arXiv subjects

Sarah Morell

Publications and source records attributed to Sarah Morell.

4 recordsLinked to original sources

arXiv2026

Unsplittable Transshipments

We introduce the Unsplittable Transshipment Problem in directed graphs with multiple sources and sinks. An unsplittable transshipment routes given supplies and demands using at most one path for each source-sink pair. Although they are a natural generalization of single source unsplittable flows, unsplittable transshipments raise interesting new challenges and require novel algorithmic techniques. As our main contribution, we give a nontrivial generalization of a seminal result of Dinitz, Garg, and Goemans (1999) by showing how to efficiently turn a given transshipment $x$ into an unsplittable transshipment $y$ with $y_a<x_a+d_{\max}$ for all arcs $a$, where $d_{\max}$ is the maximum demand (or supply) value. Further results include bounds on the number of rounds required to satisfy all demands, where each round consists of an unsplittable transshipment that routes a subset of the demands while respecting arc capacity constraints.

cs.DS↗

arXiv2024

The Submodular Santa Claus Problem

We consider the problem of allocating indivisible resources to players so as to maximize the minimum total value any player receives. This problem is sometimes dubbed the Santa Claus problem and its different variants have been subject to extensive research towards approximation algorithms over the past two decades. In the case where each player has a potentially different additive valuation function, Chakrabarty, Chuzhoy, and Khanna [FOCS'09] gave an $O(n^{\epsilon})$-approximation algorithm with polynomial running time for any constant $\epsilon > 0$ and a polylogarithmic approximation algorithm in quasi-polynomial time. We show that the same can be achieved for monotone submodular valuation functions, improving over the previously best algorithm due to Goemans, Harvey, Iwata, and Mirrokni [SODA'09], which has an approximation ratio of more than $\sqrt{n}$. Our result builds up on a sophisticated LP relaxation, which has a recursive block structure that allows us to solve it despite having exponentially many variables and constraints.

cs.DS↗

arXiv2019

Minimum-cost integer circulations in given homology classes

Let $D$ be a directed graph cellularly embedded in a surface together with non-negative cost on its arcs. Given any integer circulation in $D$, we study the problem of finding a minimum-cost non-negative integer circulation in $D$ that is homologous over the integers to the given circulation. A special case of this problem arises in recent work on the stable set problem for graphs with bounded odd cycle packing number, in which the surface is non-orientable (Conforti et al., SODA'20). For orientable surfaces, polynomial-time algorithms have been obtained for different variants of this problem. We complement these results by showing that the convex hull of feasible solutions has a very simple polyhedral description. In contrast, only little seems to be known about the case of non-orientable surfaces. We show that the problem is strongly NP-hard for general non-orientable surfaces, and give the first polynomial-time algorithm for surfaces of fixed genus. For the latter, we provide a characterization of homology (over the integers) that allows us to recast the problem as a special integer program, which can be efficiently solved using proximity results and dynamic programming.

cs.DM↗

arXiv2018

Diversity maximization in doubling metrics

Diversity maximization is an important geometric optimization problem with many applications in recommender systems, machine learning or search engines among others. A typical diversification problem is as follows: Given a finite metric space $(X,d)$ and a parameter $k \in \mathbb{N}$, find a subset of $k$ elements of $X$ that has maximum diversity. There are many functions that measure diversity. One of the most popular measures, called remote-clique, is the sum of the pairwise distances of the chosen elements. In this paper, we present novel results on three widely used diversity measures: Remote-clique, remote-star and remote-bipartition. Our main result are polynomial time approximation schemes for these three diversification problems under the assumption that the metric space is doubling. This setting has been discussed in the recent literature. The existence of such a PTAS however was left open. Our results also hold in the setting where the distances are raised to a fixed power $q\geq 1$, giving rise to more variants of diversity functions, similar in spirit to the variations of clustering problems depending on the power applied to the distances. Finally, we provide a proof of NP-hardness for remote-clique with squared distances in doubling metric spaces.

cs.DM↗