arXiv subjects

Akbar Rafiey

Publications and source records attributed to Akbar Rafiey.

16 recordsLinked to original sources

arXiv2026

The Price of Privacy For Approximating Max-CSP

We study approximation algorithms for Maximum Constraint Satisfaction Problems (Max-CSPs) under differential privacy (DP) where the constraints are considered sensitive data. Information-theoretically, we aim to classify the best approximation ratios possible for a given privacy budget $\varepsilon$. In the high-privacy regime ($\varepsilon \ll 1$), we show that any $\varepsilon$-DP algorithm cannot beat a random assignment by more than $O(\varepsilon)$ in the approximation ratio. We devise a polynomial-time algorithm which matches this barrier under the assumptions that the instances are bounded-degree and triangle-free. Finally, we show that one or both of these assumptions can be removed for specific CSPs--such as Max-Cut or Max $k$-XOR--albeit at the cost of computational efficiency.

cs.DS↗

arXiv2026

StepJack: Benchmarking Computer-Use Agent Safety Against Multi-Step Indirect Prompt Injection

Computer-use agents (CUAs) face a growing threat from indirect prompt injection, where adversarial instructions are planted in the environment such as web pages. In this paper, we introduce multi-step indirect prompt injection, a new attack class against CUAs in which the adversarial goal is decomposed into multiple innocuous-looking sub-steps and distributed across a chain of pages referenced along the agent's navigation path. We develop a pipeline to automatically decompose an adversarial goal under the constraint that the execution of the decomposed sub-steps must achieve the original goal while optimizing the innocuousness of each decomposed sub-step. With this pipeline, we build StepJack, a CUA safety benchmark with 480 test examples. On this benchmark, we evaluate six state-of-the-art CUAs and find that at a fixed decomposition depth, multi-step attacks raise attack success rate (ASR) on three of six CUAs, by up to 31.2 points (e.g., GPT-5.4-mini: 41.7% at single-step to 72.9% at three-step); averaged over the five CUAs that can reliably follow the reference chain (all but EvoCUA-32B), ASR rises from 31.3% at single-step to 36.9% at three-step. Dataset and code are available at https://github.com/BorealisAI/StepJack.

cs.CR↗

arXiv2026

The Ideal Membership Problem and Abelian Groups

Given polynomials $f_0,\dots, f_k$ the Ideal Membership Problem, IMP for short, asks if $f_0$ belongs to the ideal generated by $f_1,\dots, f_k$. In the search version of this problem the task is to find a proof of this fact. The IMP is a well-known fundamental problem with numerous applications. For instance, it underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. Although the IMP is in general intractable, in many important cases it can be efficiently solved. Mastrolilli [SODA'19] initiated a systematic study of IMPs for ideals arising from Constraint Satisfaction Problems (CSPs), parameterized by constraint languages, denoted IMP($Γ$). The ultimate goal of this line of research is to classify all such IMPs accordingly to their complexity. Mastrolilli achieved this goal for IMPs arising from CSP($Γ$) where $Γ$ is a Boolean constraint language, while Bulatov and Rafiey [STOC'22] advanced these results to several cases of CSPs over finite domains. In this paper we consider IMPs arising from CSPs over `affine' constraint languages, in which constraints are subgroups (or their cosets) of direct products of Abelian groups. This kind of CSPs include systems of linear equations and are considered one of the most important types of tractable CSPs. Some special cases of the problem have been considered before by Bharathi and Mastrolilli [MFCS'21] for linear equation modulo 2, and by Bulatov and Rafiey [STOC'22] to systems of linear equations over $GF(p)$, $p$ prime. Here we prove that if $Γ$ is an affine constraint language then IMP($Γ$) is solvable in polynomial time assuming the input polynomial has bounded degree.

