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Multi-agent hierarchical reinforcement learning (MAHRL) has been studied as an effective means to solve intelligent decision problems in complex and large-scale environments. However, most current MAHRL algorithms follow the traditional way of using reward functions in reinforcement learning, which limits their use to a single task. This study aims to design a multi-agent cooperative algorithm with logic reward shaping (LRS), which uses a more flexible way of setting the rewards, allowing for the effective completion of multi-tasks. LRS uses Linear Temporal Logic (LTL) to express the internal logic relation of subtasks within a complex task. Then, it evaluates whether the subformulae of the LTL expressions are satisfied based on a designed reward structure. This helps agents to learn to effectively complete tasks by adhering to the LTL expressions, thus enhancing the interpretability and credibility of their decisions. To enhance coordination and cooperation among multiple agents, a value iteration technique is designed to evaluate the actions taken by each agent. Based on this evaluation, a reward function is shaped for coordination, which enables each agent to evaluate its status and complete the remaining subtasks through experiential learning. Experiments have been conducted on various types of tasks in the Minecraft-like environment. The results demonstrate that the proposed algorithm can improve the performance of multi-agents when learning to complete multi-tasks.

The extraction of biomedical data has significant academic and practical value in contemporary biomedical sciences. In recent years, drug repositioning, a cost-effective strategy for drug development by discovering new indications for approved drugs, has gained increasing attention. However, many existing drug repositioning methods focus on mining information from adjacent nodes in biomedical networks without considering the potential inter-relationships between the feature spaces of drugs and diseases. This can lead to inaccurate encoding, resulting in biased mined drug-disease association information. To address this limitation, we propose a new model called Dual-Feature Drug Repurposing Neural Network (DFDRNN). DFDRNN allows the mining of two features (similarity and association) from the drug-disease biomedical networks to encode drugs and diseases. A self-attention mechanism is utilized to extract neighbor feature information. It incorporates two dual-feature extraction modules: the single-domain dual-feature extraction (SDDFE) module for extracting features within a single domain (drugs or diseases) and the cross-domain dual-feature extraction (CDDFE) module for extracting features across domains. By utilizing these modules, we ensure more appropriate encoding of drugs and diseases. A cross-dual-domain decoder is also designed to predict drug-disease associations in both domains. Our proposed DFDRNN model outperforms six state-of-the-art methods on four benchmark datasets, achieving an average AUROC of 0.946 and an average AUPR of 0.597. Case studies on two diseases show that the proposed DFDRNN model can be applied in real-world scenarios, demonstrating its significant potential in drug repositioning.

The Critical Node Problem (CNP) is concerned with identifying the critical nodes in a complex network. These nodes play a significant role in maintaining the connectivity of the network, and removing them can negatively impact network performance. CNP has been studied extensively due to its numerous real-world applications. Among the different versions of CNP, CNP-1a has gained the most popularity. The primary objective of CNP-1a is to minimize the pair-wise connectivity in the remaining network after deleting a limited number of nodes from a network. Due to the NP-hard nature of CNP-1a, many heuristic/metaheuristic algorithms have been proposed to solve this problem. However, most existing algorithms start with a random initialization, leading to a high cost of obtaining an optimal solution. To improve the efficiency of solving CNP-1a, a knowledge-guided genetic algorithm named K2GA has been proposed. Unlike the standard genetic algorithm framework, K2GA has two main components: a pretrained neural network to obtain prior knowledge on possible critical nodes, and a hybrid genetic algorithm with local search for finding an optimal set of critical nodes based on the knowledge given by the trained neural network. The local search process utilizes a cut node-based greedy strategy. The effectiveness of the proposed knowledgeguided genetic algorithm is verified by experiments on 26 realworld instances of complex networks. Experimental results show that K2GA outperforms the state-of-the-art algorithms regarding the best, median, and average objective values, and improves the best upper bounds on the best objective values for eight realworld instances., Comment: 14 pages, 13 figures

Given a graph, the minimum dominating set (MinDS) problem is to identify a smallest set $D$ of vertices such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The MinDS problem is a classic $\mathcal{NP}$-hard problem and has been extensively studied because of its many disparate applications in network analysis. To solve this problem efficiently, many heuristic approaches have been proposed to obtain a good solution within an acceptable time limit. However, existing MinDS heuristic algorithms are always limited by various tie-breaking cases when selecting vertices, which slows down the effectiveness of the algorithms. In this paper, we design an efficient local search algorithm for the MinDS problem, named DmDS -- a dual-mode local search framework that probabilistically chooses between two distinct vertex-swapping schemes. We further address limitations of other algorithms by introducing vertex selection criterion based on the frequency of vertices added to solutions to address tie-breaking cases, and a new strategy to improve the quality of the initial solution via a greedy-based strategy integrated with perturbation. We evaluate DmDS against the state-of-the-art algorithms on seven datasets, consisting of 346 instances (or families) with up to tens of millions of vertices. Experimental results show that DmDS obtains the best performance in accuracy for almost all instances and finds much better solutions than state-of-the-art MinDS algorithms on a broad range of large real-world graphs., Comment: 11 pages, 2 figures, 13 tables

