arXiv subjects

Jatin Yadav

Publications and source records attributed to Jatin Yadav.

10 recordsLinked to original sources

arXiv2026

Finding Representative and Approximately Efficient Committees

In approval-based committee voting, proportional approval voting (PAV) is a well-studied rule that combines proportional representation with Pareto efficiency. However, computing a PAV committee is NP-hard, raising a natural question: Can the proportionality and efficiency properties of PAV be achieved via computationally efficient procedures? We make two contributions toward answering this question. First, building on the known proportionality guarantees of the local-search-based variant of PAV (or local PAV), we systematically study its efficiency properties. We show that local PAV committees are weakly Pareto optimal, meaning that no other committee is strictly preferred by every voter. We also identify limitations: Local PAV guarantees only a $2$-approximation to fractional Pareto optimality ($2$-fPO) and a $2/3$-approximation to the optimal PAV score, and both bounds are tight. In contrast, global PAV is Pareto optimal and satisfies the stronger $\alpha^\star$-fPO guarantee, where $\alpha^\star \approx 1.346$ is the unique solution of $\int_0^{\alpha^\star} \frac{1-e^{-y}}{y} \, dy = 1$, and this approximation is tight. Second, we design a polynomial-time algorithm that combines the best of these guarantees. The committee returned by our algorithm satisfies EJR$+$ (a proportionality guarantee), $\alpha^\star$-fPO, and weak Pareto optimality. It also achieves a $0.79$-approximation to the optimal PAV score, matching the best possible polynomial-time approximation assuming $P \neq NP$. Our algorithm works by pipage rounding a concave relaxation of the PAV objective and using that committee to initialize local PAV, thereby combining global approximation guarantees with local search stability.

cs.GT↗

arXiv2026

Easier, but Not Easy: Nash Welfare under Lexicographic Valuations

Maximizing Nash welfare over indivisible goods is a central problem in resource allocation. For additive valuations, the best-known approximation factor is roughly $e^{-1/e}\approx0.692$, and the problem is APX-hard. We study Nash welfare maximization under lexicographic valuations, where every good is worth more than the total value of all lower-ranked goods. This large-gap structure makes preferences almost ordinal, which might suggest that the problem becomes easy. We show, however, that the picture is more nuanced: although lexicographic valuations enable stronger algorithmic guarantees, they retain significant computational hardness. Our first main result is a $(1/\sqrt{2}-\epsilon)\approx(0.707-\epsilon)$-approximation algorithm for weighted Nash welfare under lexicographic valuations, improving over the inherited guarantee of roughly $e^{-1/e}$. The algorithm rounds the configuration LP for Nash welfare, for which we show a matching integrality gap of $\sqrt{2}$. Our second main contribution is an exact algorithmic polynomial time framework for ordered lexicographic instances and doubling lexicographic instances. We introduce a domination-based branch-and-prune method for which we prove a mutual-exclusion property between sibling subtrees and use a matrix-based leaf-counting argument to bound the pruned recursion tree by a polynomial when the number of agents is constant. Finally, we show that large gaps do not eliminate hardness, as Nash welfare maximization is NP-hard even for ordered lexicographic valuations, and it is NP-hard to obtain a $0.9996$-approximation even for doubling lexicographic valuations. Thus, lexicographic valuations make Nash welfare maximization easier, but not easy: they admit tighter approximation and exact algorithms in important cases, yet still require intricate techniques and preserve some of the hardness of the general additive setting.

cs.GT↗

arXiv2026

Hitting Axis-Parallel Segments with Weighted Points

We study a geometric hitting-set problem in which the input consists of a set $P$ of weighted points and a family $S=H\cup V$ of axis-parallel segments in the plane. The goal is to select a minimum-weight subset of $P$ that hits every segment in $S$. Even restricted geometric hitting-set problems are known to be computationally hard, and for axis-parallel segments the standard decomposition into horizontal and vertical sub-instances yields only a simple factor-$2$ approximation. We present an LP-rounding algorithm that breaks the factor-2 barrier. For the weighted problem, we obtain a randomized $(1+2/e)$-approximation by combining systematic rounding on horizontal lines with an exact repair step on residual vertical sub-instances. In the unweighted case, a sharper analysis gives a $(1+1/(e-1))$-approximation. Finally, we consider the case where one of the sub-instances consists of lines instead of line segments, a problem considered by Fekete et al. (Geometric Hitting Set for Segments of Few Orientations, Theor. Comp. Sys., 62 (2) 2018),. In this case, we improve their result to obtain an approximation factor of $1+1/e$ and show that the problem is APX-hard. We also present algorithms for the generalization to $d$ orientations, as well as PTASes for bounded-complexity subclasses of the unweighted Hitting Set problem.

