arXiv subjects

Anand Louis

Publications and source records attributed to Anand Louis.

At least 19 recordsLinked to original sources

arXiv2026

Improved Certificates for Independence Number in Semirandom Hypergraphs

We study the problem of efficiently certifying upper bounds on independence number of $\ell$-uniform hypergraphs in semirandom models. This is a notoriously hard problem, with efficient algorithms failing to approximate the independence number within an $n^{1-\epsilon}$ factor in worst-case. A folklore reduction to graph case yields a weak $O(\sqrt{n/p})$ bound, and spectral certificates[GKM22] achieve $O(\sqrt{n}.polylog(n)/p^{2/\ell})$. In this work, we prove sharper bounds that eliminate logarithmic factors in $n$ and nearly attain the optimal threshold of $O(\sqrt{n}/p^{1/\ell})$. We also show matching low-degree polynomial lower bounds. Our certificates are designed using the proofs-to-algorithms paradigm via degree-$2\ell$ Sum-of-Squares(SoS) relaxation. The technically challenging case is odd-arity hypergraphs, where we employ a tensor-based analysis reducing the problem to bounding operator norm of random chaos matrices. Previous bounds[AMP21,RT23] have a logarithmic dependence, which we remove using recent matrix concentration inequalities[BBvH23,BLNvH25]; we believe this maybe useful in other hypergraph problems. Since we deploy our certificates in SoS framework, the bounds continue to hold for monotone adversaries. Additionally, we construct a 'quiet' planted distribution supported on independent sets of size $k=o(\sqrt{n}/p^{1/\ell})$ that is low-degree indistinguishable from random hypergraphs. Prior to this work, the problem of constructing a quiet planted distribution in sparse regimes was open even for graphs[JPR+22,Pot22]. This is in contrast to recovering a planted independent set, where the threshold is $k \gtrsim \sqrt{n}/p^{1/2(\ell-1)}$ (matching lower bounds in a concurrent work[FS26]). As application, our certificates combine with an SoS relaxation of an r-coloring system to recover a planted r-colorable subhypergraph under strong adversaries of [LPR25].

cs.DS↗

arXiv2025

Group Fair Matchings using Convex Cost Functions

We consider the problem of assigning items to platforms where each item has a utility associated with each of the platforms to which it can be assigned. Each platform has a soft constraint over the total number of items it serves, modeled via a convex cost function. Additionally, items are partitioned into groups, and each platform also incurs group-specific convex cost over the number of items from each group that can be assigned to the platform. These costs promote group fairness by penalizing imbalances, yielding a soft variation of fairness notions introduced in prior work, such as Restricted Dominance and Minority protection. Restricted Dominance enforces upper bounds on group representation, while Minority protection enforces lower bounds. Our approach replaces such hard constraints with cost-based penalties, allowing more flexible trade-offs. Our model also captures Nash Social Welfare kind of objective. The cost of an assignment is the sum of the values of all the cost functions across all the groups and platforms. The objective is to find an assignment that minimizes the cost while achieving a total utility that is at least a user-specified threshold. The main challenge lies in balancing the overall platform cost with group-specific costs, both governed by convex functions, while meeting the utility constraint. We present an efficient polynomial-time approximation algorithm, supported by theoretical guarantees and experimental evaluation. Our algorithm is based on techniques involving linear programming and network flows. We also provide an exact algorithm for a special case with uniform utilities and establish the hardness of the general problem when the groups can intersect arbitrarily.

