arXiv subjects

Hadas Shachnai

Publications and source records attributed to Hadas Shachnai.

At least 19 recordsLinked to original sources

arXiv2026

Online Contract Design

We initiate the study of online contracts, which integrate the game-theoretic considerations of economic contract theory, with the algorithmic and informational challenges of online algorithm design. Our starting point is the classic online setting with preemption of Buchbinder et al. [SODA'15], in which a hiring principal faces a sequence of adversarial agent arrivals. Upon arrival, the principal must decide whether to tentatively accept the agent to their team, and whether to dismiss previous tentative choices. Dismissal is irrevocable, giving the setting its online decision-making flavor. In our setting, the agents are rational players: once the team is finalized, a game is played where the principal offers contracts (performance-based payment schemes), and each agent decides whether or not to work. Working agents reward the principal, and the goal is to choose a team that maximizes the principal's utility. Our main positive result is a 1/2-competitive algorithm when agent rewards are additive, which matches the best-possible competitive ratio. Our algorithm is randomized and this is necessary, as we show that no deterministic algorithm can attain a bounded competitive ratio. Moreover, if agent rewards are allowed to exhibit combinatorial structure known as XOS, even randomized algorithms might fail. En route to our competitive algorithm, we develop the technique of balance points, which can be useful for further exploration of online contracts in the adversarial model.

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arXiv2025

You (Almost) Can't Beat Brute Force for 3-Matroid Intersection

The $\ell$-matroid intersection ($\ell$-MI) problem asks if $\ell$ given matroids share a common basis. Already for $\ell = 3$, notable canonical NP-complete special cases are $3$-Dimensional Matching and Hamiltonian Path on directed graphs. However, while these problems admit exponential-time algorithms that improve the simple brute force, the fastest known algorithm for $3$-MI is exactly brute force with runtime $2^{n}/poly(n)$, where $n$ is the number of elements. Our first result shows that in fact, brute force cannot be significantly improved, by ruling out an algorithm for $\ell$-MI with runtime $o\left(2^{n-5 \cdot n^{\frac{1}{\ell-1}} \cdot \log (n)}\right)$, for any fixed $\ell\geq 3$. We further obtain: (i) an algorithm that solves $\ell$-MI faster than brute force in time $2^{n-Ω\left(\log^2 (n)\right)} $ (ii) a parameterized running time lower bound of $2^{(\ell-2) \cdot k \cdot \log k} \cdot poly(n)$ for $\ell$-MI, where the parameter $k$ is the rank of the matroids. We obtain these two results by generalizing the Monotone Local Search technique of Fomin et al. (J. ACM'19). Broadly speaking, our generalization converts any parameterized algorithm for a subset problem into an exponential-time algorithm which is faster than brute-force.

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arXiv2025

An Algorithm-to-Contract Framework without Demand Queries

Consider costly and time-consuming tasks that add up to the success of a project, and must be fitted into a given time-frame. This is an instance of the classic budgeted maximization (knapsack) problem, which admits an FPTAS. Now assume an agent is performing these tasks on behalf of a principal, who is the one to reap the rewards if the project succeeds. The principal must design a contract to incentivize the agent. Is there still an approximation scheme? In this work we lay the foundations for an algorithm-to-contract framework, which transforms algorithms for combinatorial problems to handle contract design problems subject to the same combinatorial constraints. Our approach diverges from previous works in avoiding the assumption of demand oracle access. As an example, for budgeted maximization, we show how to "lift" the classic FPTAS to the best-possible (approximately-IC) FPTAS for the contract problem. We establish this through our local-to-global framework, in which the local step is to approximately solve a two-sided strengthened variant of the demand problem. The global step then utilizes the local one to find the approximately optimal contract. We apply our framework to a host of combinatorial constraints: multi-dimensional budgets, budgeted matroid, and budgeted matching constraints. In all cases we essentially match the best purely algorithmic approximation. Separately, we also develop a method for multi-agent contract settings. Our method yields the first approximation schemes for multi-agent contract settings that go beyond additive reward functions.