cs.CG↗

arXiv2026

FPT Approximation Schemes for Min-Sum Radii and Min-Sum Diameters Clustering

In the classical Min-Sum Radii problem (MSR) we are given a set $X$ of $n$ points in a metric space and a positive integer $k\in [n]$. Our goal is to partition $X$ into $k$ subsets (the clusters) so as to minimize the sum of the radii of these clusters. The Min-Sum Diameters problem (MSD) is defined analogously, where instead of the radii of the clusters we consider their diameters. For both problems we present FPT approximation schemes for the natural parameter $k$. Specifically, given $ε>0$, we show how to compute $(1+ε)$-approximations for both MSD and MSR in time $(1/ε)^kn^{O(1)}$ and $(1/ε)^{O(k/ε\log 1/ε)}n^{poly(1/ε)}$ respectively. The previous best FPT approximation algorithms for these problems have approximation factors $4+ε$ and $2+ε$, respectively, and finding an FPT approximation scheme for both these problems had been outstanding open problems.

cs.DS↗

arXiv2025

The Landscape of Almost Equitable Allocations

Equitability is a fundamental notion in fair division which requires that all agents derive equal value from their allocated bundles. We study, for general (possibly non-monotone) valuations, a popular relaxation of equitability known as equitability up to one item (EQ1). An EQ1 allocation may fail to exist even with additive non-monotone valuations; for instance, when there are two agents, one valuing every item positively and the other negatively. This motivates a mild and natural assumption: all agents agree on the sign of their value for the grand bundle. Under this assumption, we prove the existence and provide an efficient algorithm for computing EQ1 allocations for two agents with general valuations. When there are more than two agents, we show the existence and polynomial-time computability of EQ1 allocations for valuation classes beyond additivity and monotonicity, in particular for (1) doubly monotone valuations and (2) submodular (resp. supermodular) valuations where the value for the grand bundle is nonnegative (resp. nonpositive) for all agents. Furthermore, we settle an open question of Bil`o et al. by showing that an EQ1 allocation always exists for nonnegative(resp. nonpositive) valuations, i.e., when every agent values each subset of items nonnegatively (resp. nonpositively). Finally, we complete the picture by showing that for general valuations with more than two agents, EQ1 allocations may not exist even when agents agree on the sign of the grand bundle, and that deciding the existence of an EQ1 allocation is computationally intractable.

cs.GT↗

arXiv2025

Best-of-Both-Worlds Guarantees with Fairer Endings

Fair allocation of indivisible goods is a fundamental problem at the interface of economics and computer science. Traditional approaches focus either on randomized allocations that are fair in expectation or deterministic allocations that are approximately fair. Recent work reconciles both these approaches via best-of-both-worlds guarantees, wherein one seeks randomized allocations that are fair in expectation (ex-ante fair) while being supported on approximately fair allocations (ex-post fair). Prior work has shown that under additive valuations, there always exists a randomized allocation that is ex-ante stochastic-dominance envy-free (sd-EF) and ex-post envy-free up to one good (EF1). Our work is motivated by the goal of achieving stronger ex-post fairness guarantees such as envy-freeness up to any good (EFX) along with meaningful ex-ante guarantees. We make the following contributions: 1) We first consider lexicographic preferences, a subdomain of additive valuations where ex-post EFX allocations always exist and can be computed efficiently. On the negative side, we show that ex-ante sd-EF is fundamentally incompatible with ex-post EFX, prompting a relaxation of the ex-ante benchmark. We then present a poly. time algorithm that achieves ex-post EFX and PO together with ex-ante 9/10-EF. Our algorithm uses dependent rounding and leverages structural properties of EFX and PO allocations. 2)For monotone valuations, we study EFX-with-charity: a relaxation of EFX where some goods remain unallocated, with no agent envying the unallocated pool. We show that ex-post EFX-with-charity can be achieved alongside ex-ante 0.5-EF. 3)Finally, for subadditive valuations, we strengthen our previous ex-post guarantee to EFX-with-bounded-charity, where at most n-1 goods (n= no. of agents) remain unallocated, at the price of weakening the ex-ante guarantee to 0.5-proportionality.

cs.GT↗

arXiv2025

Approximating One-Sided and Two-Sided Nash Social Welfare With Capacities

We study the problem of maximizing Nash social welfare, which is the geometric mean of agents' utilities, in two well-known models. The first model involves one-sided preferences, where a set of indivisible items is allocated among a group of agents (commonly studied in fair division). The second model deals with two-sided preferences, where a set of workers and firms, each having numerical valuations for the other side, are matched with each other (commonly studied in matching-under-preferences literature). We study these models under capacity constraints, which restrict the number of items (respectively, workers) that an agent (respectively, a firm) can receive. We develop constant-factor approximation algorithms for both problems under a broad class of valuations. Specifically, our main results are the following: (a) For any $ε> 0$, a $(6+ε)$-approximation algorithm for the one-sided problem when agents have submodular valuations, and (b) a $1.33$-approximation algorithm for the two-sided problem when the firms have subadditive valuations. The former result provides the first constant-factor approximation algorithm for Nash welfare in the one-sided problem with submodular valuations and capacities, while the latter result improves upon an existing $\sqrt{OPT}$-approximation algorithm for additive valuations. Our result for the two-sided setting also establishes a computational separation between the Nash and utilitarian welfare objectives. We also complement our algorithms with hardness-of-approximation results. Additionally, for the case of additive valuations, we modify the configuration LP of Feng and Li [ICALP 2024] to obtain an $(e^{1/e}+ε)-$ approximation algorithm for weighted two-sided Nash social welfare under capacity constraints.

