scholarly journals Linear programming in the semi-streaming model with application to the maximum matching problem

2013 ◽  
Vol 222 ◽  
pp. 59-79 ◽  
Author(s):  
Kook Jin Ahn ◽  
Sudipto Guha
Keyword(s):  
Linear Programming ◽  
Maximum Matching ◽  
Matching Problem

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.

The Maximum Matching Problem Based on Self-Assembly Model of Molecular Beacon

2018 ◽  
Vol 10 (2) ◽  
pp. 213-218 ◽  
Author(s):  
Jing Yang ◽  
Zhixiang Yin ◽  
Kaifeng Huang ◽  
Jianzhong Cui
Keyword(s):  
Self Assembly ◽  
Molecular Beacon ◽  
Maximum Matching ◽  
Assembly Model ◽  
Matching Problem

Innovation of Matching Structures Through Clustering and Reconstructing Basic Operation Actions in the Form Layer

2017 ◽  
Author(s):  
Li Yu-Tong ◽  
Wang Yuxin
Keyword(s):  
Basic Mechanism ◽  
Maximum Matching ◽  
Basic Operation ◽  
Function Structure ◽  
Matching Problem ◽  
Direct Function ◽  
Matching Process ◽  
Order Relations ◽  
Functional Reasoning ◽  
Essential Knowledge

Due to a lack of essential knowledge to support functional reasoning from the design requirements of the kinematic compound mechanisms to their constituent mechanisms, the creative conceptual design of kinematic compound mechanisms based on functional synthesis approach is still a challenging task. Through introducing the dynamic partition-matching process between the function layer and the form layer to substitute for the direct function-structure matching in the FBS model, the function-structure matching problem corresponding to deficient functional reasoning knowledge for kinematic compound mechanisms is solved by the authors. The following challenge is how to cluster the divided subset of basic operation actions generated in the form layer during the partition-matching process into a well-organized and complete kinematic behavior that can be matched by the sub-function in the function layer and implemented by a structure in the database. The adopted strategies in this paper are: through defining the correlation indexes between basic operation actions, the basic operation action and its realized function behavior, and its embodied structure, as well as its dynamic behavior characteristics, the clustering possibility for a group of basic operation actions is determined. With the aid of the compatibility conditions between basic operation actions in the form layer and the consistency of the order relations between basic operation actions in the function layer and the form layer respectively, the consistency of the order relations among basic operation actions between the sub-functions in the function layer and the sub-behaviors in the form layer are guaranteed. Then, the optimal matching structures corresponding to the sub-functions in the function layer are determined based on the maximum matching coefficients of basic operation actions. In this way, the subsets of basic operation actions in the form layer are clustered into a number of complete behaviors that can be realized by mechanisms in the structure database and matched by the sub-functions in the function layer. Since multiple functional behaviors of each constituent basic mechanism take part in matching, some novel schemes of compound mechanisms with fewer and simpler constituent mechanisms to implement the overall function may be dug out.

scholarly journals New graph algorithms via polyhedral techniques

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jakub Tarnawski
Keyword(s):  
Linear Programming ◽  
Perfect Matching ◽  
Traveling Salesman ◽  
Benchmark Problem ◽  
Matching Problem ◽  
Graph Problems ◽  
Weighted Directed Graph ◽  
Asymmetric Traveling Salesman Problem ◽  
Problem Instances ◽  
New Algorithms

Abstract This article gives a short overview of my dissertation, where new algorithms are given for two fundamental graph problems. We develop novel ways of using linear programming formulations, even exponential-sized ones, to extract structure from problem instances and to guide algorithms in making progress. The first part of the dissertation addresses a benchmark problem in combinatorial optimization: the asymmetric traveling salesman problem (ATSP). It consists in finding the shortest tour that visits all vertices of a given edge-weighted directed graph. A ρ-approximation algorithm for ATSP is one that runs in polynomial time and always produces a tour at most ρ times longer than the shortest tour. Finding such an algorithm with constant ρ had been a long-standing open problem. Here we give such an algorithm. The second part of the dissertation addresses the perfect matching problem. We have known since the 1980s that it has efficient parallel algorithms if the use of randomness is allowed. However, we do not know if randomness is necessary – that is, whether the matching problem is in the class NC. We show that it is in the class quasi-NC. That is, we give a deterministic parallel algorithm that runs in poly-logarithmic time on quasi-polynomially many processors.

Sign in / Sign up

Export Citation Format

Share Document