cs.CC↗

arXiv2026

Geometric Algorithms for Neural Combinatorial Optimization with Constraints

Self-Supervised Learning (SSL) for Combinatorial Optimization (CO) is an emerging paradigm for solving combinatorial problems using neural networks. In this paper, we address a central challenge of SSL for CO: solving problems with discrete constraints. We design an end-to-end differentiable framework that enables us to solve discrete constrained optimization problems with neural networks. Concretely, we leverage algorithmic techniques from the literature on convex geometry and Carathéodory's theorem to decompose neural network outputs into convex combinations of polytope corners that correspond to feasible sets. This decomposition-based approach enables self-supervised training but also ensures efficient quality-preserving rounding of the neural net output into feasible solutions. Extensive experiments in cardinality-constrained optimization show that our approach can consistently outperform neural baselines. We further provide worked-out examples of how our method can be applied beyond cardinality-constrained problems to a diverse set of combinatorial optimization tasks, including finding independent sets in graphs, and solving matroid-constrained problems.

cs.LG↗

arXiv2025

Differentiable Extensions with Rounding Guarantees for Combinatorial Optimization over Permutations

Continuously extending combinatorial optimization objectives is a powerful technique commonly applied to the optimization of set functions. However, few such methods exist for extending functions on permutations, despite the fact that many combinatorial optimization problems, such as the quadratic assignment problem (QAP) and the traveling salesperson problem (TSP), are inherently optimization over permutations. We present Birkhoff Extension (BE), an almost-everywhere-differentiable continuous polytime-computable extension of any real-valued function on permutations to doubly stochastic matrices. Key to this construction is our introduction of a continuous variant of the well-known Birkhoff decomposition. Our extension has several nice properties making it appealing for optimization problems. First, BE provides a rounding guarantee, namely any solution to the extension can be efficiently rounded to a permutation without increasing the function value. Furthermore, an approximate solution in the relaxed case will give rise to an approximate solution in the space of permutations. Second, using BE, any real-valued optimization objective on permutations can be extended to an almost-everywhere-differentiable objective function over the space of doubly stochastic matrices. This makes our BE amenable to not only gradient-descent based optimization, but also unsupervised neural combinatorial optimization where training often requires a differentiable loss. Third, based on the above properties, we present a simple optimization procedure which can be readily combined with existing optimization approaches to offer local improvements (i.e., the quality of the final solution is no worse than the initial solution). Finally, we also adapt our extension to optimization problems over a class of trees, such as Steiner tree and optimization-based hierarchical clustering.

cs.DS↗

arXiv2025

Explaining GNN Explanations with Edge Gradients

In recent years, the remarkable success of graph neural networks (GNNs) on graph-structured data has prompted a surge of methods for explaining GNN predictions. However, the state-of-the-art for GNN explainability remains in flux. Different comparisons find mixed results for different methods, with many explainers struggling on more complex GNN architectures and tasks. This presents an urgent need for a more careful theoretical analysis of competing GNN explanation methods. In this work we take a closer look at GNN explanations in two different settings: input-level explanations, which produce explanatory subgraphs of the input graph, and layerwise explanations, which produce explanatory subgraphs of the computation graph. We establish the first theoretical connections between the popular perturbation-based and classical gradient-based methods, as well as point out connections between other recently proposed methods. At the input level, we demonstrate conditions under which GNNExplainer can be approximated by a simple heuristic based on the sign of the edge gradients. In the layerwise setting, we point out that edge gradients are equivalent to occlusion search for linear GNNs. Finally, we demonstrate how our theoretical results manifest in practice with experiments on both synthetic and real datasets.

