we show that, for any non-negative submodular function, an $α$-approximation algorithm for the continuous relaxation implies a randomized $(α- \eps)$-approximation algorithm for the discrete problem. We use this relation to improve the best known approximation ratio for the problem to $1/4- \eps$, for any $\eps > 0$, and to obtain a nearly optimal $(1-e^{-1}-\eps)-$approximation ratio for the monotone case, for any $\eps>0$. We further show that the probabilistic domain defined by a continuous solution can be reduced to yield a polynomial size domain, given an oracle for the extension by expectation. This leads to a deterministic version of our technique.

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