arXiv subjects

Qingjie Ye

Publications and source records attributed to Qingjie Ye.

7 recordsLinked to original sources

arXiv2026

Tight bounds for generalized power domination in regular graphs

Dorbec et al. [SIAM J. Discrete Math., 27 (2013)] conjectured that, for all integers $k\geq1$ and $r\geq3$, every connected $r$-regular graph $G$ of order $n$, other than $K_{r,r}$, satisfies $γ_{P,k}(G)\leq n/(r+1)$. After disproving this conjecture, Chen et al.[Graphs Combin., 38 (2022)] proposed a corresponding conjecture for claw-free regular graphs. In this paper, we prove this conjecture: for integers $k\geq\ell\geq1$, every connected claw-free $(k+\ell+1)$-regular graph $G$ of order $n$ satisfies $γ_{P,k}(G)\leq n/(k+\ell+2)$, and this bound is tight. Moreover, without the claw-free assumption, we show that, for each fixed integer $k\geq1$, the supremum of $γ_{P,k}(G)/\lvert V(G)\rvert$ over all connected $r$-regular graphs $G$ is asymptotic to $(\ln r)/r$ as $r\to\infty$.

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arXiv2026

On walk domination: Between different types of walks and $m_3$-path

Given two non-adjacent vertices $ u $ and $ v $, we say a $uv$-walk $ W $ dominates a $uv$-walk $ W' $ if every internal vertex of $ W' $ is adjacent to some internal vertex of $ W $ or belongs to $ W $. A class of walks $\mathbf{A}$ dominates a class of walks $\mathbf{B}$ if for every pair of non-adjacent vertices $u,v$ in the graph, every $uv$-walk in $\mathbf{A}$ dominates every $uv$-walk in $\mathbf{B}$. This paper investigates the domination relationships among various types of walks connecting two non-adjacent vertices in a graph. In particular, we focus on a problem proposed by Tondato (2024). We study the domination between different walk types (shortest paths, toll walks, weakly toll walks, $l_k$-paths for $k\in \left\{2,3\right\}$) and $m_3$-paths. Furthermore, we show how these relationships give rise to characterizations of graph classes.

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arXiv2026

Online Trading as a Secretary Problem Variant

This paper studies an online trading variant of the classical secretary problem, called secretary problem variant trading (SPVT), from the perspective of an intermediary who facilitates trade between a seller and $n$ buyers (collectively referred to as agents). The seller has an item, and each buyer demands the item. These agents arrive sequentially in a uniformly random order to meet the intermediary, each revealing their valuation of the item upon arrival. After each arrival, the intermediary must make an immediate and irrevocable decision before the next agent appears. The intermediary's objective is to maximize the price of the agent who ultimately holds the item at the end of the process. We evaluate the performance of online algorithms for SPVT using two notions of competitive ratio: strong and weak. The strong notion benchmarks the online algorithm against a powerful offline optimum: the highest price among the $n+1$ agents. We propose an online algorithm for SPVT achieving a strong competitive ratio of $\frac{4e^2}{e^2+1} \approx 3.523$, which is the best possible even when the seller's price may be zero. This tight ratio closes the gap between the previous best upper bound of $4.189$ and lower bound of $3.258$. In contrast, the weak notion restricts the offline optimal algorithm to the given arrival order. The offline algorithm can no longer alter the predetermined arrival order to always place the item in the hands of the agent offering the highest price. Against this weaker benchmark, we design a simple online algorithm for SPVT, achieving a weak competitive ratio of $2$. We further investigate the special case in which the seller's price is zero. For this special SPVT, we develop a double-threshold algorithm achieving a weak competitive ratio of at most $1.83683$ and establish a lower bound of $1.76239$.

