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We study the budgeted laminar matroid independent set problem. The input is a ground set, where each element has a cost and a non-negative profit, along with a laminar matroid over the elements and a budget. The goal is to select a maximum profit independent set of the matroid whose total cost is bounded by the budget. Several well known special cases, where we have, e.g., no matroid constraint (the classic knapsack problem) or a uniform matroid constraint (knapsack with a cardinality constraint), admit a fully polynomial-time approximation scheme (FPTAS). In contrast, the budgeted matroid independent set (BMI) problem with a general matroid has an efficient polynomial-time approximation scheme (EPTAS) but does not admit an FPTAS. This implies an EPTAS for our problem, which is the best known result prior to this work. We present an FPTAS for budgeted laminar matroid independent set, improving the previous EPTAS for this matroid family and generalizing the FPTAS known for knapsack with a cardinality constraint and multiple-choice knapsack. Our scheme is based on a simple dynamic program which utilizes the tree-like structure of laminar matroids.

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.

In this paper we derive tight lower bounds resolving the hardness status of several fundamental weighted matroid problems. One notable example is budgeted matroid independent set, for which we show there is no fully polynomial-time approximation scheme (FPTAS), indicating the Efficient PTAS of [Doron-Arad, Kulik and Shachnai, SOSA 2023] is the best possible. Furthermore, we show that there is no pseudo-polynomial time algorithm for exact weight matroid independent set, implying the algorithm of [Camerini, Galbiati and Maffioli, J. Algorithms 1992] for representable matroids cannot be generalized to arbitrary matroids. Similarly, we show there is no Fully PTAS for constrained minimum basis of a matroid and knapsack cover with a matroid, implying the existing Efficient PTAS for the former is optimal. For all of the above problems, we obtain unconditional lower bounds in the oracle model, where the independent sets of the matroid can be accessed only via a membership oracle. We complement these results by showing that the same lower bounds hold under standard complexity assumptions, even if the matroid is encoded as part of the instance. All of our bounds are based on a specifically structured family of paving matroids.

We study budgeted variants of well known maximization problems with multiple matroid constraints. Given an $\ell$-matchoid $\cm$ on a ground set $E$, a profit function $p:E \rightarrow \mathbb{R}_{\geq 0}$, a cost function $c:E \rightarrow \mathbb{R}_{\geq 0}$, and a budget $B \in \mathbb{R}_{\geq 0}$, the goal is to find in the $\ell$-matchoid a feasible set $S$ of maximum profit $p(S)$ subject to the budget constraint, i.e., $c(S) \leq B$. The {\em budgeted $\ell$-matchoid} (BM) problem includes as special cases budgeted $\ell$-dimensional matching and budgeted $\ell$-matroid intersection. A strong motivation for studying BM from parameterized viewpoint comes from the APX-hardness of unbudgeted $\ell$-dimensional matching (i.e., $B = \infty$) already for $\ell = 3$. Nevertheless, while there are known FPT algorithms for the unbudgeted variants of the above problems, the {\em budgeted} variants are studied here for the first time through the lens of parameterized complexity. We show that BM parametrized by solution size is $W[1]$-hard, already with a degenerate single matroid constraint. Thus, an exact parameterized algorithm is unlikely to exist, motivating the study of {\em FPT-approximation schemes} (FPAS). Our main result is an FPAS for BM (implying an FPAS for $\ell$-dimensional matching and budgeted $\ell$-matroid intersection), relying on the notion of representative set $-$ a small cardinality subset of elements which preserves the optimum up to a small factor. We also give a lower bound on the minimum possible size of a representative set which can be computed in polynomial time.

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 each bin forms an independent set in the matroid. The problem is a generalization of both Group Bin Packing and Bin Packing with Cardinality Constraints. Bin Packing with Partition Matroid naturally arises in resource allocation to ensure fault tolerance and security, as well as in harvesting computing capacity. Our main result is a polynomial-time algorithm that packs the items in $OPT + o(OPT)$ bins, where OPT is the minimum number of bins required for packing the given instance. This matches the best known result for the classic Bin Packing problem up to the function hidden by o(OPT). As special cases, our result improves upon the existing APTAS for Group Bin Packing and generalizes the AFTPAS for Bin Packing with Cardinality Constraints. Our approach is based on rounding a solution for a configuration-LP formulation of the problem. The rounding takes a novel point of view of prototypes in which items are interpreted as placeholders for other items and applies fractional grouping to modify a fractional solution (prototype) into one having nice integrality properties.

