arXiv subjects
Publications and source records attributed to Pasin Manurangsi.
We prove the following asymptotically tight lower bound for $k$-color discrepancy: For any $k \geq 2$, there exists a hypergraph with $n$ hyperedges such that its $k$-color discrepancy is at least $Ω(\sqrt{n})$. This improves on the previously known lower bound of $Ω(\sqrt{n/\log k})$ due to Caragiannis et al. (arXiv:2502.10516). As an application, we show that our result implies improved lower bounds for group fair division.
In the Densest $k$-Subgraph problem, given an undirected graph $G$ and an integer $k$, the goal is to find a subgraph of $G$ on $k$ vertices that contains maximum number of edges. Even though the state-of-the-art algorithm for the problem achieves only $O(n^{1/4 + \varepsilon})$ approximation ratio (Bhaskara et al., 2010), previous attempts at proving hardness of approximation, including those under average case assumptions, fail to achieve a polynomial ratio; the best ratios ruled out under any worst case assumption and any average case assumption are only any constant (Raghavendra and Steurer, 2010) and $2^{Ω(\log^{2/3} n)}$ (Alon et al., 2011) respectively. In this work, we propose the Almost-Exponential Time Hypothesis (Almost-ETH), which postulates that no algorithm can solve 3SAT on $n$ variables in $O(2^{n^{1 - γ}})$ time for any constant $γ> 0$. Assuming Almost-ETH, we show that, for any constant $\varepsilon > 0$, there is no polynomial-time algorithm that approximates {\sc D$k$S} to within $2^{(\log n)^{1 - \varepsilon}}$ factor of the optimum. In addition, our result has perfect completeness, meaning that, assuming Almost-ETH, no polynomial-time algorithm can even distinguish between the case in which $G$ contains a $k$-clique and the case in which every induced $k$-subgraph of $G$ has density at most $2^{-(\log n)^{1 - \varepsilon}}$.
Benade et al. (EC 2024) introduced a semi-random model for non-additive fair division in which each agent starts with an arbitrary base valuation over bundles of items and independently permutes the item names uniformly at random. While they proved that envy-free allocations exist with high probability as the number of items grows for $n = 2$ agents, they left open whether the same holds for $n > 2$ agents. We resolve this open question in the affirmative: for any fixed number of agents $n$ and any number of items $m$ divisible by $n$, an envy-free allocation exists with probability $1 - o(1)$ as $m \to \infty$. Our proof combines the second moment method with a connection to the (small-set) expansion of generalized Johnson graphs, which may be of independent interest.
We introduce CALLIOPE, a top-down framework for discovering topic hierarchies in large corpora of unstructured data (e.g., LLM conversations), while supporting differential privacy (DP). Unlike traditional and even recent LLM-augmented approaches that ultimately rely on clustering vector embeddings, our framework leverages LLM capabilities directly for the entire pipeline. Experiments demonstrate that this approach provides better utility over existing (embedding-based) methods in the DP setting, especially in the high privacy regime. We evaluate the quality of CALLIOPE outputs using a variety of traditionally studied clustering metrics, as well as some novel utility metrics that we introduce.
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.
We study combinatorial optimization problems under the constraint of $ε$-differential privacy ($ε$-DP). Given the strong lower bounds for explicitly outputting solutions, we work within the implicit representation framework of Gupta et al. (SODA 2010), where a private polynomial-time randomized "encoder" generates a representation of a solution, and a "decoder" uses this representation along with the input to extract a valid final solution. In this work, we generalize this framework by allowing the encoder to run in fixed-parameter tractable time. This circumvents approximation barriers inherent to polynomial-time algorithms and obtains improved guarantees for many fundamental combinatorial optimization problems. Finally, we establish the first representation-independent lower bounds for our framework. Assuming a non-uniform variant of the Gap Exponential Time Hypothesis, for sufficiently small $ε> 0$, we prove that no $ε$-DP encoder-decoder pair can achieve certain approximation guarantees, if the decoder runs in subexponential time. We further provide representation-dependent lower bounds that hold even for larger $ε$.