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arXiv2026

On the minimum doubly resolving set problem in line graphs

Given a connected graph $G$ with at least three vertices, let $d_G(u,v)$ denote the distance between vertices $u,v\in V(G)$. A subset $S\subseteq V$ is called a doubly resolving set (DRS) of $G$ if for any two distinct vertices $u, v \in V(G)$, there exists a pair $\{x,y\}\subseteq S$ such that $d_G(u,x)-d_G(u,y)\neq d_G(v,x)-d_G(v,y)$. This paper studies the minimum cardinality of a DRS in the line graph of $G$, denoted by $Ψ(L(G))$. First, we prove that computing $Ψ(L(G))$ is NP-hard, even when $G$ is a bipartite graph. Second, we establish that $\lceil \log_2 (1+Δ(G))\rceil \le Ψ(L(G)) \le |V(G)| - 1$ holds for all $G$ with maximum degree $Δ(G)$, and show that both inequalities are tight. Finally, we determine the exact value of $Ψ(L(G))$ provided $G$ is a tree.

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arXiv2025

Supermodular Maximization with Cardinality Constraints

Let $V$ be a finite set of $n$ elements, $f: 2^V \rightarrow \mathbb{R}_+$ be a nonnegative monotone supermodular function, and $k$ be a positive integer no greater than $n$. This paper addresses the problem of maximizing $f(S)$ over all subsets $S \subseteq V$ subject to the cardinality constraint $|S| = k$ or $|S|\le k$. Let $r$ be a constant integer. The function $f$ is assumed to be {\em $r$-decomposable}, meaning there exist $m\,(\ge1)$ subsets $V_1, \dots, V_m$ of $V$, each with a cardinality at most $r$, and a corresponding set of nonnegative supermodular functions $f_i : 2^{V_i} \rightarrow \mathbb{R}_+$, $i=1,\ldots,m$ such that $f(S) =\sum_{i=1}^m f_i(S \cap V_i)$ holds for each $S \subseteq V$. Given $r$ as an input, we present a polynomial-time $O(n^{(r-1)/2})$-approximation algorithm for this maximization problem, which does not require prior knowledge of the specific decomposition. When the decomposition $(V_i,f_i)_{i=1}^m$ is known, an additional connectivity requirement is introduced to the problem. Let $G$ be the graph with vertex set $V$ and edge set $\cup_{i=1}^m \{uv:u,v\in V_i,u\neq v\}$. The cardinality constrained solution set $S$ is required to induce a connected subgraph in $G$. This model generalizes the well-known problem of finding the densest connected $k$-subgraph. We propose a polynomial time $O(n^{(r-1)/2})$-approximation algorithm for this generalization. Notably, this algorithm gives an $O(n^{1/2})$-approximation for the densest connected $k$-subgraph problem, improving upon the previous best-known approximation ratio of $O(n^{2/3})$.

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arXiv2021

A bridge between the minimal doubly resolving set problem in (folded) hypercubes and the coin weighing problem

In this paper, we consider the minimal doubly resolving set problem in Hamming graphs, hypercubes and folded hypercubes. We prove that the minimal doubly resolving set problem in hypercubes is equivalent to the coin weighing problem. Then we answer an open question on the minimal doubly resolving set problem in hypercubes. We disprove a conjecture on the metric dimension problem in folded hypercubes and give some asymptotic results for the metric dimension and the minimal doubly resolving set problems in Hamming graphs and folded hypercubes by establishing connections between these problems. Using the Lindström's method for the coin weighing problem, we give an efficient algorithm for the minimal doubly resolving set problem in hypercubes and report some new upper bounds. We also prove that the minimal doubly resolving set problem is NP-hard even restrict on split graphs, bipartite graphs and co-bipartite graphs.

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arXiv2020

Generalized power domination in claw-free regular graphs

In this paper, we give a series of couterexamples to negate a conjecture and hence answer an open question on the $k$-power domination of regular graphs (see [P. Dorbec et al., SIAM J. Discrete Math., 27 (2013), pp. 1559-1574]). Furthermore, we focus on the study of $k$-power domination of claw-free graphs. We show that for $l\in\{2,3\}$ and $k\ge l$, the $k$-power domination number of a connected claw-free $(k+l+1)$-regular graph on $n$ vertices is at most $\frac{n}{k+l+2}$, and this bound is tight.

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