cs.LG↗

arXiv2024

Decomposable Submodular Maximization in Federated Setting

Submodular functions, as well as the sub-class of decomposable submodular functions, and their optimization appear in a wide range of applications in machine learning, recommendation systems, and welfare maximization. However, optimization of decomposable submodular functions with millions of component functions is computationally prohibitive. Furthermore, the component functions may be private (they might represent user preference function, for example) and cannot be widely shared. To address these issues, we propose a {\em federated optimization} setting for decomposable submodular optimization. In this setting, clients have their own preference functions, and a weighted sum of these preferences needs to be maximized. We implement the popular {\em continuous greedy} algorithm in this setting where clients take parallel small local steps towards the local solution and then the local changes are aggregated at a central server. To address the large number of clients, the aggregation is performed only on a subsampled set. Further, the aggregation is performed only intermittently between stretches of parallel local steps, which reduces communication cost significantly. We show that our federated algorithm is guaranteed to provide a good approximate solution, even in the presence of above cost-cutting measures. Finally, we show how the federated setting can be incorporated in solving fundamental discrete submodular optimization problems such as Maximum Coverage and Facility Location.

cs.DS↗

arXiv2022

Toward a Dichotomy for Approximation of $H$-coloring

Given two (di)graphs G, H and a cost function $c:V(G)\times V(H) \to \mathbb{Q}_{\geq 0}\cup\{+\infty\}$, in the minimum cost homomorphism problem, MinHOM(H), goal is finding a homomorphism $f:V(G)\to V(H)$ (a.k.a H-coloring) that minimizes $\sum\limits_{v\in V(G)}c(v,f(v))$. The complexity of exact minimization of this problem is well understood [34], and the class of digraphs H, for which the MinHOM(H) is polynomial time solvable is a small subset of all digraphs. In this paper, we consider the approximation of MinHOM within a constant factor. For digraphs, MinHOM(H) is not approximable if H contains a digraph asteroidal triple (DAT). We take a major step toward a dichotomy classification of approximable cases. We give a dichotomy classification for approximating the MinHOM(H) when H is a graph. For digraphs, we provide constant factor approximation algorithms for two important classes of digraphs, namely bi-arc digraphs (digraphs with a conservative semi-lattice polymorphism or min-ordering), and k-arc digraphs (digraphs with an extended min-ordering). Specifically, we show that: 1. Dichotomy for Graphs: MinHOM(H) has a $2|V(H)|$-approximation algorithm if graph H admits a conservative majority polymorphims (i.e. H is a bi-arc graph), otherwise, it is inapproximable; 2. MinHOM(H) has a $|V(H)|^2$-approximation algorithm if H is a bi-arc digraph; 3. MinHOM(H) has a $|V(H)|^2$-approximation algorithm if H is a k-arc digraph. In conclusion, we show the importance of these results and provide insights for achieving a dichotomy classification of approximable cases. Our constant factors depend on the size of H. However, the implementation of our algorithms provides a much better approximation ratio. It leaves open to investigate a classification of digraphs H, where MinHOM(H) admits a constant factor approximation algorithm that is independent of H.

cs.DS↗

arXiv2022

Private Boosted Decision Trees via Smooth Re-Weighting

Protecting the privacy of people whose data is used by machine learning algorithms is important. Differential Privacy is the appropriate mathematical framework for formal guarantees of privacy, and boosted decision trees are a popular machine learning technique. So we propose and test a practical algorithm for boosting decision trees that guarantees differential privacy. Privacy is enforced because our booster never puts too much weight on any one example; this ensures that each individual's data never influences a single tree "too much." Experiments show that this boosting algorithm can produce better model sparsity and accuracy than other differentially private ensemble classifiers.

cs.LG↗

arXiv2022

Sparsification of Decomposable Submodular Functions

Submodular functions are at the core of many machine learning and data mining tasks. The underlying submodular functions for many of these tasks are decomposable, i.e., they are sum of several simple submodular functions. In many data intensive applications, however, the number of underlying submodular functions in the original function is so large that we need prohibitively large amount of time to process it and/or it does not even fit in the main memory. To overcome this issue, we introduce the notion of sparsification for decomposable submodular functions whose objective is to obtain an accurate approximation of the original function that is a (weighted) sum of only a few submodular functions. Our main result is a polynomial-time randomized sparsification algorithm such that the expected number of functions used in the output is independent of the number of underlying submodular functions in the original function. We also study the effectiveness of our algorithm under various constraints such as matroid and cardinality constraints. We complement our theoretical analysis with an empirical study of the performance of our algorithm.