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arXiv2025

Analysis of Two-variable Recurrence Relations with Application to Parameterized Approximations

In this paper we introduce randomized branching as a tool for parameterized approximation and develop the mathematical machinery for its analysis. Our algorithms improve the best known running times of parameterized approximation algorithms for Vertex Cover and $3$-Hitting Set for a wide range of approximation ratios. One notable example is a simple parameterized random $1.5$-approximation algorithm for Vertex Cover, whose running time of $\tilde{O}(1.01657^k)$ substantially improves the best known runnning time of $\tilde{O}(1.0883^k)$ [Brankovic and Fernau, 2013]. For $3$-Hitting Set we present a parameterized random $2$-approximation algorithm with running time of $\tilde{O}(1.0659^k)$, improving the best known $\tilde{O}(1.29^k)$ algorithm of [Brankovic and Fernau, 2012]. The running times of our algorithms are derived from an asymptotic analysis of a wide class of two-variable recurrence relations of the form: $$p(b,k) = \min_{1\leq j \leq N} \sum_{i=1}^{r_j} \barγ_i^j \cdot p(b-\bar{b}^j_i, k-\bar{k}_i^j),$$ where $\bar{b}^j$ and $\bar{k}^j$ are vectors of natural numbers, and $\barγ^j$ is a probability distribution over $r_j$ elements, for $1\leq j \leq N$. Our main theorem asserts that for any $α>0$, $$\lim_{k \rightarrow \infty } \frac{1}{k} \cdot \ln p(\lfloor{αk}\rfloor,k) = -\max_{1\leq j \leq N} M_j,$$ where $M_j$ depends only on $α$, $\barγ^j$, $\bar{b}^j$ and $\bar{k}^j$, and can be efficiently calculated by solving a simple numerical optimization problem. To prove the theorem we show an equivalence between the recurrence and a stochastic process. We analyze this process using the {\em method of types}, by introducing an adaptation of Sanov's theorem to our setting. We believe our novel analysis of recurrence relations which is of independent interest is a main contribution of this paper.

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arXiv2025

Finding Possible Winners in Spatial Voting with Incomplete Information

We consider a spatial voting model where both candidates and voters are positioned in the $d$-dimensional Euclidean space, and each voter ranks candidates based on their proximity to the voter's ideal point. We focus on the scenario where the given information about the locations of the voters' ideal points is incomplete; for each dimension, only an interval of possible values is known. In this context, we investigate the computational complexity of determining the possible winners under positional scoring rules. Our results show that the possible winner problem in one dimension is solvable in polynomial time for all $k$-truncated voting rules with constant $k$. Moreover, for some scoring rules for which the possible winner problem is NP-complete, such as approval voting for any dimension or $k$-approval for $d \geq 2$ dimensions, we give an FPT algorithm parameterized by the number of candidates. Finally, we classify tractable and intractable settings of the weighted possible winner problem in one dimension, and resolve the computational complexity of the weighted case for all two-valued positional scoring rules when $d=1$.

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arXiv2024

Spatial Voting with Incomplete Voter Information

We consider spatial voting where candidates are located in the Euclidean $d$-dimensional space, and each voter ranks candidates based on their distance from the voter's ideal point. We explore the case where information about the location of voters' ideal points is incomplete: for each dimension, we are given an interval of possible values. We study the computational complexity of finding the possible and necessary winners for positional scoring rules. Our results show that we retain tractable cases of the classic model where voters have partial-order preferences. Moreover, we show that there are positional scoring rules under which the possible-winner problem is intractable for partial orders, but tractable in the one-dimensional spatial setting. We also consider approval voting in this setting. We show that for up to two dimensions, the necessary-winner problem is tractable, while the possible-winner problem is hard for any number of dimensions.

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arXiv2024

An EPTAS for Cardinality Constrained Multiple Knapsack via Iterative Randomized Rounding

