scholarly journals Dynamic Approximate Vertex Cover and Maximum Matching

2010 ◽  
pp. 341-345 ◽  
Author(s):  
Krzysztof Onak ◽  
Ronitt Rubinfeld
Keyword(s):  
Vertex Cover ◽  
Maximum Matching

scholarly journals Deterministic Dynamic Matching in O(1) Update Time

Algorithmica ◽  
2019 ◽  
Vol 82 (4) ◽  
pp. 1057-1080 ◽  
Author(s):  
Sayan Bhattacharya ◽  
Deeparnab Chakrabarty ◽  
Monika Henzinger
Keyword(s):  
Computer Science ◽  
Vertex Cover ◽  
Approximation Ratio ◽  
Approximate Algorithm ◽  
Maximum Matching ◽  
Matching Problem ◽  
Dynamic Algorithm ◽  
Dynamic Matching ◽  
Deterministic Dynamic ◽  
Oblivious Adversary

Abstract We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld (in: Proceedings of the ACM symposium on theory of computing (STOC), 2010), this problem has received significant attention in recent years. Very recently, extending the framework of Baswana et al. (in: Proceedings of the IEEE symposium on foundations of computer science (FOCS), 2011) , Solomon (in: Proceedings of the IEEE symposium on foundations of computer science (FOCS), 2016) gave a randomized dynamic algorithm for this problem that has an approximation ratio of 2 and an amortized update time of O(1) with high probability. This algorithm requires the assumption of an oblivious adversary, meaning that the future sequence of edge insertions/deletions in the graph cannot depend in any way on the algorithm’s past output. A natural way to remove the assumption on oblivious adversary is to give a deterministic dynamic algorithm for the same problem in O(1) update time. In this paper, we resolve this question. We present a new deterministic fully dynamic algorithm that maintains a O(1)-approximate minimum vertex cover and maximum fractional matching, with an amortized update time of O(1). Previously, the best deterministic algorithm for this problem was due to Bhattacharya et al. (in: Proceedings of the ACM-SIAM symposium on discrete algorithms (SODA), 2015); it had an approximation ratio of $$(2+\varepsilon )$$(2+ε) and an amortized update time of $$O(\log n/\varepsilon ^2)$$O(logn/ε2). Our result can be generalized to give a fully dynamic $$O(f^3)$$O(f3)-approximate algorithm with $$O(f^2)$$O(f2) amortized update time for the hypergraph vertex cover and fractional hypergraph matching problem, where every hyperedge has at most f vertices.

scholarly journals ANALYZING NONBLOCKING MULTILOG NETWORKS WITH THE KÖNIG–EGEVARÝ THEOREM

2009 ◽  
Vol 01 (01) ◽  
pp. 127-139 ◽  
Author(s):  
HUNG Q. NGO ◽  
THANH-NHAN NGUYEN ◽  
DUC T. HA
Keyword(s):  
Bipartite Graph ◽  
Vertex Cover ◽  
Window Size ◽  
Maximum Matching ◽  
Switching Network ◽  
Good Tool ◽  
Photonic Switching ◽  
Multicast Networks ◽  
First Time ◽  
By Products

When analyzing a nonblocking switching network, the typical problem is to find a route for a new request through the network without disturbing existing routes. By solving this problem, we can derive how many hardware components of a certain type (Banyan planes in a multi-log network, for instance) are needed for the network to be nonblocking. This scenario appears in virtually all combinations of switching environments: strictly, widesense or rearrangeably nonblocking, unicast or multicast switching, and circuit, multirate, or photonic switching. In this paper, we show that the König–Egevarý theorem is a very good tool which helps solve the above prototypical problem. The idea is to somehow "represent" the potential blocking connections as edges of a bipartite graph. The maximum number of blocking connections roughly corresponds to the size of a maximum matching in that bipartite graph. The size of any vertex cover, by the König–Egevarý theorem, is an upper bound on the maximum number of blocking connections. Thus, by specifying a small vertex cover, we can derive the sufficient number of hardware components for the network to be nonblocking. We illustrate the technique by analyzing crosstalk-free and non-crosstalk-free widesense nonblocking multicast multi-log networks. Particularly, for the first time in the literature we derive conditions for the d-ary multi-log network to be crosstalk-free multicast widesense nonblocking under the window algorithm for any given window size. Several by-products follow from our approach and results. Firstly, our results allow for computing the best window size minimizing the fabric cost, showing that the multi-log network is a good candidate for crosstalk-free multicast switching architectures. Secondly, the analytical approach also gives a much simpler proof of the known case when the network is not required to be crosstalk-free. Thirdly, we show that several known results for the multi-log multicast networks under the so-called fanout constraints are simple corollaries of our results.

Complexity of the Maximum k-Path Vertex Cover Problem

2020 ◽  
Vol E103.A (10) ◽  
pp. 1193-1201
Author(s):  
Eiji MIYANO ◽  
Toshiki SAITOH ◽  
Ryuhei UEHARA ◽  
Tsuyoshi YAGITA ◽  
Tom C. van der ZANDEN
Keyword(s):  
Vertex Cover ◽  
Vertex Cover Problem ◽  
Cover Problem

scholarly journals Approximation algorithm for minimum connected 3-path vertex cover

2020 ◽  
Vol 287 ◽  
pp. 77-84
Author(s):  
Pengcheng Liu ◽  
Zhao Zhang ◽  
Xianyue Li ◽  
Weili Wu
Keyword(s):  
Approximation Algorithm ◽  
Vertex Cover

scholarly journals Characterizations of maximum matching graphs of certain types

2005 ◽  
Vol 290 (2-3) ◽  
pp. 283-289 ◽  
Author(s):  
Yan Liu
Keyword(s):  
Maximum Matching

scholarly journals From the Quasi-Total Strong Differential to Quasi-Total Italian Domination in Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1036
Author(s):  
Abel Cabrera Martínez ◽  
Alejandro Estrada-Moreno ◽  
Juan Alberto Rodríguez-Velázquez
Keyword(s):  
Vertex Cover ◽  
Domination Number ◽  
Theoretical Computer Science ◽  
Discrete Mathematics ◽  
Special Issue ◽  
Total Domination Number ◽  
Theoretical Computer ◽  
Graph Parameters ◽  
Semitotal Domination ◽  
Domination In Graphs

This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex x∈V(G) of a graph G, the neighbourhood of x is denoted by N(x). The neighbourhood of a set X⊆V(G) is defined to be N(X)=⋃x∈XN(x), while the external neighbourhood of X is defined to be Ne(X)=N(X)∖X. Now, for every set X⊆V(G) and every vertex x∈X, the external private neighbourhood of x with respect to X is defined as the set Pe(x,X)={y∈V(G)∖X:N(y)∩X={x}}. Let Xw={x∈X:Pe(x,X)≠⌀}. The strong differential of X is defined to be ∂s(X)=|Ne(X)|−|Xw|, while the quasi-total strong differential of G is defined to be ∂s*(G)=max{∂s(X):X⊆V(G)andXw⊆N(X)}. We show that the quasi-total strong differential is closely related to several graph parameters, including the domination number, the total domination number, the 2-domination number, the vertex cover number, the semitotal domination number, the strong differential, and the quasi-total Italian domination number. As a consequence of the study, we show that the problem of finding the quasi-total strong differential of a graph is NP-hard.

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