cs.GT↗

arXiv2024

On Sparsest Cut and Conductance in Directed Polymatroidal Networks

We consider algorithms and spectral bounds for sparsest cut and conductance in directed polymatrodal networks. This is motivated by recent work on submodular hypergraphs \cite{Yoshida19,LiM18,ChenOT23,Veldt23} and previous work on multicommodity flows and cuts in polymatrodial networks \cite{ChekuriKRV15}. We obtain three results. First, we obtain an $O(\sqrt{\log n})$-approximation for sparsest cut and point out how this generalizes the result in \cite{ChenOT23}. Second, we consider the symmetric version of conductance and obtain an $O(\sqrt{OPT \log r})$ approximation where $r$ is the maximum degree and we point out how this generalizes previous work on vertex expansion in graphs. Third, we prove a non-constructive Cheeger like inequality that generalizes previous work on hypergraphs. We provide a unified treatment via line-embeddings which were shown to be effective for submodular cuts in \cite{ChekuriKRV15}.

cs.DS↗

arXiv2024

Improved linearly ordered colorings of hypergraphs via SDP rounding

We consider the problem of linearly ordered (LO) coloring of hypergraphs. A hypergraph has an LO coloring if there is a vertex coloring, using a set of ordered colors, so that (i) no edge is monochromatic, and (ii) each edge has a unique maximum color. It is an open question as to whether or not a 2-LO colorable 3-uniform hypergraph can be LO colored with 3 colors in polynomial time. Nakajima and \v{Z}ivn\'{y} recently gave a polynomial-time algorithm to color such hypergraphs with $\widetilde{O}(n^{1/3})$ colors and asked if SDP methods can be used directly to obtain improved bounds. Our main result is to show how to use SDP-based rounding methods to produce an LO coloring with $\widetilde{O}(n^{1/5})$ colors for such hypergraphs. We show how to reduce the problem to cases with highly structured SDP solutions, which we call balanced hypergraphs. Then, we discuss how to apply classic SDP-rounding tools to obtain improved bounds.

cs.DS↗

arXiv2023

New Approximation Bounds for Small-Set Vertex Expansion

The vertex expansion of the graph is a fundamental graph parameter. Given a graph $G=(V,E)$ and a parameter $\delta \in (0,1/2]$, its $\delta$-Small-Set Vertex Expansion (SSVE) is defined as \[ \min_{S : |S| = \delta |V|} \frac{|{\partial^V(S)}|}{ \min \{ |S|, |S^c| \} } \] where $\partial^V(S)$ is the vertex boundary of a set $S$. The SSVE~problem, in addition to being of independent interest as a natural graph partitioning problem, is also of interest due to its connections to the Strong Unique Games problem. We give a randomized algorithm running in time $n^{{\sf poly}(1/\delta)}$, which outputs a set $S$ of size $\Theta(\delta n)$, having vertex expansion at most \[ \max\left(O(\sqrt{\phi^* \log d \log (1/\delta)}) , \tilde{O}(d\log^2(1/\delta)) \cdot \phi^* \right), \] where $d$ is the largest vertex degree of the graph, and $\phi^*$ is the optimal $\delta$-SSVE. The previous best-known guarantees for this were the bi-criteria bounds of $\tilde{O}(1/\delta)\sqrt{\phi^* \log d}$ and $\tilde{O}(1/\delta)\phi^* \sqrt{\log n}$ due to Louis-Makarychev [TOC'16]. Our algorithm uses the basic SDP relaxation of the problem augmented with ${\rm poly}(1/\delta)$ rounds of the Lasserre/SoS hierarchy. Our rounding algorithm is a combination of the rounding algorithms of Raghavendra-Tan [SODA'12] and Austrin-Benabbas-Georgiou [SODA'13]. A key component of our analysis is novel Gaussian rounding lemma for hyperedges which might be of independent interest.

cs.DS↗

arXiv2023

Optimizing Group-Fair Plackett-Luce Ranking Models for Relevance and Ex-Post Fairness

