Figure - available from: Theory of Computing Systems
This content is subject to copyright. Terms and conditions apply.
![Illustrative example of a configuration graph. (a) Temporal graph instance I=(G,k,ℓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I=(\mathcal {G},k,\ell )$\end{document} from Fig. 1 with G=(G1,G2,G3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {G}=(G_{1},G_{2},G_{3})$\end{document}, k = 2, and ℓ = 1. (b) Configuration graph of I from (a); a directed s-t path is highlighted corresponding to the solution depicted in Fig. 1](publication/360268197/figure/fig2/AS:11431281085973068@1663984171840/Illustrative-example-of-a-configuration-graph-a-Temporal-graph-instance.png)
Illustrative example of a configuration graph. (a) Temporal graph instance I=(G,k,ℓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I=(\mathcal {G},k,\ell )$\end{document} from Fig. 1 with G=(G1,G2,G3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {G}=(G_{1},G_{2},G_{3})$\end{document}, k = 2, and ℓ = 1. (b) Configuration graph of I from (a); a directed s-t path is highlighted corresponding to the solution depicted in Fig. 1
Source publication
The NP-complete Vertex Cover problem asks to cover all edges of a graph by a small (given) number of vertices. It is among the most prominent graph-algorithmic problems. Following a recent trend in studying temporal graphs (a sequence of graphs, so-called layers, over the same vertex set but, over time, changing edge sets), we initiate the study of...
Citations
... Such data can be modeled by a temporal graph, that is, a multi-layer graph in which the layers are ordered linearly [16,17,[30][31][32][33]. The goal in clustering a temporal graph is to find a clustering that slowly evolves over time consistently with the graph [16][17][18][34][35][36]. ...
Motivated by the recent rapid growth of research for algorithms to cluster multi-layer and temporal graphs, we study extensions of the classical Cluster Editing problem. In Multi-Layer Cluster Editing we receive a set of graphs on the same vertex set, called layers and aim to transform all layers into cluster graphs (disjoint unions of cliques) that differ only slightly. More specifically, we want to mark at most d vertices and to transform each layer into a cluster graph using at most k edge additions or deletions per layer so that, if we remove the marked vertices, we obtain the same cluster graph in all layers. In Temporal Cluster Editing we receive a sequence of layers and we want to transform each layer into a cluster graph so that consecutive layers differ only slightly. That is, we want to transform each layer into a cluster graph with at most k edge additions or deletions and to mark a distinct set of d vertices in each layer so that each two consecutive layers are the same after removing the vertices marked in the first of the two layers. We study the combinatorial structure of the two problems via their parameterized complexity with respect to the parameters d and k, among others. Despite the similar definition, the two problems behave quite differently: In particular, Multi-Layer Cluster Editing is fixed-parameter tractable with running time kO(k+d)sO(1) for inputs of size s, whereas Temporal Cluster Editing is W[1]-hard with respect to k even if d=3.
... In another approach, MVCP is presented in an approach that solves the used graph by separating it into subgraphs [41]. MVCP is solved in the graph divided into layers with the multistage vertex-cover approach [42]. In the approach, the solution is reasonable under some constraints unless there are too many (too different) mismatched nodes in both layers. ...
The minimum vertex-cover problem (MVCP) is an NP-complete optimization problem widely used in areas such as graph theory, social network, security and transportation, etc. Different approaches and algorithms have been proposed in the literature to solve this problem, since MVCP is an optimization problem, the solutions developed for this problem could be more intuitive and give results under certain constraints. In addition, the proposed solution methods for this problem could be more effective, and the determined solution sets change in each iteration. The algorithms/methods developed for solving MVCP are mostly based on heuristic or greedy approaches. This study presents the Malatya vertex-cover algorithm, which provides an efficient solution and a robust approach based on the Malatya centrality value algorithm for MVCP. Although MVCP is an NP-complete problem that cannot be solved in polynomial time, the proposed method offers a polynomial solution to this problem, and the obtained solutions are optimum or near-optimum (optimal solution). This algorithm consists of two basic steps. In the first step, the Malatya centrality values of the nodes in the graph are calculated using the Malatya centrality algorithm. The Malatya centrality value of the nodes in any graph is the summation of the ratio of the node’s degree to the adjacent nodes’ degrees for each node. In the second step, nodes are selected for the MVCP solution based on the node with the maximum Malatya centrality value (Ψ) in the graph is selected and added to the solution set. Then this node and the edges incident on this node are removed from the graph. For the graph consisting of the remaining nodes, Malatya centrality values are calculated again, and the selection process is continued. The process is terminated when all edges in the graph are covered. The proposed algorithm has been tested on artificial, actual graphs and large-scale random graphs produced with the Erdos–Renyi model. When the results are examined, the proposed algorithm yields a robust solution set in polynomial time and polynomial space independent of constraints. In addition, the successful test results in the sample graphs and the analysis of the proposed approach show the effectiveness/superiority of the Malatya centrality algorithm and the proposed method.
