94 research outputs found
Total vertex irregularity strength of interval graphs
A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers). For a simple graph G = (V, E) with vertex set V and edge set E, a labeling φ : V ∪E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x ∈ V (G) under a total k-labeling φ is defined as wt(x) = φ(x)+ P y∈N(x) φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is defined to be a vertex irregular total labeling of a graph G, if wt(x) 6= wt(y) holds for every two different vertices x and y of G. The minimum k for which a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs are presented for interval graphs.Publisher's Versio
A unified analysis of likelihood-based estimators in the Plackett--Luce model
The Plackett--Luce model is a popular approach for ranking data analysis,
where a utility vector is employed to determine the probability of each outcome
based on Luce's choice axiom. In this paper, we investigate the asymptotic
theory of utility vector estimation by maximizing different types of
likelihood, such as the full-, marginal-, and quasi-likelihood. We provide a
rank-matching interpretation for the estimating equations of these estimators
and analyze their asymptotic behavior as the number of items being compared
tends to infinity. In particular, we establish the uniform consistency of these
estimators under conditions characterized by the topology of the underlying
comparison graph sequence and demonstrate that the proposed conditions are
sharp for common sampling scenarios such as the nonuniform random hypergraph
model and the hypergraph stochastic block model; we also obtain the asymptotic
normality of these estimators and discuss the trade-off between statistical
efficiency and computational complexity for practical uncertainty
quantification. Both results allow for nonuniform and inhomogeneous comparison
graphs with varying edge sizes and different asymptotic orders of edge
probabilities. We verify our theoretical findings by conducting detailed
numerical experiments.Comment: 42 pages, corrected typos, added the supplementary file containing
all remaining proof
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Algorithms for nonuniform networks
In this thesis, observations on structural properties of natural networks are taken as a starting point for developing efficient algorithms for natural instances of different graph problems. The key areas discussed are sampling, clustering, routing, and pattern mining for large, nonuniform graphs. The results include observations on structural effects together with algorithms that aim to reveal structural properties or exploit their presence in solving an interesting graph problem.
Traditionally networks were modeled with uniform random graphs, assuming that each vertex was equally important and each edge equally likely to be present. Within the last decade, the approach has drastically changed due to the numerous observations on structural complexity in natural networks, many of which proved the uniform model to be inadequate for some contexts.
This quickly lead to various models and measures that aim to characterize topological properties of different kinds of real-world networks also beyond the uniform networks. The goal of this thesis is to utilize such observations in algorithm design, in addition to empowering the process of network analysis. Knowing that a graph exhibits certain characteristics allows for more efficient storage, processing, analysis, and feature extraction.
Our emphasis is on local methods that avoid resorting to information of the graph structure that is not relevant to the answer sought. For example, when seeking for the cluster of a single vertex, we compute it without using any global knowledge of the graph, iteratively examining the vicinity of the seed vertex. Similarly we propose methods for sampling and spanning-tree construction according to certain criteria on the outcome without requiring knowledge of the graph as a whole.
Our motivation for concentrating on local methods is two-fold: one driving factor is the ever-increasing size of real-world problems, but an equally important fact is the nonuniformity present in many natural graph instances; properties that hold for the entire graph are often lost when only a small subgraph is examined.reviewe
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
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