In learning-to-rank (LTR), optimizing only the relevance (or the expected ranking utility) can cause representational harm to certain categories of items. Moreover, if there is implicit bias in the relevance scores, LTR models may fail to optimize for true relevance. Previous works have proposed efficient algorithms to train stochastic ranking models that achieve fairness of exposure to the groups ex-ante (or, in expectation), which may not guarantee representation fairness to the groups ex-post, that is, after realizing a ranking from the stochastic ranking model. Typically, ex-post fairness is achieved by post-processing, but previous work does not train stochastic ranking models that are aware of this post-processing. In this paper, we propose a novel objective that maximizes expected relevance only over those rankings that satisfy given representation constraints to ensure ex-post fairness. Building upon recent work on an efficient sampler for ex-post group-fair rankings, we propose a group-fair Plackett-Luce model and show that it can be efficiently optimized for our objective in the LTR framework. Experiments on three real-world datasets show that our group-fair algorithm guarantees fairness alongside usually having better relevance compared to the LTR baselines. In addition, our algorithm also achieves better relevance than post-processing baselines, which also ensures ex-post fairness. Further, when implicit bias is injected into the training data, our algorithm typically outperforms existing LTR baselines in relevance.

cs.LG↗

arXiv2023

Sampling Individually-Fair Rankings that are Always Group Fair

Rankings on online platforms help their end-users find the relevant information -- people, news, media, and products -- quickly. Fair ranking tasks, which ask to rank a set of items to maximize utility subject to satisfying group-fairness constraints, have gained significant interest in the Algorithmic Fairness, Information Retrieval, and Machine Learning literature. Recent works, however, identify uncertainty in the utilities of items as a primary cause of unfairness and propose introducing randomness in the output. This randomness is carefully chosen to guarantee an adequate representation of each item (while accounting for the uncertainty). However, due to this randomness, the output rankings may violate group fairness constraints. We give an efficient algorithm that samples rankings from an individually-fair distribution while ensuring that every output ranking is group fair. The expected utility of the output ranking is at least $\alpha$ times the utility of the optimal fair solution. Here, $\alpha$ depends on the utilities, position-discounts, and constraints -- it approaches 1 as the range of utilities or the position-discounts shrinks, or when utilities satisfy distributional assumptions. Empirically, we observe that our algorithm achieves individual and group fairness and that Pareto dominates the state-of-the-art baselines.

cs.CY↗

arXiv2022

Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes

There are a lot of recent works on generalizing the spectral theory of graphs and graph partitioning to hypergraphs. There have been two broad directions toward this goal. One generalizes the notion of graph conductance to hypergraph conductance [LM16, CLTZ18]. In the second approach one can view a hypergraph as a simplicial complex and study its various topological properties [LM06, MW09, DKW16, PR17] and spectral properties [KM17, DK17, KO18a, KO18b, Opp20]. In this work, we attempt to bridge these two directions of study by relating the spectrum of {\em up-down walks} and {\em swap-walks} on the simplicial complex to hypergraph expansion. In surprising contrast to random-walks on graphs, we show that the spectral gap of swap-walks and up-down walks between level $m$ and $l$ with $1 < m \leq l$ can not be used to infer any bounds on hypergraph conductance. Moreover, we show that the spectral gap of swap-walks between $X(1)$ and $X(k-1)$ can not be used to infer any bounds on hypergraph conductance, whereas we give a Cheeger-like inequality relating the spectral of walks between level $1$ and $l$ for any $l \leq k$ to hypergraph expansion. This is a surprising difference between swaps-walks and up-down walks! Finally, we also give a construction to show that the well-studied notion of {\em link expansion} in simplicial complexes can not be used to bound hypergraph expansion in a Cheeger-like manner.

cs.DS↗

arXiv2022

Online Algorithms for Matchings with Proportional Fairness Constraints and Diversity Constraints