... , I τ ) of instances of some problem P as input, and it asks for a "robust" sequence of solutions to the instances in the sense that any two consecutive solutions are similar. Several classical problems have been studied in the multistage model, both from an approximate [1][2][3][4] and from a parameterized [8,19,20,23,31,34] algorithmics point of view. While E E-MstP and V V-MstP adhere to the original multistage model, our two problems E∩E-MstP and V∩V-MstP can be seen as a novel and natural variation of the multistage model by replacing the goal of consecutive similarity with consecutive dissimilarity. ...
... Proof The proof is in line with the proof of [23,Proposition 4.2]. We sketch the proof in the general setup Π -MstP. ...
Addressing a quest by Gupta et al. (in: Proceedings of the 41st international colloquium on automata, languages, and programming (ICALP 2014), vol 8572 of LNCS. Springer, pp 563–575, 2014), we provide a first, comprehensive study of finding a short s – t path in the multistage graph model, referred to as the Multistage s – t Path problem. Herein, given a sequence of graphs over the same vertex set but changing edge sets, the task is to find short s – t paths in each graph (“snapshot”) such that in the found path sequence the consecutive s – t paths are “similar”. We measure similarity by the size of the symmetric difference of either the vertex set (vertex-similarity) or the edge set (edge-similarity) of any two consecutive paths. We prove that these two variants of Multistage s – t Path are already NP -hard for an input sequence of only two snapshots and maximum vertex degree four. Motivated by this fact and natural applications of this scenario e.g. in traffic route planning, we perform a parameterized complexity analysis. Among other results, for both variants, vertex- and edge-similarity, we prove parameterized hardness ( W[1] -hardness) regarding the parameter path length (solution size). As a further conceptual investigation, we then modify the multistage model by asking for dissimilar consecutive paths. As one of the main technical results (employing so-called representative sets known from non-temporal settings), we prove that dissimilarity allows for fixed-parameter tractability for the parameter solution size, contrasting with our W[1]-hardness proof of the corresponding similarity case. We also provide partially positive results concerning efficient and effective data reduction (kernelization).
... Fluschnik et al. [12] and Heeger et al. [13] studied the parameterized complexity of a "multistage" variant of Vertex Cover. Here, one is given a temporal graph and seeks a sequence of vertex covers of size at most k (one for each layer of the temporal graph) such that consecutive vertex covers are in some sense similar, e.g., the symmetric difference of two consecutive vertex covers is upper-bounded by some value [12] or the sum of symmetric differences of all consecutive vertex covers is upper-bounded by some value [13]. ...
... Fluschnik et al. [12] and Heeger et al. [13] studied the parameterized complexity of a "multistage" variant of Vertex Cover. Here, one is given a temporal graph and seeks a sequence of vertex covers of size at most k (one for each layer of the temporal graph) such that consecutive vertex covers are in some sense similar, e.g., the symmetric difference of two consecutive vertex covers is upper-bounded by some value [12] or the sum of symmetric differences of all consecutive vertex covers is upper-bounded by some value [13]. ...
We study the network untangling problem introduced by Rozenshtein, Tatti, and Gionis [DMKD 2021], which is a variant of Vertex Cover on temporal graphs -- graphs whose edge set changes over discrete time steps. They introduce two problem variants. The goal is to select at most k time intervals for each vertex such that all time-edges are covered and (depending on the problem variant) either the maximum interval length or the total sum of interval lengths is minimized. This problem has data mining applications in finding activity timelines that explain the interactions of entities in complex networks. Both variants of the problem are NP-hard. In this paper, we initiate a multivariate complexity analysis involving the following parameters: number of vertices, lifetime of the temporal graph, number of intervals per vertex, and the interval length bound. For both problem versions, we (almost) completely settle the parameterized complexity for all combinations of those four parameters, thereby delineating the border of fixed-parameter tractability.