cs.GT↗

arXiv2025

Robust-Sorting and Applications to Ulam-Median

Sorting is one of the most basic primitives in many algorithms and data analysis tasks. Comparison-based sorting algorithms, like quick-sort and merge-sort, are known to be optimal when the outcome of each comparison is error-free. However, many real-world sorting applications operate in scenarios where the outcome of each comparison can be noisy. In this work, we explore settings where a bounded number of comparisons are potentially corrupted by erroneous agents, resulting in arbitrary, adversarial outcomes. We model the sorting problem as a query-limited tournament graph where edges involving erroneous nodes may yield arbitrary results. Our primary contribution is a randomized algorithm inspired by quick-sort that, in expectation, produces an ordering close to the true total order while only querying $\tilde{O}(n)$ edges. We achieve a distance from the target order $π$ within $(3 + ε)|B|$, where $B$ is the set of erroneous nodes, balancing the competing objectives of minimizing both query complexity and misalignment with $π$. Our algorithm needs to carefully balance two aspects: identify a pivot that partitions the vertex set evenly and ensure that this partition is "truthful" and yet query as few "triangles" in the graph $G$ as possible. Since the nodes in $B$ can potentially hide in an intricate manner, our algorithm requires several technical steps. Additionally, we demonstrate significant implications for the Ulam-$k$-Median problem, a classical clustering problem where the metric is defined on the set of permutations on a set of $d$ elements. Chakraborty, Das, and Krauthgamer gave a $(2-\varepsilon)$ FPT approximation algorithm for this problem, where the running time is super-linear in both $n$ and $d$. We use our robust sorting framework to give the first $(2-\varepsilon)$ FPT linear time approximation algorithm for this problem.

cs.DS↗

arXiv2024

FPT Approximation for Capacitated Sum of Radii

We consider the capacitated clustering problem in general metric spaces where the goal is to identify $k$ clusters and minimize the sum of the radii of the clusters (we call this the Capacitated-$k$-sumRadii problem). We are interested in fixed-parameter tractable (FPT) approximation algorithms where the running time is of the form $f(k) \cdot \text{poly}(n)$, where $f(k)$ can be an exponential function of $k$ and $n$ is the number of points in the input. In the uniform capacity case, Bandyapadhyay et al. recently gave a $4$-approximation algorithm for this problem. Our first result improves this to an FPT $3$-approximation and extends to a constant factor approximation for any $L_p$ norm of the cluster radii. In the general capacities version, Bandyapadhyay et al. gave an FPT $15$-approximation algorithm. We extend their framework to give an FPT $(4 + \sqrt{13})$-approximation algorithm for this problem. Our framework relies on a novel idea of identifying approximations to optimal clusters by carefully pruning points from an initial candidate set of points. This is in contrast to prior results that rely on guessing suitable points and building balls of appropriate radii around them. On the hardness front, we show that assuming the Exponential Time Hypothesis, there is a constant $c > 1$ such that any $c$-approximation algorithm for the non-uniform capacity version of this problem requires running time $2^{Ω\left(\frac{k}{polylog(k)} \right)}$.

cs.DS↗

arXiv2022

Efficient Algorithms for Sorting in Trees

Sorting is a foundational problem in computer science that is typically employed on sequences or total orders. More recently, a more general form of sorting on partially ordered sets (or posets), where some pairs of elements are incomparable, has been studied. General poset sorting algorithms have a lower-bound query complexity of $Ω(wn + n \log n)$, where $w$ is the width of the poset. We consider the problem of sorting in trees, a particular case of partial orders, and parametrize the complexity with respect to $d$, the maximum degree of an element in the tree, as $d$ is usually much smaller than $w$ in trees. For example, in complete binary trees, $d = Θ(1), w = Θ(n)$. We present a randomized algorithm for sorting a tree poset in worst-case expected $O(dn\log n)$ query and time complexity. This improves the previous upper bound of $O(dn \log^2 n)$. Our algorithm is the first to be optimal for bounded-degree trees. We also provide a new lower bound of $Ω(dn + n \log n)$ for the worst-case query complexity of sorting a tree poset. Finally, we present the first deterministic algorithm for sorting tree posets that has lower total complexity than existing algorithms for sorting general partial orders.

cs.DS↗