arXiv subjects

Harshul Sagar

Publications and source records attributed to Harshul Sagar.

3 recordsLinked to original sources

arXiv2026

Canonical Cut Hierarchy in Almost-Linear Time via Reduction to Max-Flow

The canonical cut hierarchy recursively partitions a graph along Tutte-Nash-Williams partitions, choosing the partition with the most number of parts at each step. We give a randomized almost linear-time algorithm for computing this hierarchy in undirected graphs with polynomially bounded capacities via reduction to max-flow. This yields randomized almost linear-time algorithms for computing partition connectivity, strength, both arboricity and fractional arboricity, both integral and fractional spanning tree packing numbers, and implicit representations of the principal partition sequence and a maximum fractional spanning tree packing.

cs.DS↗

arXiv2026

Path Length Existence Problems in Bounded-Treewidth Probabilistic Graphs

We consider the problem of computing the probability of existence of a source-to-target path or walk satisfying a given length constraint in an input probabilistic graph. The length constraints we consider can require the length to be at least $\ell$, at most $\ell$, or exactly $\ell$, for $\ell$ a target length given as input. The probabilistic graphs that we consider are directed graphs where each edge is annotated with an independent probability of existence. As this task is generally intractable (#P-hard), we aim for algorithms applying to input graphs that are assumed to be of bounded treewidth. We classify the complexity landscape of these problems: - We give an XP algorithm parameterised by treewidth to compute the probability that an input probabilistic graph contains a source-to-target path/walk of length $\geq \ell$, and do the same for a path/walk of length $\leq \ell$. - We show that the four problems mentioned in the last item are W[1]-hard parameterised by pathwidth (hence not FPT under the standard assumption that FPT$\neq$W[1]). This is in contrast to the case of non-probabilistic graphs, where the corresponding problems were known to be FPT parameterised by treewidth; and to the connectivity problem (and more generally MSO evaluation) on probabilistic graphs, where the same was known. - Surprisingly, we show that it is #P-hard to compute the probability that an input graph contains a source-to-target path/walk of length exactly $\ell$, even on graphs of pathwidth $2$. By contrast, all our problems are tractable on graphs of treewidth $1$. - Last, we investigate the complexity of our problems when the edges can carry weights. This causes all our problems to become #P-hard already on pathwidth-2 graphs, and causes specifically the ``exactly $\ell$'' walk problem to be #P-hard even on pathwidth-1 graphs.

cs.DB↗

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↗