cs.DS↗

arXiv2021

On the Complexity of CSP-based Ideal Membership Problems

In this paper we consider the Ideal Membership Problem (IMP for short), in which we are given real polynomials $f_0,f_1,\dots, f_k$ and the question is to decide whether $f_0$ belongs to the ideal generated by $f_1,\dots,f_k$. In the more stringent version the task is also to find a proof of this fact. The IMP underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. In the majority of such applications the IMP involves so called combinatorial ideals that arise from a variety of discrete combinatorial problems. This restriction makes the IMP significantly easier and in some cases allows for an efficient algorithm to solve it. The first part of this paper follows the work of Mastrolilli [SODA'19] who initiated a systematic study of IMPs arising from Constraint Satisfaction Problems (CSP) of the form $CSP(Γ)$, that is, CSPs in which the type of constraints is limited to relations from a set $Γ$. We show that many CSP techniques can be translated to IMPs thus allowing us to significantly improve the methods of studying the complexity of the IMP. We also develop universal algebraic techniques for the IMP that have been so useful in the study of the CSP. This allows us to prove a general necessary condition for the tractability of the IMP, and three sufficient ones. The sufficient conditions include IMPs arising from systems of linear equations over $GF(p)$, $p$ prime, and also some conditions defined through special kinds of polymorphisms. Our work has several consequences and applications in terms of bit complexity of sum-of-squares (SOS) proofs and their automatizability, and studying (construction of) theta bodies of combinatorial problems.

cs.CC↗

arXiv2020

Fast and Private Submodular and $k$-Submodular Functions Maximization with Matroid Constraints

The problem of maximizing nonnegative monotone submodular functions under a certain constraint has been intensively studied in the last decade, and a wide range of efficient approximation algorithms have been developed for this problem. Many machine learning problems, including data summarization and influence maximization, can be naturally modeled as the problem of maximizing monotone submodular functions. However, when such applications involve sensitive data about individuals, their privacy concerns should be addressed. In this paper, we study the problem of maximizing monotone submodular functions subject to matroid constraints in the framework of differential privacy. We provide $(1-\frac{1}{\mathrm{e}})$-approximation algorithm which improves upon the previous results in terms of approximation guarantee. This is done with an almost cubic number of function evaluations in our algorithm. Moreover, we study $k$-submodularity, a natural generalization of submodularity. We give the first $\frac{1}{2}$-approximation algorithm that preserves differential privacy for maximizing monotone $k$-submodular functions subject to matroid constraints. The approximation ratio is asymptotically tight and is obtained with an almost linear number of function evaluations.

cs.DS↗

arXiv2020

Bi-Arc Digraphs and Conservative Polymorphisms

In this paper we study the class of bi-arc digraphs, important from two seemingly unrelated perspectives. On the one hand, they are precisely the digraphs that admit certain polymorphisms of interest in the study of constraint satisfaction problems; on the other hand, they are a very broad generalization of interval graphs. Bi-arc digraphs is the class of digraphs that admit conservative semilattice polymorphisms. There is much interest in understanding structures that admit particular types of polymorphisms, and especially in their recognition algorithms. (Such problems are referred to as metaproblems.) Surprisingly, the class of bi-arc digraphs also describes the class of digraphs that admit certain other kinds of conservative polymorphisms. Thus solving the recognition problem for bi-arc digraphs solves the metaproblem for digraphs for several types of conservative polymorphisms. The complexity of the recognition problem for digraphs with conservative semilattice polymorphisms was an open problem, while it was known to be NP-complete for certain more complex relational structures. We complement our result by providing a complete dichotomy classification of which general relational structures have polynomial or NP-complete recognition problems for the existence of conservative semilattice polymorphisms. Bi-arc digraphs also generalizes the class of interval graphs; in fact it reduces to the class of interval graphs for symmetric and reflexive digraphs. It is much broader than interval graphs and includes other generalizations of interval graphs such as co-threshold tolerance graphs and adjusted interval digraphs. Yet, it is still a reasonable extension of interval graphs, in the sense that it keeps much of the appeal of interval graphs. Our main result is a forbidden obstruction characterization of, and a polynomial recognition for, the class of bi-arc digraphs.