In [Math. Oper. Res., 2011], Fleischer et al. introduced a powerful technique for solving the generic class of separable assignment problems (SAP), in which a set of items of given values and weights needs to be packed into a set of bins subject to separable assignment constraints, so as to maximize the total value. The approach of Fleischer at al. relies on solving a configuration LP and sampling a configuration for each bin independently based on the LP solution. While there is a SAP variant for which this approach yields the best possible approximation ratio, for various special cases, there are discrepancies between the approximation ratios obtained using the above approach and the state-of-the-art approximations. This raises the following natural question: Can we do better by iteratively solving the configuration LP and sampling a few bins at a time? To assess the potential gain from iterative randomized rounding, we consider as a case study one interesting SAP variant, namely, Uniform Cardinality Constrained Multiple Knapsack, for which we answer this question affirmatively. The input is a set of items, each has a value and a weight, and a set of uniform capacity bins. The goal is to assign a subset of the items of maximum total value to the bins such that $(i)$ the capacity of any bin is not exceeded, and $(ii)$ the number of items assigned to each bin satisfies a given cardinality constraint. While the technique of Fleischer et al. yields a $\left(1-\frac{1}{e}\right)$-approximation for the problem, we show that iterative randomized rounding leads to an efficient polynomial time approximation scheme (EPTAS), thus essentially resolving the complexity status of the problem. Our analysis of iterative randomized rounding can be useful for solving other SAP variants.

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arXiv2024

Lower Bounds for Matroid Optimization Problems with a Linear Constraint

We study a family of matroid optimization problems with a linear constraint (MOL). In these problems, we seek a subset of elements which optimizes (i.e., maximizes or minimizes) a linear objective function subject to (i) a matroid independent set, or a matroid basis constraint, (ii) additional linear constraint. A notable member in this family is budgeted matroid independent set (BM), which can be viewed as classic $0/1$-knapsack with a matroid constraint. While special cases of BM, such as knapsack with cardinality constraint and multiple-choice knapsack, admit a fully polynomial-time approximation scheme (Fully PTAS), the best known result for BM on a general matroid is an Efficient PTAS. Prior to this work, the existence of a Fully PTAS for BM, and more generally, for any problem in the family of MOL problems, has been open. In this paper, we answer this question negatively by showing that none of the (non-trivial) problems in this family admits a Fully PTAS. This resolves the complexity status of several well studied problems. Our main result is obtained by showing first that exact weight matroid basis (EMB) does not admit a pseudo-polynomial time algorithm. This distinguishes EMB from the special cases of $k$-subset sum and EMB on a linear matroid, which are solvable in pseudo-polynomial time. We then obtain unconditional hardness results for the family of MOL problems in the oracle model (even if randomization is allowed), and show that the same results hold when the matroids are encoded as part of the input, assuming $P \neq NP$. For the hardness proof of EMB, we introduce the $Π$-matroid family. This intricate subclass of matroids, which exploits the interaction between a weight function and the matroid constraint, may find use in tackling other matroid optimization problems.

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arXiv2024

Approximations and Hardness of Packing Partially Ordered Items

Motivated by applications in production planning and storage allocation in hierarchical databases, we initiate the study of covering partially ordered items (CPO). Given a capacity $k \in \mathbb{Z}^+$, and a directed graph $G=(V,E)$ where each vertex has a size in $\{0,1, \ldots,k\}$, we seek a collection of subsets of vertices $S_1, \ldots, S_m$ that cover all the vertices, such that for any $1 \leq j \leq m$, the total size of vertices in $S_j$ is bounded by $k$, and there are no edges from $V \setminus S_j$ to $S_j$. The objective is to minimize the number of subsets $m$. CPO is closely related to the rule caching problem (RCP) that is of wide interest in the networking area. The input for RCP is a directed graph $G=(V,E)$, a profit function $p:V \rightarrow \mathbb{Z}_{0}^+$, and $k \in \mathbb{Z}^+$. The output is a subset $S \subseteq V$ of maximum profit such that $|S| \leq k$ and there are no edges from $V \setminus S$ to $S$. Our main result is a $2$-approximation algorithm for CPO on out-trees, complemented by an asymptotic $1.5$-hardness of approximation result. We also give a two-way reduction between RCP and the densest $k$-subhypergraph problem, surprisingly showing that the problems are equivalent w.r.t. polynomial-time approximation within any factor $ρ\geq 1$. This implies that RCP cannot be approximated within factor $|V|^{1-\eps}$ for any fixed $\eps>0$, under standard complexity assumptions. Prior to this work, RCP was just known to be strongly NP-hard. We further show that there is no EPTAS for the special case of RCP where the profits are uniform, assuming Gap-ETH. Since this variant admits a PTAS, we essentially resolve the complexity status of this problem.