In online facility location, points arrive one at a time, and the algorithm must either open a facility at the arriving point or route the point to an existing facility. The Meyerson sketch opens a facility at each arriving point with probability proportional to the point's distance to the closest open facility, and requires no state beyond the set of open centers. Due to its simplicity, space efficiency, and strong guarantees against the offline optimum, the Meyerson sketch has become a workhorse of streaming and online clustering. In many such applications, however, the set of open facilities are visible to the process that generates the stream, which can adaptively select future points based on the algorithm's past random choices, voiding its classical guarantees. In this work, we quantify the effect of such adaptivity. We compare an adaptively generated run of the sketch against an \emph{oblivious replay}, an independent execution, with fresh coins, on the very same generated sequence, and study the \emph{adaptivity ratio} of expected adaptive cost to expected replay cost. We determine the worst-case ratio in both directions: adaptivity can neither inflate nor deflate the expected cost, or the number of open facilities, by more than an $O(\logΔ/\log\logΔ)$ factor, where $Δ$ is the aspect ratio of the input points (the ratio of the largest to the smallest pairwise distance). This is asymptotically tight as there are deterministic generators on the real line that inflate or deflate the cost by an $Ω(\logΔ/\log\logΔ)$ factor. We show that these robustness guarantees carry over to Meyerson-based sketches for approximate $k$ clustering with sketch size $O(k\,\mathrm{polylog}(n))$.
We study the question of answering linear queries with differential privacy using few (expected) random bits. We provide a randomness-efficient analog of the $\| \cdot \|_K$-norm mechanism of Hardt and Talwar [HT10]. For the $\ell_\infty$-error, our algorithm can answer $d$ linear queries with $O(d / \varepsilon)$ error using $O(\log d)$ random bits, improving upon algorithms of Canonne et al. and Ghentiyala [CSV25, Ghe26]; this is optimal when $\varepsilon \le 1/d$. We also provide a computationally efficient version of our algorithm, albeit with an $O(\log d)$ multiplicative increase in the error.
We study differentially private (DP) $k$-means and $k$-median clustering in the online streaming setting. In this model, points arrive sequentially, and at each time step, we need to output a set of $k$ centers that optimizes the clustering objective for all points seen so far. We give a generic reduction that transforms the (sensitive) input stream into a private stream, which is a semi-coreset of the input stream. This implies that any (non-private) online clustering algorithm, run as a post-processing step, can achieve good utility for the original clustering objective. Our algorithm matches or improves upon the approximation ratio, space usage, and running time of existing algorithms [Epasto et al., 2026, Dupré la Tour et al., 2024]. A key aspect of our reduction is that it inherits desirable properties of the underlying non-private clustering algorithm, such as consistency [Lattanzi and Vassilvitskii, 2017]--a property not satisfied by previous DP algorithms.
Differential privacy protects individual voting records by injecting randomness into the published outcome, but this noise can lead to erroneous results when an election is close. We study how precise central differential privacy and local differential privacy can be for common voting rules, including Plurality, Condorcet, Maximin, Plurality with Runoff, and Single Transferable Vote (STV). Our measure of precision is the margin of victory needed for a private mechanism to return the same winner as the non-private rule with high probability. We give private algorithms for publishing the winner and prove upper bounds on the required margin for these algorithms. We also prove lower bounds showing that nontrivial margins are necessary; many of these bounds match the corresponding upper bounds up to logarithmic factors. For STV, an information-theoretic upper bound matches the lower bound, but we prove that this guarantee cannot be achieved in polynomial time unless NP $\subseteq$ BPP. This gives a rare example of a computationally tractable task that becomes intractable when one simultaneously requires differential privacy and utility.