cs.DS↗

arXiv2019

Counting maximal near perfect matchings in quasirandom and dense graphs

A maximal $\varepsilon$-near perfect matching is a maximal matching which covers at least $(1-\varepsilon)|V(G)|$ vertices. In this paper, we study the number of maximal near perfect matchings in generalized quasirandom and dense graphs. We provide tight lower and upper bounds on the number of $\varepsilon$-near perfect matchings in generalized quasirandom graphs. Moreover, based on these results, we provide a deterministic polynomial time algorithm that for a given dense graph $G$ of order $n$ and a real number $\varepsilon>0$, returns either a conclusion that $G$ has no $\varepsilon$-near perfect matching, or a positive non-trivial number $\ell$ such that the number of maximal $\varepsilon$-near perfect matchings in $G$ is at least $n^{\ell n}$. Our algorithm uses algorithmic version of Szemerédi Regularity Lemma, and has $O(f(\varepsilon)n^{5/2})$ time complexity. Here $f(\cdot)$ is an explicit function depending only on $\varepsilon$.

math.CO↗

arXiv2016

Ordering with precedence constraints and budget minimization

We introduce a variation of the scheduling with precedence constraints problem that has applications to molecular folding and production management. We are given a bipartite graph $H=(B,S)$. Vertices in $B$ are thought of as goods or services that must be \emph{bought} to produce items in $S$ that are to be \emph{sold}. An edge from $j\in S$ to $i\in B$ indicates that the production of $j$ requires the purchase of $i$. Each vertex in $B$ has a cost, and each vertex in $S$ results in some gain. The goal is to obtain an ordering of $B\cup S$ that respects the precedence constraints and maximizes the minimal net profit encountered as the vertices are processed. We call this optimal value the \emph{budget} or \emph{capital} investment required for the bipartite graph, and refer to our problem as \emph{the bipartite graph ordering problem}. The problem is equivalent to a version of an NP-complete molecular folding problem that has been studied recently [12]. Work on the molecular folding problem has focused on heuristic algorithms and exponential-time exact algorithms for the un-weighted problem where costs are $\pm 1$ and when restricted to graphs arising from RNA folding. The bipartite graph present work seeks exact algorithms for solving the bipartite ordering problem. We demonstrate an algorithm that computes the optimal ordering in time $O^*(2^n)$ when $n$ is the number of vertices in the input bipartite graph. Our main result is a general strategy that can be used to find an optimal ordering in polynomial time for bipartite graphs that satisfy certain properties. We apply the technique to a variety of graph classes, obtaining polynomial-time solutions to the bipartite graph ordering problem for bipartite permutation graphs, trivially perfect, co-bipartite graphs, and trees.

cs.DM↗

arXiv2015

Approximation Algorithms for Generalized MST and TSP in Grid Clusters

We consider a special case of the generalized minimum spanning tree problem (GMST) and the generalized travelling salesman problem (GTSP) where we are given a set of points inside the integer grid (in Euclidean plane) where each grid cell is $1 \times 1$. In the MST version of the problem, the goal is to find a minimum tree that contains exactly one point from each non-empty grid cell (cluster). Similarly, in the TSP version of the problem, the goal is to find a minimum weight cycle containing one point from each non-empty grid cell. We give a $(1+4\sqrt{2}+ε)$ and $(1.5+8\sqrt{2}+ε)$-approximation algorithm for these two problems in the described setting, respectively. Our motivation is based on the problem posed in [7] for a constant approximation algorithm. The authors designed a PTAS for the more special case of the GMST where non-empty cells are connected end dense enough. However, their algorithm heavily relies on this connectivity restriction and is unpractical. Our results develop the topic further.

cs.DM↗