Matching problems with group-fairness constraints and diversity constraints have numerous applications such as in allocation problems, committee selection, school choice, etc. Moreover, online matching problems have lots of applications in ad allocations and other e-commerce problems like product recommendation in digital marketing. We study two problems involving assigning {\em items} to {\em platforms}, where items belong to various {\em groups} depending on their attributes; the set of items are available offline and the platforms arrive online. In the first problem, we study online matchings with {\em proportional fairness constraints}. Here, each platform on arrival should either be assigned a set of items in which the fraction of items from each group is within specified bounds or be assigned no items; the goal is to assign items to platforms in order to maximize the number of items assigned to platforms. In the second problem, we study online matchings with {\em diversity constraints}, i.e. for each platform, absolute lower bounds are specified for each group. Each platform on arrival should either be assigned a set of items that satisfy these bounds or be assigned no items; the goal is to maximize the set of platforms that get matched. We study approximation algorithms and hardness results for these problems. The technical core of our proofs is a new connection between these problems and the problem of matchings in hypergraphs. Our experimental evaluation shows the performance of our algorithms on real-world and synthetic datasets exceeds our theoretical guarantees.

cs.DS↗

arXiv2022

Socially Fair Center-based and Linear Subspace Clustering

Center-based clustering (e.g., $k$-means, $k$-medians) and clustering using linear subspaces are two most popular techniques to partition real-world data into smaller clusters. However, when the data consists of sensitive demographic groups, significantly different clustering cost per point for different sensitive groups can lead to fairness-related harms (e.g., different quality-of-service). The goal of socially fair clustering is to minimize the maximum cost of clustering per point over all groups. In this work, we propose a unified framework to solve socially fair center-based clustering and linear subspace clustering, and give practical, efficient approximation algorithms for these problems. We do extensive experiments to show that on multiple benchmark datasets our algorithms either closely match or outperform state-of-the-art baselines.

cs.LG↗

arXiv2022

Individual Fairness under Varied Notions of Group Fairness in Bipartite Matching - One Framework to Approximate Them All

We study the probabilistic assignment of items to platforms that satisfies both group and individual fairness constraints. Each item belongs to specific groups and has a preference ordering over platforms. Each platform enforces group fairness by limiting the number of items per group that can be assigned to it. There could be multiple optimal solutions that satisfy the group fairness constraints, but this alone ignores item preferences. Our approach explores a `best of both worlds fairness' solution to get a randomized matching, which is ex-ante individually fair and ex-post group-fair. Thus, we seek a `probabilistic individually fair' distribution over `group-fair' matchings where each item has a `high' probability of matching to one of its top choices. This distribution is also ex-ante group-fair. Users can customize fairness constraints to suit their requirements. Our first result is a polynomial-time algorithm that computes a distribution over `group-fair' matchings such that the individual fairness constraints are approximately satisfied and the expected size of a matching is close to OPT. We empirically test this on real-world datasets. We present two additional polynomial-time bi-criteria approximation algorithms that users can choose from to balance group fairness and individual fairness trade-offs. For disjoint groups, we provide an exact polynomial-time algorithm adaptable to additional lower `group fairness' bounds. Extending our model, we encompass `maxmin group fairness,' amplifying underrepresented groups, and `mindom group fairness,' reducing the representation of dominant groups.'

cs.AI↗

arXiv2022

Approximating CSPs with Outliers

Constraint satisfaction problems (CSPs) are ubiquitous in theoretical computer science. We study the problem of StrongCSPs, i.e. instances where a large induced sub-instance has a satisfying assignment. More formally, given a CSP instance $\Psi(V, E, [k], \{\Pi_{ij}\}_{(i,j) \in E})$ consisting of a set of vertices $V$, a set of edges $E$, alphabet $[k]$, a constraint $\Pi_{ij} \subset [k] \times [k]$ for each $(i,j) \in E$, the goal of this problem is to compute the largest subset $S \subseteq V$ such that the instance induced on $S$ has an assignment that satisfies all the constraints. In this paper, we study approximation algorithms for Unique Games and related problems under the StrongCSP framework when the underlying constraint graph satisfies mild expansion properties. In particular, we show that given a Strong Unique Games instance whose optimal solution $S^*$ is supported on a regular low threshold rank graph, there exists an algorithm that runs in time exponential in the threshold rank, and recovers a large satisfiable sub-instance whose size is independent on the label set size and maximum degree of the graph. Our algorithm combines the techniques of Barak-Raghavendra-Steurer (FOCS'11), Guruswami-Sinop (FOCS'11) with several new ideas and runs in time exponential in the threshold rank of the optimal set. A key component of our algorithm is a new threshold rank based spectral decomposition, which is used to compute a "large" induced subgraph of "small" threshold rank; our techniques build on the work of Oveis Gharan and Rezaei (SODA'17) and could be of independent interest.

