Sequential metric dimension for random graphs

AbstractIn the localization game on a graph, the goal is to find a fixed but unknown target node $v^\star$ with the least number of distance queries possible. In the jth step of the game, the player queries a single node $v_j$ and receives, as an answer to their query, the distance between the nodes $v_j$ and $v^\star$ . The sequential metric dimension (SMD) is the minimal number of queries that the player needs to guess the target with absolute certainty, no matter where the target is.The term SMD originates from the related notion of metric dimension (MD), which can be defined the same way as the SMD except that the player’s queries are non-adaptive. In this work we extend the results of Bollobás, Mitsche, and Prałat [4] on the MD of Erdős–Rényi graphs to the SMD. We find that, in connected Erdős–Rényi graphs, the MD and the SMD are a constant factor apart. For the lower bound we present a clean analysis by combining tools developed for the MD and a novel coupling argument. For the upper bound we show that a strategy that greedily minimizes the number of candidate targets in each step uses asymptotically optimal queries in Erdős–Rényi graphs. Connections with source localization, binary search on graphs, and the birthday problem are discussed.

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Computation of the Double Metric Dimension in Convex Polytopes

Journal of Mathematics ◽

10.1155/2021/9958969 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Liying Pan ◽

Muhammad Ahmad ◽

Zohaib Zahid ◽

Sohail Zafar

Keyword(s):

Source Localization ◽

Computer Network ◽

Convex Polytopes ◽

The Internet ◽

Metric Dimension ◽

Human Beings ◽

Source Detection ◽

Resolving Set ◽

Detection Problem ◽

Rumor Spreading

A source detection problem in complex networks has been studied widely. Source localization has much importance in order to model many real-world phenomena, for instance, spreading of a virus in a computer network, epidemics in human beings, and rumor spreading on the internet. A source localization problem is to identify a node in the network that gives the best description of the observed diffusion. For this purpose, we select a subset of nodes with least size such that the source can be uniquely located. This is equivalent to find the minimal doubly resolving set of a network. In this article, we have computed the double metric dimension of convex polytopes R n and Q n by describing their minimal doubly resolving sets.

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Near-Optimal Bayesian Ambiguity Sets for Distributionally Robust Optimization

Management Science ◽

10.1287/mnsc.2018.3140 ◽

2019 ◽

Vol 65 (9) ◽

pp. 4242-4260 ◽

Cited By ~ 8

Author(s):

Vishal Gupta

Keyword(s):

Robust Optimization ◽

Relative Size ◽

Constant Factor ◽

Bayesian Robustness ◽

Distributionally Robust Optimization ◽

Dimensional Parameter ◽

Asymptotically Optimal ◽

Small Constant ◽

Distributionally Robust ◽

Optimal Set

We propose a Bayesian framework for assessing the relative strengths of data-driven ambiguity sets in distributionally robust optimization (DRO) when the underlying distribution is defined by a finite-dimensional parameter. The key idea is to measure the relative size between a candidate ambiguity set and a specific asymptotically optimal set. As the amount of data grows large, this asymptotically optimal set is the smallest convex ambiguity set that satisfies a novel Bayesian robustness guarantee that we introduce. This guarantee is defined with respect to a given class of constraints and is a Bayesian analog of more common frequentist feasibility guarantees from the DRO literature. Using this framework, we prove that many popular existing ambiguity sets are significantly larger than the asymptotically optimal set for constraints that are concave in the ambiguity. By contrast, we construct new ambiguity sets that are tractable, satisfy our Bayesian robustness guarantee, and are at most a small, constant factor larger than the asymptotically optimal set; we call these sets Bayesian near-optimal. We further prove that asymptotically, solutions to DRO models with our Bayesian near-optimal sets enjoy strong frequentist robustness properties, despite their smaller size. Finally, our framework yields guidelines for practitioners selecting between competing ambiguity set proposals in DRO. Computational evidence in portfolio allocation using real and simulated data confirms that our framework, although motivated by asymptotic analysis in a Bayesian setting, provides practical insight into the performance of various DRO models with finite data under frequentist assumptions. This paper was accepted by Yinyu Ye, optimization.

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