arXiv subjects

Havu Miikonen

Publications and source records attributed to Havu Miikonen.

5 recordsLinked to original sources

arXiv2026

On the Vertices That Belong to All Minimum Identifying Codes

Identifying codes in graphs have been widely studied since their introduction by Karpovsky, Chakrabarty and Levitin in 1998. In this paper, we consider the vertices that are in every minimum identifying code in a graph. There are two types of such vertices: \emph{always-forced} vertices that belong to all identifying codes (minimum or not) and \emph{min-forced} vertices that belong to all minimum identifying codes. A vertex is called \emph{proper-min-forced} if it is min-forced but not always-forced. We show an upper bound $2n/3$ for the number of such proper-min-forced vertices in a closed-twin-free graph of order $n$. Moreover, for integers $n$ divisible by three, we construct an infinite family of graphs in which there are $2n/3-1$ such vertices. In addition, we determine the maximum number of edges in a graph of even order such that the graph contains proper-min-forced vertices. We also show that the decision problem of determining whether a given vertex in a graph is proper-min-forced is co-NP-hard.

math.CO↗

arXiv2026

On the Maximum Number of Vertices that Belong to Every Metric Basis

Metric bases of graphs have been widely studied since their introduction in the 1970's by Slater and, independently, by Harary and Melter. In this paper, we concentrate on the existence of vertices in a graph $G$ that belong to all metric bases of $G$. We call these basis forced vertices, and denote the number of them by $\mathrm{bf}(G)$. We show that $\mathrm{bf}(G)\le 2/3(n-k-1)$ for any connected nontrivial graph $G$ of order $n$ having $k$ vertices in each metric basis. In addition, we show that this bound can be attained. Furthermore, the previous result implies the bound $\mathrm{bf}(G)\le 2/5(n-1)$ formulated in terms of the order $n$ of the graph for any nontrivial connected graph $G$. This result answers a question posed by Bagheri et al. in 2016. Moreover, we provide a complete realization of the parameters $n$, $\dim(G)$ and $\mathrm{bf}(G) \ge 1$ within the previous bounds. We consider some extremal cases related to basis forced vertices in a graph, in particular, we give a full characterization of the graphs with $\mathrm{bf}(G) = 2$ and $\dim(G) = n-4$.

math.CO↗

arXiv2026

New Results on Vertices that Belong to Every Minimum Locating-Dominating Code

Locating-dominating codes have been studied widely since their introduction in the 1980s by Slater and Rall. In this paper, we concentrate on vertices that must belong to all minimum locating-dominating codes in a graph. We call them \emph{min-forced vertices}. We show that the number of min-forced vertices in a connected nontrivial graph of order $n$ is bounded above by $\frac{2}{3}\left(n -γ^{LD}(G)\right)$, where $γ^{LD}(G)$ denotes the cardinality of a minimum locating-dominating code. This implies that the maximum ratio between the number of min-forced vertices and the order of a connected nontrivial graph is at most $\frac{2}{5}$. Moreover, both of these bounds can be attained. In particular, the ratio $\frac{2}{5}$ is obtained by paths of order $5m$ having a unique minimum locating-dominating code of size $2m$. Furthermore, as a natural extension, we determine the number of different minimum locating-dominating codes in paths of all orders. In addition, we show that deciding whether a vertex is min-forced is co-NP-hard.

math.CO↗

arXiv2024

A Link between Coding Theory and Cross-Validation with Applications

How many different binary classification problems a single learning algorithm can solve on a fixed data with exactly zero or at most a given number of cross-validation errors? While the number in the former case is known to be limited by the no-free-lunch theorem, we show that the exact answers are given by the theory of error detecting codes. As a case study, we focus on the AUC performance measure and leave-pair-out cross-validation (LPOCV), in which every possible pair of data with different class labels is held out at a time. We show that the maximal number of classification problems with fixed class proportion, for which a learning algorithm can achieve zero LPOCV error, equals the maximal number of code words in a constant weight code (CWC), with certain technical properties. We then generalize CWCs by introducing light CWCs, and prove an analogous result for nonzero LPOCV errors and light CWCs. Moreover, we prove both upper and lower bounds on the maximal numbers of code words in light CWCs. Finally, as an immediate practical application, we develop new LPOCV based randomization tests for learning algorithms that generalize the classical Wilcoxon-Mann-Whitney U test.

cs.LG↗

arXiv2021

Massively Parallel Correlation Clustering in Bounded Arboricity Graphs

Identifying clusters of similar elements in a set is a common task in data analysis. With the immense growth of data and physical limitations on single processor speed, it is necessary to find efficient parallel algorithms for clustering tasks. In this paper, we study the problem of correlation clustering in bounded arboricity graphs with respect to the Massively Parallel Computation (MPC) model. More specifically, we are given a complete graph where the edges are either positive or negative, indicating whether pairs of vertices are similar or dissimilar. The task is to partition the vertices into clusters with as few disagreements as possible. That is, we want to minimize the number of positive inter-cluster edges and negative intra-cluster edges. Consider an input graph $G$ on $n$ vertices such that the positive edges induce a $λ$-arboric graph. Our main result is a 3-approximation ($\textit{in expectation}$) algorithm to correlation clustering that runs in $\mathcal{O}(\log λ\cdot \textrm{poly}(\log \log n))$ MPC rounds in the $\textit{strongly sublinear memory regime}$. This is obtained by combining structural properties of correlation clustering on bounded arboricity graphs with the insights of Fischer and Noever (SODA '18) on randomized greedy MIS and the $\texttt{PIVOT}$ algorithm of Ailon, Charikar, and Newman (STOC '05). Combined with known graph matching algorithms, our structural property also implies an exact algorithm and algorithms with $\textit{worst case}$ $(1+ε)$-approximation guarantees in the special case of forests, where $λ=1$.

cs.DC↗