We study the $d$-dimensional Vector Bin Packing ($d$VBP) problem, a generalization of Bin Packing with central applications in resource allocation and scheduling. In $d$VBP, we are given a set of items, each of which is characterized by a $d$-dimensional volume vector; the objective is to partition the items into a minimum number of subsets (bins), such that the total volume of items in each subset is at most $1$ in each dimension. Our main result is an asymptotic approximation algorithm for $d$VBP that yields a ratio of $(1+\ln d-χ(d) +\varepsilon)$ for all $d \in \mathbb{N}$ and any $\varepsilon > 0$; here, $χ(d)$ is some strictly positive function. This improves upon the best known asymptotic ratio of $ \left(1+ \ln d +\varepsilon\right)$ due to Bansal, Caprara and Sviridenko (SICOMP 2010) for any $d >3$. By slightly modifying our algorithm to include an initial matching phase and applying a tighter analysis we obtain an asymptotic approximation ratio of $\left(\frac{4}{3}+\varepsilon\right)$ for the special case of $d=2$, thus substantially improving the previous best ratio of $\left(\frac{3}{2}+\varepsilon\right)$ due to Bansal, Elias and Khan (SODA 2016). Our algorithm iteratively solves a configuration LP relaxation for the residual instance (from previous iterations) and samples a small number of configurations based on the solution for the configuration LP. While iterative rounding was already used by Karmarkar and Karp (FOCS 1982) to establish their celebrated result for classic (one-dimensional) Bin Packing, iterative randomized rounding is used here for the first time in the context of (Vector) Bin Packing. Our results show that iterative randomized rounding is a powerful tool for approximating $d$VBP, leading to simple algorithms with improved approximation guarantees.

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.

We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size n which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized α-approximation algorithm that runs in c^k⋅n^𝒪(1) time, where k is the solution size, can be used to derive an α-approximation randomized algorithm that runs in dⁿ⋅n^𝒪(1) time, where d is the unique value in (1, 1+{c-1}/α) such that 𝒟(1/α‖{d-1}/{c-1}) = {ln c}/α and 𝒟(a‖b) is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for α = 1, and is strictly better when α > 1, for any c > 1. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time. We use an approximate variant of the exhaustive search as a benchmark for our algorithm. We show that the classic 2ⁿ⋅n^𝒪(1) exhaustive search can be adapted to an α-approximate exhaustive search that runs in time (1+exp(-α⋅ℋ(1/(α))))ⁿ⋅n^𝒪(1), where ℋ is the entropy function. Furthermore, we provide a lower bound stating that the running time of this α-approximate exhaustive search is the best achievable running time in an oracle model. When compared to approximate exhaustive search, and to other techniques, the running times obtained by approximate monotone local search are strictly better for any α ≥ 1, c > 1. We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, 3-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a 1.1-approximation algorithm for Vertex Cover with running time 1.114ⁿ⋅n^𝒪(1), improving upon the previously best known 1.1-approximation running in time 1.127ⁿ⋅n^𝒪(1) by Bourgeois et al. [DAM 2011]., LIPIcs, Vol. 244, 30th Annual European Symposium on Algorithms (ESA 2022), pages 50:1-50:19

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 $\widetilde{O}(|D|^{\frac{\omega}{2} n}\mathrm{polylog}(M))$ time, where $\omega \in [2,2.372)$ is the exponent of currently fastest matrix multiplication algorithm., Comment: We improved constant c, 29 pages, 5 colored figures

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.

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}.

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.

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-ε)-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., LIPIcs, Vol. 204, 29th Annual European Symposium on Algorithms (ESA 2021), pages 41:1-41:16

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., Preliminary version to appear in European Symposium on Algorithms 2019

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.

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 $\alpha$-approximation algorithm for the continuous relaxation implies a randomized $(\alpha - \eps)$-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., Comment: A preliminary version of this paper appeared in the Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, New York, January 2009