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Given a graph G = (V , E) , the problem of Graph Burning is to find a sequence of nodes from V , called a burning sequence, to burn the whole graph. This is a discrete-step process, and at each step, an unburned vertex is selected as an agent to spread fire to its neighbors by marking it as a burnt node. A burnt node spreads the fire to its neighbors at the next consecutive step. The goal is to find the burning sequence of minimum length. The Graph Burning problem is NP-Hard for general graphs and even for binary trees. A few approximation results are known, including a 3 -approximation algorithm for general graphs and a 2 -approximation algorithm for trees. The Graph Burning on directed graphs is more challenging than on undirected graphs. In this paper, we propose (1) A 2. 7 5 -approximation algorithm for a cactus graph (undirected), (2) A 3 -approximation algorithm for multi-rooted directed trees (polytree) and (3) A 1. 9 0 5 -approximation algorithm for the single-rooted directed tree (arborescence). We implement all three approximation algorithms, and the results are shown for randomly generated cactus graphs, multi-rooted directed trees and single-rooted directed trees. [ABSTRACT FROM AUTHOR]

Many real-world applications arising from social networks, personalized recommendations, and others, require extracting a relatively small but broadly representative portion of massive data sets. Such problems can often be formulated as maximizing a monotone set function with cardinality constraints. In this paper, we consider a streaming model where elements arrive quickly over time, and create an effective, and low-memory algorithm. First, we provide the first single-pass linear-time algorithm, which is a a deterministic algorithm, achieves an approximation ratio of [ γ 4 a (1 + γ + γ 2 + γ 3) − ] for any ≥ 0 with a query complexity of ⌈ n / a ⌉ + a and a memory complexity of O (a k log (k) log (1 /)) , where a is a positive integer and γ is the submodularity ratio. However, the algorithm may produce less-than-ideal results. Our next result is to describe a multi-streaming algorithm, which is the first deterministic algorithm to attain an approximation ratio of 1 − e − γ − with linear query complexity. [ABSTRACT FROM AUTHOR]

Correlation clustering problem (CorCP) is a classical clustering problem, which clusters data based on the similarity of data set, and has many applications in interaction networks, cross-lingual link detection, and communication networks, etc. In this paper, we study a practical generalization of the CorCP, called the capacitated correlation clustering problem (the capacitated CorCP), by constructing a labeled complete graph. On this labeled complete graph, each vertex represents a piece of data. If two pieces of data are similar, then the edge between the corresponding vertices is marked by a positive label +. Otherwise, this edge is marked by a negative label -. The objective of the capacitated CorCP is to group some similar data sets into one cluster as far as possible, while satisfying the cluster capacity constraint. To achieve this objective, we shall partition the vertex set of the labeled complete graph into several clusters, each cluster's size subjecting to an upper bound, so as to minimize the number of disagreements. Here the number of disagreements is defined as the total number of the edges with positive labels between clusters and the edges with negative labels within clusters. Different with the previous algorithm in [18], which subjects to the constraint on the cluster size by a penalty measure, we design an algorithm for the capacitated CorCP to directly output a feasible solution by iteratively constructing clusters based on a preset threshold. Through carefully setting the threshold and sophisticatedly analyzing, our algorithm is proved to have an improved approximation ratio of 5.37. In addition, we also conduct a series of numerical experiments to demonstrate the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study a discrete momentum consensus-based optimization (Momentum-CBO) algorithm which corresponds to a second-order generalization of the discrete first-order CBO [S.-Y. Ha, S. Jin and D. Kim, Convergence of a first-order consensus-based global optimization algorithm, Math. Models Methods Appl. Sci. 30 (2020) 2417–2444]. The proposed algorithm can be understood as the modification of ADAM-CBO, replacing the normalization term by unity. For the proposed Momentum-CBO, we provide a sufficient framework which guarantees the convergence of algorithm toward a global minimum of the objective function. Moreover, we present several experimental results showing that Momentum-CBO has an improved success rate of finding the global minimum compared to vanilla-CBO and show the stability of Momentum-CBO under different initialization schemes. We also show that Momentum-CBO can be used as the alternative of ADAM-CBO which does not have a proper convergence analysis. Finally, we give an application of Momentum-CBO for Lyapunov function approximation using symbolic regression techniques. [ABSTRACT FROM AUTHOR]

For an m -edge connected simple graph G , finding a spanning tree of G with the maximum number of leaves is MAXSNP-complete. The problem remains NP-complete even if G is planar and the maximal degree of G is at most four. Lu and Ravi gave the first known polynomial-time approximation algorithms with approximation factors 5 and 3. Later, they obtained a 3 -approximation algorithm that runs in near-linear time. The best known result is Solis-Oba, Bonsma, and Lowski's O (m) -time 2 -approximation algorithm. We show an alternative simple O (m) -time 2 -approximation algorithm whose analysis is simpler. This paper is dedicated to the cherished memory of our dear friend, Professor Takao Nishizeki. [ABSTRACT FROM AUTHOR]

In this paper, we present an approximation algorithm for the maximum independent set (MIS) problem over the class of B 1 -VPG graphs when the input is specified by a B 1 -VPG representation. We obtain a 6 (log n) 2 -approximation algorithm running in O (n (log n) 3) time. This is an improvement over the previously best n -approximation algorithm [J. Fox and J. Pach, Computing the independence number of intersection graphs, in Proc. Twenty-Second Annual ACM-SIAM Symp. Discrete Algorithms (SODA 2011), 2011, pp. 1161–1165, doi:10.1137/1.9781611973082.87] (for some fixed > 0) designed for some subclasses of string graphs, on B 1 -VPG graphs. [ABSTRACT FROM AUTHOR]

For a simple, undirected, connected graph G , a function h : V (G) → { 0 , 1 , 2 , 3 } which satisfies the following conditions is called a total Roman {3}-dominating function (TR3DF) of G with weight h (V) = ∑ p ∈ V h (p) : (C1) For every vertex u ∈ V if h (u) = 0 , then u has m (m ≥ 1) neighbors such that whose sum is at least 3, and if h (u) = 1 , then u has n (n ≥ 1) neighbors such that whose sum is at least 2. (C2) The subgraph induced by the set of vertices labeled one, two or three has no isolated vertices. For a graph G , the smallest possible weight of a TR3DF of G denoted γ t { R 3 } (G) is known as the total Roman { 3 } -domination number of G. The problem of determining γ t { R 3 } (G) of a graph G is called minimum total Roman {3}-domination problem (MTR3DP). In this paper, we show that the problem of deciding if G has a TR3DF of weight at most l for chordal graphs is NP-complete. We also show that MTR3DP is polynomial time solvable for bounded treewidth graphs, chain graphs and threshold graphs. We design a 3 (ln (Δ − 0. 5) + 1. 5) -approximation algorithm for the MTR3DP and show that the same cannot have (1 − δ) ln | V | ratio approximation algorithm for any δ > 0 unless NP ⊆ DTIME (| V | O (log log | V |)). Next, we show that MTR3DP is APX-complete for graphs with Δ = 4. We also show that the domination and total Roman {3}-domination problems are not equivalent in computational complexity aspects. Finally, we present an integer linear programming formulation for MTR3DP. [ABSTRACT FROM AUTHOR]