arXiv subjects

Nitya Raju

Publications and source records attributed to Nitya Raju.

5 recordsLinked to original sources

arXiv2026

The Dirichlet Mechanism for rounding with strong negative correlation, with applications

Many optimization and scheduling problems can be abstracted in terms of a bipartite ``assignment graph" $G = (U \cup V, E)$, where the goal is to select exactly one edge for each right-node. For example, a right-node may correspond to a job, and a left-node to a possible machine assignment. A common strategy to solve such problems is to obtain a fractional relaxation $x_e$ for each edge $e$, and then have each right-node independently select an edge with probability $x_e$. However, this may cause the left-nodes to become unevenly loaded, leading to suboptimal solutions for some problems. To address this, a number of algorithms for dependent rounding with strong negative correlation have been developed, e.g. Bansal, Srinivasan & Svensson (2021), Im & Shadloo (2020), Im & Li (2023), Harris (2024), Naor, Srinivasan & Wajc (2025). We introduce a new method for this, which we call the \emph{Dirichlet mechanism}. It is based on having each left-node draw Dirichlet random variables for its edges, and then having each right-node select an edge based on these values. This achieves quantitatively stronger negative correlation than previous algorithms, and is also simpler since it avoids the need for a tie-breaking mechanism. We illustrate the mechanism with improved approximation ratios for two problems. For oblivious online dependent rounding, we achieve a $0.68$-approximation which improves upon the previous $0.652$-approximation of Naor, Srinivasan & Wajc (2025). For the problem of scheduling jobs on unrelated machines to minimize weighted completion time, we achieve a $1.387$-approximation which improves upon the $1.398$-approximation of Harris (2024). (A recent algorithm of Li (2025) based on iterated rounding also provides a $1.36$-approximation if the weights of each job are independent of machine.)

cs.DS↗

arXiv2026

Robust Stable Matchings: Dealing with Changes in Preferences

We study stable matchings that are robust to preference changes in the two-sided stable matching setting of Gale and Shapley [GS62]. Given two instances $A$ and $B$ on the same set of agents, a matching is said to be robust if it is stable under both instances. This notion captures desirable robustness properties in matching markets where preferences may evolve, be misreported, or be subject to uncertainty. While the classical theory of stable matchings reveals rich lattice, algorithmic, and polyhedral structure for a single instance, it is unclear which of these properties persist when stability is required across multiple instances. Our work initiates a systematic study of the structural and computational behavior of robust stable matchings under increasingly general models of preference changes. We analyze robustness under a hierarchy of perturbation models: 1. a single upward shift in one agent's preference list, 2. an arbitrary permutation change by a single agent, and 3. arbitrary preference changes by multiple agents on both sides. For each regime, we characterize when: 1. the set of robust stable matchings forms a sublattice, 2. the lattice of robust stable matchings admits a succinct Birkhoff partial order enabling efficient enumeration, 3. worker-optimal and firm-optimal robust stable matchings can be computed efficiently, and 4. the robust stable matching polytope is integral (by studying its LP formulation). We provide explicit counterexamples demonstrating where these structural and geometric properties break down, and complement these results with XP-time algorithms running in $O(n^k)$ time, parameterized by $k$, the number of agents whose preferences change. Our results precisely delineate the boundary between tractable and intractable cases for robust stable matchings.

cs.GT↗

arXiv2025

Dimension-Free Correlated Sampling for the Hypersimplex

Sampling from multiple distributions so as to maximize overlap has been studied by statisticians since the 1950s. Since the 2000s, such correlated sampling from the probability simplex has been a powerful building block in disparate areas of theoretical computer science. We study a generalization of this problem to sampling sets from given vectors in the hypersimplex, i.e., outputting sets of size (at most) some $k$ in $[n]$, while maximizing the sampled sets' overlap. Specifically, the expected difference between two output sets should be at most $\alpha$ times their input vectors' $\ell_1$ distance. A value of $\alpha=O(\log n)$ is known to be achievable, due to Chen et al.~(ICALP'17). We improve this factor to $O(\log k)$, independent of the ambient dimension~$n$. Our algorithm satisfies other desirable properties, including (up to a $\log^* n$ factor) input-sparsity sampling time, logarithmic parallel depth and dynamic update time, as well as preservation of submodular objectives. Anticipating broader use of correlated sampling algorithms for the hypersimplex, we present applications of our algorithm to online paging, offline approximation of metric multi-labeling and swift multi-scenario submodular welfare approximating reallocation.

cs.DS↗

arXiv2023

Stable Matching: Dealing with Changes in Preferences

We study stable matchings that are robust to preference changes in the two-sided stable matching setting of Gale and Shapley[GS62]. Given two instances $A$ and $B$ on the same set of agents, a matching is said to be robust if it is stable under both instances. While prior work has considered the case where a single agent changes preferences between $A$ and $B$, we allow multiple agents on both sides to update their preferences and ask whether three central properties of stable matchings extend to robust stable matchings: (i) Can a robust stable matching be found in polynomial time? (ii) Does the set of robust stable matchings form a lattice? (iii) Is the fractional robust stable matching polytope integral? We show that all three properties hold when any number of agents on one side change preferences, as long as at most one agent on the other side does. For the case where two or more agents on both sides change preferences, we construct examples showing that both the lattice structure and polyhedral integrality fail-identifying this setting as a sharp threshold. We also present an XP-time algorithm for the general case, which implies a polynomial-time algorithm when the number of agents with changing preferences is constant. While these results establish the tractability of these regimes, closing the complexity gap in the fully general setting remains an interesting open question.

cs.DM↗

arXiv2022

A Structural and Algorithmic Study of Stable Matching Lattices of "Nearby" Instances, with Applications

Recently MV18 identified and initiated work on the new problem of understanding structural relationships between the lattices of solutions of two "nearby" instances of stable matching. They also gave an application of their work to finding a robust stable matching. However, the types of changes they allowed in going from instance $A$ to $B$ were very restricted, namely any one agent executes an upward shift. In this paper, we allow any one agent to permute its preference list arbitrarily. Let $M_A$ and $M_B$ be the sets of stable matchings of the resulting pair of instances $A$ and $B$, and let $\mathcal{L}_A$ and $\mathcal{L}_B$ be the corresponding lattices of stable matchings. We prove that the matchings in $M_A \cap M_B$ form a sublattice of both $\mathcal{L}_A$ and $\mathcal{L}_B$ and those in $M_A \setminus M_B$ form a join semi-sublattice of $\mathcal{L}_A$. These properties enable us to obtain a polynomial time algorithm for not only finding a stable matching in $M_A \cap M_B$, but also for obtaining the partial order, as promised by Birkhoff's Representation Theorem, thereby enabling us to generate all matchings in this sublattice. Our algorithm also helps solve a version of the robust stable matching problem. We discuss another potential application, namely obtaining new insights into the incentive compatibility properties of the Gale-Shapley Deferred Acceptance Algorithm.

cs.DM↗