cs.DS↗

arXiv2022

Exact recovery algorithm for Planted Bipartite Graph in Semi-random Graphs

The problem of finding the largest induced balanced bipartite subgraph in a given graph is NP-hard. This problem is closely related to the problem of finding the smallest Odd Cycle Transversal. In this work, we consider the following model of instances: starting with a set of vertices $V$, a set $S \subseteq V$ of $k$ vertices is chosen and an arbitrary $d$-regular bipartite graph is added on it; edges between pairs of vertices in $S \times (V \setminus S)$ and $(V \setminus S) \times (V \setminus S)$ are added with probability $p$. Since for $d=0$, the problem reduces to recovering a planted independent set, we don't expect efficient algorithms for $k=o(\sqrt{n})$. This problem is a generalization of the planted balanced biclique problem where the bipartite graph induced on $S$ is a complete bipartite graph; [Lev18] gave an algorithm for recovering $S$ in this problem when $k=\Omega(\sqrt{n})$. Our main result is an efficient algorithm that recovers (w.h.p.) the planted bipartite graph when $k=\Omega_p(\sqrt{n \log n})$ for a large range of parameters. Our results also hold for a natural semi-random model of instances, which involve the presence of a monotone adversary. Our proof shows that a natural SDP relaxation for the problem is integral by constructing an appropriate solution to it's dual formulation. Our main technical contribution is a new approach for constructing the dual solution where we calibrate the eigenvectors of the adjacency matrix to be the eigenvectors of the dual matrix. We believe that this approach may have applications to other recovery problems in semi-random models as well. When $k=\Omega(\sqrt{n})$, we give an algorithm for recovering $S$ whose running time is exponential in the number of small eigenvalues in graph induced on $S$; this algorithm is based on subspace enumeration techniques due to the works of [KT07,ABS10,Kol11].

cs.DS↗

arXiv2022

Sampling Ex-Post Group-Fair Rankings

Randomized rankings have been of recent interest to achieve ex-ante fairer exposure and better robustness than deterministic rankings. We propose a set of natural axioms for randomized group-fair rankings and prove that there exists a unique distribution $D$ that satisfies our axioms and is supported only over ex-post group-fair rankings, i.e., rankings that satisfy given lower and upper bounds on group-wise representation in the top-$k$ ranks. Our problem formulation works even when there is implicit bias, incomplete relevance information, or only ordinal ranking is available instead of relevance scores or utility values. We propose two algorithms to sample a random group-fair ranking from the distribution $D$ mentioned above. Our first dynamic programming-based algorithm samples ex-post group-fair rankings uniformly at random in time $O(k^2\ell)$, where $\ell$ is the number of groups. Our second random walk-based algorithm samples ex-post group-fair rankings from a distribution $\delta$-close to $D$ in total variation distance and has expected running time $O^*(k^2\ell^2)$, when there is a sufficient gap between the given upper and lower bounds on the group-wise representation. The former does exact sampling, but the latter runs significantly faster on real-world data sets for larger values of $k$. We give empirical evidence that our algorithms compare favorably against recent baselines for fairness and ranking utility on real-world data sets.

