Object Reachability via Swaps under Strict and Weak Preferences

Sen Huang; Mingyu Xiao

Home/Archive/arXiv

arXiv · 1909.07557

Object Reachability via Swaps under Strict and Weak Preferences

Sen Huang·Mingyu Xiao

View original source ↗Read source document ↗Export metadata ↓

Abstract

The \textsc{Housing Market} problem is a widely studied resource allocation problem. In this problem, each agent can only receive a single object and has preferences over all objects. Starting from an initial endowment, we want to reach a certain assignment via a sequence of rational trades. We first consider whether an object is reachable for a given agent under a social network, where a trade between two agents is allowed if they are neighbors in the network and no participant has a deficit from the trade. Assume that the preferences of the agents are strict (no tie among objects is allowed). This problem is polynomially solvable in a star-network and NP-complete in a tree-network. It is left as a challenging open problem whether the problem is polynomially solvable when the network is a path. We answer this open problem positively by giving a polynomial-time algorithm. Then we show that when the preferences of the agents are weak (ties among objects are allowed), the problem becomes NP-hard even when the network is a path. In addition, we consider the computational complexity of finding different optimal assignments for the problem under the network being a path or a star.

Explore related subjects

cs.DS cs.AI

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Sen Huang, Mingyu Xiao. 2020-07-22. Object Reachability via Swaps under Strict and Weak Preferences. https://arxiv.org/abs/1909.07557

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Marked Edge Walk: A Novel MCMC Algorithm for Sampling of Graph Partitions

Novel Markov Chain Monte Carlo (MCMC) methods have enabled the generation of large ensembles of redistricting plans modeled as a graph partitioning problem. However, existing algorithms such as Reversible Recombination (RevReCom) and Metropolized Forest Recombination (MFR) have strong preferences for distributions related to the spanning tree measure. In this paper we introduce the Marked Edge Walk (MEW), a novel Markov chain proposal for sampling from the space of graph partitions. The walk operates on the space of spanning trees with marked edges, allowing for calculable transition probabilities for use in the Metropolis-Hastings algorithm. Empirical results on real-world dual graphs show convergence under a broad class of target distributions less constrained by spanning tree counts, including policy-based distributions, such as competitiveness on New Hampshire that are independent of spanning trees, and compactness and partisan symmetry distributions on New Hampshire and Texas that, while related to spanning trees, can now be properly targeted with a smaller degree of spanning tree bias, which represents an advancement in flexible ensemble generation.

cs.DS↗

arXiv2026

The Price of Privacy For Approximating Max-CSP

We study approximation algorithms for Maximum Constraint Satisfaction Problems (Max-CSPs) under differential privacy (DP) where the constraints are considered sensitive data. Information-theoretically, we aim to classify the best approximation ratios possible for a given privacy budget $\varepsilon$. In the high-privacy regime ($\varepsilon \ll 1$), we show that any $\varepsilon$-DP algorithm cannot beat a random assignment by more than $O(\varepsilon)$ in the approximation ratio. We devise a polynomial-time algorithm which matches this barrier under the assumptions that the instances are bounded-degree and triangle-free. Finally, we show that one or both of these assumptions can be removed for specific CSPs--such as Max-Cut or Max $k$-XOR--albeit at the cost of computational efficiency.

cs.DS↗

arXiv2026

Distributed Santa Claus via Global Rounding

In this paper, we initiate the study of a new class of problems in the CONGEST model: Mixed packing and covering linear programs (LP). Previously, the design of optimization algorithms in the distributed setting was heavily focused on solving linear programs with either packing or covering constraints. We are the first to explore the class of linear programs with both packing and covering constraints by providing a general-purpose CONGEST solver for such LPs and by studying the sequentially well-studied Santa Claus problem as a central representative. This NP-hard problem can be modeled as a bipartite graph of children and gifts where an edge indicates that a child desires a gift. The goal is to assign the gifts to the children such that the least happy child is as happy as possible. Even though this is a well-studied problem in the sequential setting, we provide the first results in the distributed setting. In particular, we show that the complexity of computing an $\mathcal{O}(\log n/\log \log n)$-approximation is $\widehat Θ(\sqrt n+D)$ rounds, where our $\widetildeΩ(\sqrt n+D)$-round lower bound even holds for any approximation.

cs.DS↗