We study the maximum coverage problem for geometric set systems: given a set of points, a set of geometric objects, and a number $k$, select $k$ objects maximizing the number of points inside their union. - We present a polynomial-time approximation algorithm with approximation factor strictly better than $1-1/e$ for any set system with linear 2-shallow cell complexity (or any set system that can be decomposed into a constant number of such set systems). The result also holds for the weighted maximum coverage problem, where objects have weights and we want to select objects with total weight within a given budget. The result applies to many types of geometric objects, including pseudodisks in 2D, fat axis-aligned rectangles in 2D, similar-size fat triangles in 2D, axis-aligned unit cubes in 3D. - For small $k$, we obtain a $(1-ε)$-approximation algorithm more generally for any set system with constant VC dimension, running in time exponential in $\tilde{O}(k/ε)$. This simplifies and improves Badanidiyuru, Kleinberg, and Lee's parameterized approximation scheme [SoCG'12] running in time exponential in $\tilde{O}(k^2/ε^5)$. - A continuous version of the geometric maximum coverage problem asks for $k$ objects maximizing the volume of their union. We give better approximation algorithms for this problem for certain families of objects; e.g., we obtain an EPTAS for fat convex objects in any constant dimension. - We complement our algorithms with several hardness results, e.g., APX-hardness for fat axis-aligned rectangles in 2D, $(1-1/e+ε)$-approximation hardness for axis-aligned boxes in a dimension dependent on $ε$, and a lower bound ruling out $n^{\mathop{\rm poly}(1/ε)}$-time PTASs for the continuous problem for axis-aligned boxes in 3D.
We study the complexity of computing the Threshold dimension of a hypothesis class and its variant, the Extended threshold dimension. For the latter, we prove that it is both NP-hard and co-NP-hard, which (partially) answers an open question of Dmitriev et al. (SODA 2026). Furthermore, by relating the problem to a variant of Maximum Balanced Biclique, we prove strong hardness of approximation for both dimensions, including in the parameterized setting. As an intermediate result, we also prove hardness (of approximation) results for computing the ladder index and the semi-ladder index (Fabianski et al., STACS 2019), which have recently been used in the design of fixed-parameter tractable algorithms.
We study the problem of generating synthetic data under differential privacy. We establish fixed-parameter tractability (FPT) for this problem where the parameter is the treewidth of the query family's incidence graph. Our algorithms attain optimal error rates across all regimes and are realized by two different approaches: the first is based on linear programming (LP) and the FPT of the separation problem for the LP dual; the second is based on a subsampled private multiplicative weights method, where we obtain FPT for sampling from Gibbs distributions. Both approaches are unified by a dynamic programming framework over a tree decomposition.
We study the Densest At-Least-$k$-Subgraph (DAL$k$S) problem, in which we are given an undirected graph $G$ and an integer $k$, and the goal is to find a subgraph of $G$ with at least $k$ vertices with maximum density. The best-known algorithm, independently discovered by Khuller and Saha (2009) and by Andersen (2007), yields a 2-approximation for DAL$k$S in polynomial time. In this note, we provide a (simple) reduction from Densest $k$-Subgraph (D$k$S) to Densest At-Least-$k$-Subgraph, which shows that, if D$k$S is hard to approximate to within any constant factor, then DAL$k$S is hard to approximate to within $(3/2 - \varepsilon)$ factor for every $\varepsilon > 0$. This holds in both the normal (non-parameterized) and the parameterized (by $k$) settings. We then generalize the reduction to provide a tight $(2 - \varepsilon)$ factor hardness of approximating Densest At-Least-$k$-Subgraph, albeit under a stronger hypothesis which roughly states that Densest $k$-Subgraph is hard to approximate to within $k^{1 - δ}$ factor for any constant $δ> 0$. Once again, this extends naturally to the parameterized setting. Previously, $(2 - \varepsilon)$ factor inapproximability for DAL$k$S was only known under the Small Set Expansion Hypothesis (Bergner, 2013; Manurangsi, 2017), which does not apply to the parameterized version of the problem. Furthermore, we show that the exact version of DAL$k$S is W[1]-hard (parameterized by $k$).