cs.LG↗

arXiv2021

Matchings with Group Fairness Constraints: Online and Offline Algorithms

We consider the problem of assigning items to platforms in the presence of group fairness constraints. In the input, each item belongs to certain categories, called classes in this paper. Each platform specifies the group fairness constraints through an upper bound on the number of items it can serve from each class. Additionally, each platform also has an upper bound on the total number of items it can serve. The goal is to assign items to platforms so as to maximize the number of items assigned while satisfying the upper bounds of each class. In some cases, there is a revenue associated with matching an item to a platform, then the goal is to maximize the revenue generated. This problem models several important real-world problems like ad-auctions, scheduling, resource allocations, school choice etc.We also show an interesting connection to computing a generalized maximum independent set on hypergraphs and ranking items under group fairness constraints. We show that if the classes are arbitrary, then the problem is NP-hard and has a strong inapproximability. We consider the problem in both online and offline settings under natural restrictions on the classes. Under these restrictions, the problem continues to remain NP-hard but admits approximation algorithms with small approximation factors. We also implement some of the algorithms. Our experiments show that the algorithms work well in practice both in terms of efficiency and the number of items that get assigned to some platform.

cs.DS↗

arXiv2021

Independent Sets in Semi-random Hypergraphs

A set of vertices in a hypergraph is called an independent set if no hyperedge is completely contained inside the set. Given a hypergraph, computing its largest size independent set is an NP-hard problem. In this work, we study the independent set problem on hypergraphs in a natural semi-random family of instances. Our semi-random model is inspired by the Feige-Kilian model [FK01]. This popular model has also been studied in the works of [FK01, Ste17, MMT20] etc. McKenzie, Mehta, and Trevisan [MMT20] gave algorithms for computing independent sets in such a semi-random family of graphs. The algorithms by McKenzie et al. [MMT20] are based on rounding a "crude-SDP". We generalize their results and techniques to hypergraphs for an analogous family of hypergraph instances. Our algorithms are based on rounding the "crude-SDP" of McKenzie et al. [MMT20], augmented with "Lasserre/SoS like" hierarchy of constraints. Analogous to the results of McKenzie et al. [MMT20], we study the ranges of input parameters where we can recover the planted independent set or a large independent set.

cs.DS↗

arXiv2021

On the Problem of Underranking in Group-Fair Ranking

Search and recommendation systems, such as search engines, recruiting tools, online marketplaces, news, and social media, output ranked lists of content, products, and sometimes, people. Credit ratings, standardized tests, risk assessments output only a score, but are also used implicitly for ranking. Bias in such ranking systems, especially among the top ranks, can worsen social and economic inequalities, polarize opinions, and reinforce stereotypes. On the other hand, a bias correction for minority groups can cause more harm if perceived as favoring group-fair outcomes over meritocracy. In this paper, we formulate the problem of underranking in group-fair rankings, which was not addressed in previous work. Most group-fair ranking algorithms post-process a given ranking and output a group-fair ranking. We define underranking based on how close the group-fair rank of each item is to its original rank, and prove a lower bound on the trade-off achievable for simultaneous underranking and group fairness in ranking. We give a fair ranking algorithm that takes any given ranking and outputs another ranking with simultaneous underranking and group fairness guarantees comparable to the lower bound we prove. Our algorithm works with group fairness constraints for any number of groups. Our experimental results confirm the theoretical trade-off between underranking and group fairness, and also show that our algorithm achieves the best of both when compared to the state-of-the-art baselines.

cs.IR↗

arXiv2021

Group Fairness for Knapsack Problems

We study the knapsack problem with group fairness constraints. The input of the problem consists of a knapsack of bounded capacity and a set of items, each item belongs to a particular category and has and associated weight and value. The goal of this problem is to select a subset of items such that all categories are fairly represented, the total weight of the selected items does not exceed the capacity of the knapsack,and the total value is maximized. We study the fairness parameters such as the bounds on the total value of items from each category, the total weight of items from each category, and the total number of items from each category. We give approximation algorithms for these problems. These fairness notions could also be extended to the min-knapsack problem. The fair knapsack problems encompass various important problems, such as participatory budgeting, fair budget allocation, advertising.

cs.DS↗