... For instance, there have been recent studies on Multistage Matching [1], Multistage Knapsack [3], and Online Multistage Subset Maximization [2]. In particular, due to natural parameterizations in this problem setting such as 'number of layers' or 'maximum degree of change' between the solutions for subsequent instances, also parameterized complexity studies have recently been started [8]. ...
Time-evolving or temporal graphs gain more and more popularity when studying the behavior of complex networks. In this context, the multistage view on computational problems is among the most natural frameworks. Roughly speaking, herein one studies the different (time) layers of a temporal graph (effectively meaning that the edge set may change over time, but the vertex set remains unchanged), and one searches for a solution of a given graph problem for each layer. The twist in the multistage setting is that the found solutions may not differ too much between subsequent layers. We relax on this notion by introducing a global instead of the so far local budget view. More specifically, we allow for few disruptive changes between subsequent layers but request that overall, that is, summing over all layers, the degree of change is upper-bounded. Studying several classic graph problems (both NP-hard and polynomial-time solvable ones) from a parameterized angle, we encounter both fixed-parameter tractability and parameterized hardness results. Somewhat surprisingly, we find that sometimes the global multistage versions of NP-hard problems such as Vertex Cover turn out to be computationally easier than the ones of polynomial-time solvable problems such as Matching.
We study the network untangling problem introduced by Rozenshtein et al. (Data Min. Knowl. Disc. 35(1), 213–247, 2021), which is a variant of Vertex Cover on temporal graphs–graphs whose edge set changes over discrete time steps. They introduce two problem variants. The goal is to select at most k time intervals for each vertex such that all time-edges are covered and (depending on the problem variant) either the maximum interval length or the total sum of interval lengths is minimized. This problem has data mining applications in finding activity timelines that explain the interactions of entities in complex networks. Both variants of the problem are NP-hard. In this paper, we initiate a multivariate complexity analysis involving the following parameters: number of vertices, lifetime of the temporal graph, number of intervals per vertex, and the interval length bound. For both problem versions, we (almost) completely settle the parameterized complexity for all combinations of those four parameters, thereby delineating the border of fixed-parameter tractability.
Target-based computer-assisted orchestration can be thought of as the process of searching for combinations of orchestral sounds in a database of sound samples to match a given sound (called target) while respecting specific symbolic constraints (such as the musical instruments that we can use). It is modeled as a combinatorial optimization problem, where the feasible solutions are subsets of orchestral sounds, and the function to minimize measures a distance between the chosen subset of orchestral sounds and the target. In this article, we focus on this optimization problem and analyse it through the lens of computational complexity and approximation algorithms. We first study the so-called static case where there is no temporality in the target sound (we want to reproduce a ‘single’ constant sound). We show that the problem is already NP-hard, but is solvable by a pseudo-polynomial-time algorithm. We also provide an approximation algorithm with additive error. We then consider the more general case with temporality, when the target sound can be seen as a sequence of sounds over a time horizon. We show how the previous results in the static case generalize in this temporal case. We conclude our study with some experiments on real target sounds.
We consider the multistage framework introduced in (Gupta et al., Eisenstat et al., both in ICALP 2014), where we are given a time horizon and a sequence of instances of a (static) combinatorial optimization problem (one for each time step), and the goal is to find a sequence of solutions (one for each time step) reaching a tradeoff between the quality of the solutions in each time step and the stability/similarity of the solutions in consecutive time steps. We first introduce a novel rounding scheme, tailored for multistage problems, that accounts for the moving cost (or stability revenue) of adjacent solutions. Using this rounding scheme, we propose improved approximation algorithms for the multistage variants of Prize-Collecting Steiner Tree and Prize-Collecting Traveling Salesman problems. Furthermore, we introduce a 2-approximation algorithm for multistage Multi-Cut on trees, and we also show that our general scheme can be extended to maximization problems, by providing a 0.75-approximation algorithm for the multistage variant of MaxSat.
