arXiv subjects

Magnus Lie Hetland

Publications and source records attributed to Magnus Lie Hetland.

9 recordsLinked to original sources

arXiv2022

Maximin Shares Under Cardinality Constraints

We study the problem of fair allocation of a set of indivisible items among agents with additive valuations, under cardinality constraints. In this setting, the items are partitioned into categories, each with its own limit on the number of items it may contribute to any bundle. We consider the fairness measure known as the maximin share (MMS) guarantee, and propose a novel polynomial-time algorithm for finding $1/2$-approximate MMS allocations for goods -- an improvement from the previously best available guarantee of $11/30$. For single-category instances, we show that a modified variant of our algorithm is guaranteed to produce $2/3$-approximate MMS allocations. Among various other existence and non-existence results, we show that a $(\sqrt{n}/(2\sqrt{n} - 1))$-approximate MMS allocation always exists for goods. For chores, we show similar results as for goods, with a $2$-approximate algorithm in the general case and a $3/2$-approximate algorithm for single-category instances. We extend the notions and algorithms related to ordered and reduced instances to work with cardinality constraints, and combine these with bag filling style procedures to construct our algorithms.

cs.GT↗

arXiv2022

Fair Allocation of Conflicting Items

We study fair allocation of indivisible items, where the items are furnished with a set of conflicts, and agents are not permitted to receive conflicting items. This kind of constraint captures, for example, participating in events that overlap in time, or taking on roles in the presence of conflicting interests. We demonstrate, both theoretically and experimentally, that fairness characterizations such as EF1, MMS and MNW still are applicable and useful under item conflicts. Among other existence, non-existence and computability results, we show that a $1/Δ$-approximate MMS allocation for $n$ agents may be found in polynomial time when $n>Δ>2$, for any conflict graph with maximum degree $Δ$, and that, if $n > Δ$, a 1/3-approximate MMS allocation always exists.

cs.GT↗

arXiv2021

Fairest Neighbors: Tradeoffs Between Metric Queries

Metric search commonly involves finding objects similar to a given sample object. We explore a generalization, where the desired result is a fair tradeoff between multiple query objects. This builds on previous results on complex queries, such as linear combinations. We instead use measures of inequality, like ordered weighted averages, and query existing index structures to find objects that minimize these. We compare our method empirically to linear scan and a post hoc combination of individual queries, and demonstrate a considerable speedup.

cs.DS↗

arXiv2020

Metrics and Ambits and Sprawls, Oh My

A follow-up to my previous tutorial on metric indexing, this paper walks through the classic structures, placing them all in the context of the recently proposed "sprawl of ambits" framework. The indexes are presented as configurations of a single, more general structure, all queried using the same search procedure.

cs.DS↗

arXiv2020

Optimal Metric Search Is Equivalent to the Minimum Dominating Set Problem

In metric search, worst-case analysis is of little value, as the search invariably degenerates to a linear scan for ill-behaved data. Consequently, much effort has been expended on more nuanced descriptions of what performance might in fact be attainable, including heuristic baselines like the AESA family, as well as statistical proxies such as intrinsic dimensionality. This paper gets to the heart of the matter with an exact characterization of the best performance actually achievable for any given data set and query. Specifically, linear-time objective-preserving reductions are established in both directions between optimal metric search and the minimum dominating set problem, whose greedy approximation becomes the equivalent of an oracle-based AESA, repeatedly selecting the pivot that eliminates the most of the remaining points. As an illustration, the AESA heuristic is adapted to downplay the role of previously eliminated points, yielding some modest performance improvements over the original, as well as its younger relative iAESA2.

cs.DS↗

arXiv2019

Comparison-Based Indexing From First Principles

Basic assumptions about comparison-based indexing are laid down and a general design space is derived from these. An index structure spanning this design space (the sprawl) is described, along with an associated family of partitioning predicates, or regions (the ambits), as well as algorithms for search and, to some extent, construction. The sprawl of ambits forms a unification and generalization of current indexing methods, and a jumping-off point for future designs.

cs.DS↗

arXiv2015

Ptolemaic Indexing

This paper discusses a new family of bounds for use in similarity search, related to those used in metric indexing, but based on Ptolemy's inequality, rather than the metric axioms. Ptolemy's inequality holds for the well-known Euclidean distance, but is also shown here to hold for quadratic form metrics in general, with Mahalanobis distance as an important special case. The inequality is examined empirically on both synthetic and real-world data sets and is also found to hold approximately, with a very low degree of error, for important distances such as the angular pseudometric and several Lp norms. Indexing experiments demonstrate a highly increased filtering power compared to existing, triangular methods. It is also shown that combining the Ptolemaic and triangular filtering can lead to better results than using either approach on its own.

cs.DS↗

arXiv2014

A Real-Time Spatial Index for In-Vehicle Units

We construct a spatial indexing solution for the highly constrained environment of an in-vehicle unit in a distributed vehicle tolling scheme based on satellite navigation (GNSS). We show that an immutable, purely functional implementation of a high-fanout quadtree is a simple, practical solution that satisfies all the requirements of such a system.

cs.DS↗

arXiv2013

Simulating Ability: Representing Skills in Games

Throughout the history of games, representing the abilities of the various agents acting on behalf of the players has been a central concern. With increasingly sophisticated games emerging, these simulations have become more realistic, but the underlying mechanisms are still, to a large extent, of an ad hoc nature. This paper proposes using a logistic model from psychometrics as a unified mechanism for task resolution in simulation-oriented games.

cs.GT↗