DNA computing, a nontraditional computing mechanism, provides a feasible and effective method for solving NP-hard problems because of the vast parallelism and high-density storage of DNA molecules. Although DNA computing has been exploited to solve various intractable computational problems, such as the Hamiltonian path problem, SAT problem, and graph coloring problem, there has been little discussion of designing universal DNA computing-based models, which can solve a class of problems. In this paper, by leveraging the dynamic and enzyme-free properties of DNA strand displacement, we propose a universal model named DCMSubset for solving subset problems in graph theory. The model aims to find a minimum (or maximum) set satisfying given constraints. For each element x involved in a given problem, DCMSubset uses an exclusive single-stranded DNA molecule to model x as well as a specific DNA complex to model the relationship between x and other elements. Based on the proposed model, we conducted simulation and biochemical experiments on three kinds of subset problems, a minimum dominating set, maximum independent set, and minimum vertex cover. We observed that DCMSubset can also be used to solve the graph coloring problem. Moreover, we extended DCMSubset to a model for solving the SAT problem. The results of experiments showed the feasibility and university of the proposed method. Our results highlighted the potential for DNA strand displacement to act as a computation tool to solve NP-hard problems.

The maximum independent set (MIS) problem, a classical NP-hard problem with extensive applications in various areas, aims to find the largest set of vertices with no edge among them. Due to its computational intractability, it is difficult to solve the MIS problem effectively, especially on large graphs. Employing heuristic approaches to obtain a good solution within an acceptable amount of time has attracted much attention in literature. In this paper, we propose an efficient local search framework for MIS called ARIR, which encompasses two main parts: a lightweight adaptive mechanism and a novel inexact efficient reduction rule to simplify instances. Based on ARIR, three algorithms -- ARIR-I, ARIR-II, and ARIR-III -- are developed by adopting three distinct reduction strategies. We conduct experiments on five benchmarks, encompassing 92 instances. Compared with six state-of-the-art algorithms, our ARIR-based algorithms offer the best accuracy on the majority of instances, while obtaining competitive results on the remaining instances., Comment: 11 pages, 0 figures

For the purpose of maximizing the spread of influence caused by a certain small number k of nodes in a social network, we are asked to find a k-subset of nodes (i.e., a seed set) with the best capacity to influence the nodes not in it. This problem of influence maximization (IM) has wide application, belongs to subset problems, and is NP-hard. To solve it, we should theoretically examine all seed sets and evaluate their influence spreads, which is time-consuming. Therefore, metaheuristic strategies are generally employed to gain a good seed set within a reasonable time. We observe that many algorithms for the IM problem only adopt a uniform mechanism in the whole solution search process, which lacks a response measure when the algorithm becomes trapped in a local optimum. To address this issue, we propose a phased hybrid evaluation-enhanced (PHEE) approach for IM, which utilizes two distinct search strategies to enhance the search of optimal solutions: a randomized range division evolutionary (RandRDE) algorithm to improve the solution quality, and a fast convergence strategy. Our approach is evaluated on 10 real-world social networks of different sizes and types. Experimental results demonstrate that our algorithm is efficient and obtains the best influence spread for all the datasets compared with three state-of-the-art algorithms, outperforms the time consuming CELF algorithm on four datasets, and performs worse than CELF on only two networks.

The metric (resp. edge metric) dimension of a simple connected graph $G$, denoted by dim$(G)$ (resp. edim$(G)$), is the cardinality of a smallest vertex subset $S\subseteq V(G)$ for which every two distinct vertices (resp. edges) in $G$ have distinct distances to a vertex of $S$. It is an interesting topic to discuss the relation between dim$(G)$ and edim$(G)$ for some class of graphs $G$. In this paper, we settle two open problems on this topic for a widely studied class of graphs, called unicyclic graphs. Specifically, we introduce four classes of subgraphs to characterize the structure of a unicyclic graph whose metric (resp . edge metric) dimension is equal to the lower bound on this invariant for unicyclic graphs. Based on this, we determine the exact values of dim$(G)$ and edim$(G)$ for all unicyclic graphs $G$.