We study the fair allocation of indivisible items subject to conflict constraints. In this framework, the items are represented as the vertices of a graph, with edges corresponding to conflicts between pairs of items. Each agent is assigned an independent set of items from the graph. Our goal is to achieve a fair and efficient allocation of these items. Fairness pertains to satisfying envy-freeness up to one item (EF1), while efficiency is defined by maximality, meaning that no unallocated item can be feasibly assigned to any agent. First, we explore the case of two agents. For monotone valuations, we show that a maximal EF1 allocation always exists on any graph. Our existence proof relies on a color-switching technique, which locally modifies a maximal allocation while preserving feasibility and restoring EF1. We further show that such allocations can be computed in pseudopolynomial time in general, and in polynomial time for additive valuations on arbitrary graphs, as well as for monotone valuations on interval and bipartite graphs. By contrast, once monotonicity is dropped, maximal EF1 allocations need not exist even for identical additive valuations, and deciding existence becomes NP-hard. Next, we consider the case with a general number of agents. Again, we arrive at a negative result: An EF1 and maximal allocation fails to exist even for three agents under identical monotone valuations, and determining the existence of such an allocation is NP-hard. On the positive side, we show that under identical non-monotone additive valuations on a path graph, an EF[1,1] and maximal allocation always exists. This result involves a novel application of the "cycle plus triangles" theorem.
We study the problem of Stochastic Convex Optimization (SCO) under the constraint of local Label Differential Privacy (L-LDP). In this setting, the features are considered public, but the corresponding labels are sensitive and must be randomized by each user locally before being sent to an untrusted analyzer. Prior work for SCO under L-LDP (Ghazi et al., 2021) established an excess population risk bound with a \emph{linear} dependence on the size of the label space, $K$: $O\left({\frac{K}{ε\sqrt{n}}}\right)$ in the high-privacy regime ($ε\leq 1$) and $O\left({\frac{K}{e^ε \sqrt{n}}}\right)$ in the medium-privacy regime ($1 \leq ε\leq \ln K$). This left open whether this linear cost is fundamental to the L-LDP model. In this note, we resolve this question. First, we present a novel and efficient non-interactive L-LDP algorithm that achieves an excess risk of $O\left({\sqrt{\frac{K}{εn}}}\right)$ in the high-privacy regime ($ε\leq 1$) and $O\left({\sqrt{\frac{K}{e^ε n}}}\right)$ in the medium-privacy regime ($1 \leq ε\leq \ln K$). This quadratically improves the dependency on the label space size from $O(K)$ to $O(\sqrt{K})$. Second, we prove a matching information-theoretic lower bound across all privacy regimes for any sufficiently large $n$.
We study the Maximum Balanced Biclique (MBB) problem: Given a bipartite graph $G$ with $n$ vertices on each side, find a balanced biclique in $G$ with maximum size. We give a polynomial-time $\left(\frac{n}{\widetildeΩ\left((\log n)^3\right)}\right)$-approximation algorithm for the problem, which improves upon an $\left(\frac{n}{Ω\left((\log n)^2\right)}\right)$-approximation by Chalermsook et al. (2020) and answers their open question. Furthermore, our approximation ratio matches that of the maximum clique problem by Feige (2004) up to an $O(\log \log n)$ factor.
We study privacy filters, which enable privacy accounting for differentially private (DP) mechanisms with adaptively chosen privacy characteristics. We develop a general theory that characterizes the worst-case privacy loss of an interaction involving an analyst that respects some restrictions on what queries they may issue. We apply this theory to develop residue filters, which unifies existing privacy filters. We develop the Gaussian DP (GDP) residue filter, which strictly improves upon the naïve GDP filter. We also show that residue filters capture the natural filter, which promises greater utility by leveraging exact privacy accounting techniques. Earlier privacy filters consider only simple privacy parameters such as Rényi-DP or GDP parameters. Natural filters account for the entire privacy profile of every query, promising more efficient use of a given privacy budget. We show that, contrary to other forms of DP, natural privacy filters are not free in general. We present a characterization of when a family of private queries admits free natural filters for a given budget. In particular, only families of privacy mechanisms that are totally-ordered when composed admit free natural privacy filters with respect to an arbitrary privacy budget. Finally, we show that, while the natural approximate-DP filter can fail in the presence of adaptive adversary, it cannot fail too badly: the output remains approximate-DP with parameters at most poly-logarithmically worse than the intended privacy parameters.