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arXiv2023

Tight Bounds for Budgeted Maximum Weight Independent Set in Bipartite and Perfect Graphs

We consider the classic budgeted maximum weight independent set (BMWIS) problem. The input is a graph $G = (V,E)$, a weight function $w:V \rightarrow \mathbb{R}_{\geq 0}$, a cost function $c:V \rightarrow \mathbb{R}_{\geq 0}$, and a budget $B \in \mathbb{R}_{\geq 0}$. The goal is to find an independent set $S \subseteq V$ in $G$ such that $\sum_{v \in S} c(v) \leq B$, which maximizes the total weight $\sum_{v \in S} w(v)$. Since the problem on general graphs cannot be approximated within ratio $|V|^{1-\varepsilon}$ for any $\varepsilon>0$, BMWIS has attracted significant attention on graph families for which a maximum weight independent set can be computed in polynomial time. Two notable such graph families are bipartite and perfect graphs. BMWIS is known to be NP-hard on both of these graph families; however, the best possible approximation guarantees for these graphs are wide open. In this paper, we give a tight $2$-approximation for BMWIS on perfect graphs and bipartite graphs. In particular, we give We a $(2-\varepsilon)$ lower bound for BMWIS on bipartite graphs, already for the special case where the budget is replaced by a cardinality constraint, based on the Small Set Expansion Hypothesis (SSEH). For the upper bound, we design a $2$-approximation for BMWIS on perfect graphs using a Lagrangian relaxation based technique. Finally, we obtain a tight lower bound for the capacitated maximum weight independent set (CMWIS) problem, the special case of BMWIS where $w(v) = c(v)~\forall v \in V$. We show that CMWIS on bipartite and perfect graphs is unlikely to admit an efficient polynomial-time approximation scheme (EPTAS). Thus, the existing PTAS for CMWIS is essentially the best we can expect.

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arXiv2023

Budgeted Matroid Maximization: a Parameterized Viewpoint

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.

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arXiv2023

Improved Approximation for Two-dimensional Vector Multiple Knapsack

We study the uniform $2$-dimensional vector multiple knapsack (2VMK) problem, a natural variant of multiple knapsack arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a $2$-dimensional weight vector and a positive profit, along with $m$ $2$-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized. Our main result is a $(1- \frac{\ln 2}{2} - \varepsilon)$-approximation algorithm for 2VMK, for every fixed $\varepsilon > 0$, thus improving the best known ratio of $(1 - \frac{1}{e}-\varepsilon)$ which follows as a special case from a result of [Fleischer at al., MOR 2011]. Our algorithm relies on an adaptation of the Round$\&$Approx framework of [Bansal et al., SICOMP 2010], originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to $\approx m\cdot \ln 2 \approx 0.693\cdot m$ of the bins, followed by a reduction to the ($1$-dimensional) Multiple Knapsack problem for assigning items to the remaining bins.

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arXiv2023

Bin Packing with Partition Matroid can be Approximated within $o(OPT)$ Bins

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.

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arXiv2023

Improved Approximations for Vector Bin Packing via Iterative Randomized Rounding

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.

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arXiv2023

An FPTAS for Budgeted Laminar Matroid Independent Set

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.

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arXiv2023

Approximating Bin Packing with Conflict Graphs via Maximization Techniques

We give a comprehensive study of bin packing with conflicts (BPC). The input is a set $I$ of items, sizes $s:I \rightarrow [0,1]$, and a conflict graph $G = (I,E)$. The goal is to find a partition of $I$ into a minimum number of independent sets, each of total size at most $1$. Being a generalization of the notoriously hard graph coloring problem, BPC has been studied mostly on polynomially colorable conflict graphs. An intriguing open question is whether BPC on such graphs admits the same best known approximation guarantees as classic bin packing. We answer this question negatively, by showing that (in contrast to bin packing) there is no asymptotic polynomial-time approximation scheme (APTAS) for BPC already on seemingly easy graph classes, such as bipartite and split graphs. We complement this result with improved approximation guarantees for BPC on several prominent graph classes. Most notably, we derive an asymptotic $1.391$-approximation for bipartite graphs, a $2.445$-approximation for perfect graphs, and a $\left(1+\frac{2}{e}\right)$-approximation for split graphs. To this end, we introduce a generic framework relying on a novel interpretation of BPC allowing us to solve the problem via maximization techniques. Our framework may find use in tackling BPC on other graph classes arising in applications.

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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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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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