The $k$-rainbow independent domination number of a graph $G$, denoted $\gamma_{\rm rik}(G)$, is the cardinality of a smallest set consisting of two vertex-disjoint independent sets $V_1$ and $V_2$ for which every vertex in $V(G)\setminus (V_1\cup V_2)$ has neighbors in both $V_1$ and $V_2$. This domination invariant was proposed by {\v{S}}umenjak, Rall and Tepeh in (Applied Mathematics and Computation 333(15), 2018: 353-361), which allows to reduce the problem of computing the independent domination number of the generalized prism $G {\Box} K_k$ to an integer labeling problem on $G$. They proved a Nordhaus-Gaddum-type theorem: $5\leq \gamma_{\rm rik}(G)+\gamma_{\rm rik}(\overline{G})\leq n+3$ for every graph $G$ of order $n\geq 3$, where $\overline{G}$ is the complement of $G$. In this paper, we improve this result by showing that if $G$ is not isomorphic to the 5-cycle, then $5\leq \gamma_{\rm rik}(G)+\gamma_{\rm rik}(\overline{G})\leq n+2$. Moreover, we show that the problem of deciding whether a graph has a $k$-rainbow independent dominating function of a given weight is $\mathcal{NP}$-complete. Our results respond some open questions proposed by \v{S}umenjak, et al., Comment: 9 pages, 1 figure

A total $k$-coloring of a graph is an assignment of $k$ colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The Total Coloring Conjecture (TCC) states that every simple graph $G$ has a total ($\Delta(G)+2$)-coloring, where $\Delta(G)$ is the maximum degree of $G$. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph $G$ of $\Delta(G) \geq 9$ or $\Delta(G) \in \{7, 8\}$ with some restrictions has a total $(\Delta(G) + 1)$-coloring. In particular, in [Shen and Wang, "On the 7 total colorability of planar graphs with maximum degree 6 and without 4-cycles", Graphs and Combinatorics, 25: 401-407, 2009], the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent 4-cycles or not incident with three cycles of size $p,q,\ell$ for some $\{p,q,\ell\}\in \{\{3,4,4\},\{3,3,4\}\}$.

Graph embedding has been proven to be efficient and effective in facilitating graph analysis. In this paper, we present a novel spectral framework called NOn-Backtracking Embedding (NOBE), which offers a new perspective that organizes graph data at a deep level by tracking the flow traversing on the edges with backtracking prohibited. Further, by analyzing the non-backtracking process, a technique called graph approximation is devised, which provides a channel to transform the spectral decomposition on an edge-to-edge matrix to that on a node-to-node matrix. Theoretical guarantees are provided by bounding the difference between the corresponding eigenvalues of the original graph and its graph approximation. Extensive experiments conducted on various real-world networks demonstrate the efficacy of our methods on both macroscopic and microscopic levels, including clustering and structural hole spanner detection., Comment: SDM 2018 (Full version including all proofs)

A graph $G$ is \emph{uniquely k-colorable} if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. For a plane graph $G$, two faces $f_1$ and $f_2$ of $G$ are \emph{adjacent $(i,j)$-faces} if $d(f_1)=i$, $d(f_2)=j$ and $f_1$ and $f_2$ have a common edge, where $d(f)$ is the degree of a face $f$. In this paper, we prove that every uniquely 3-colorable plane graph has adjacent $(3,k)$-faces, where $k\leq 5$. The bound 5 for $k$ is best possible. Furthermore, we prove that there exist a class of uniquely 3-colorable plane graphs having neither adjacent $(3,i)$-faces nor adjacent $(3,j)$-faces, where $i,j\in \{3,4,5\}$ and $i \neq j$. One of our constructions implies that there exist an infinite family of edge-critical uniquely 3-colorable plane graphs with $n$ vertices and $\frac{7}{3}n-\frac{14}{3}$ edges, where $n(\geq 11)$ is odd and $n\equiv 2\pmod{3}$., Comment: 7 pages, 4 figures

A tree-coloring of a maximal planar graph is a proper vertex $4$-coloring such that every bichromatic subgraph, induced by this coloring, is a tree. A maximal planar graph $G$ is tree-colorable if $G$ has a tree-coloring. In this article, we prove that a tree-colorable maximal planar graph $G$ with $\delta(G)\geq 4$ contains at least four odd-vertices. Moreover, for a tree-colorable maximal planar graph of minimum degree 4 that contains exactly four odd-vertices, we show that the subgraph induced by its four odd-vertices is not a claw and contains no triangles., Comment: 18pages,10figures

A graph $G$ is \emph{uniquely k-colorable} if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable graph for any edge $e\in E(G)$. Mel'nikov and Steinberg [L. S. Mel'nikov, R. Steinberg, One counterexample for two conjectures on three coloring, Discrete Math. 20 (1977) 203-206] asked to find an exact upper bound for the number of edges in a edge-critical 3-colorable planar graph with $n$ vertices. In this paper, we give some properties of edge-critical uniquely 3-colorable planar graphs and prove that if $G$ is such a graph with $n(\geq6)$ vertices, then $|E(G)|\leq \frac{5}{2}n-6 $, which improves the upper bound $\frac{8}{3}n-\frac{17}{3}$ given by Matsumoto [N. Matsumoto, The size of edge-critical uniquely 3-colorable planar graphs, Electron. J. Combin. 20 (3) (2013) $\#$P49]. Furthermore, we find some edge-critical 3-colorable planar graphs which have $n(=10,12, 14)$ vertices and $\frac{5}{2}n-7$ edges., Comment: 17pages,5figures