快速匹配算法在随机网络中的最优性及其在结构可控性中的应用

Two-sided profile-based optimality in the stable marriage problem

We study the problem of finding "fair" stable matchings in the Stable Marriage problem with Incomplete lists (SMI). In particular, we seek stable matchings that are optimal with respect to profile, which is a vector that indicates the number of agents who have their first-, second-, third-choice partner, etc. In a rank maximal stable matching, the maximum number of agents have their first-choice partner, and subject to this, the maximum number of agents have their second-choice partner, etc., whilst in a generous stable matching $M$, the minimum number of agents have their $d$th-choice partner…

带有优先级的多对多匹配中的临界稳定匹配

Critical Relaxed-Stable Matchings with Ties in the Many-to-Many Setting

本文提出了一种在多对多匹配中考虑优先级和下限配额的临界稳定匹配方法，证明了临界且放松稳定的匹配总是存在，并提供了最大基数的近似算法。

随机正则图在稀疏网络设计中的最优性

Optimality of Random Regular Graphs in Sparse Network Designs

本文证明随机正则图在二分和单分网络设计中达到平均度的渐近下界，实现稀疏网络的最优性能，并通过数值实验验证。

Maximum Matchings and Popularity

Let $G$ be a bipartite graph where every node has a strict ranking of its neighbors. For every node, its preferences over neighbors extend naturally to preferences over matchings. Matching $N$ is more popular than matching $M$ if the number of nodes that prefer $N$ to $M$ is more than the number that prefer $M$ to $N$. A maximum matching $M$ in $G$ is a "popular max-matching" if there is no maximum matching in $G$ that is more popular than $M$. Such matchings are relevant in applications where the set of admissible solutions is the set of maximum matchings and we wish to find a best maximum ma…

A Decomposition Algorithm to Solve the Multi-Hop Peer-to-Peer Ride-Matching Problem

In this paper, we mathematically model the multi-hop Peer-to-Peer (P2P) ride-matching problem as a binary program. We formulate this problem as a many-to-many problem in which a rider can travel by transferring between multiple drivers, and a driver can carry multiple riders. We propose a pre-processing procedure to reduce the size of the problem, and devise a decomposition algorithm to solve the original ride-matching problem to optimality by means of solving multiple smaller problems. We conduct extensive numerical experiments to demonstrate the computational efficiency of the proposed algor…

The complexity of the Multiple Pattern Matching Problem for random strings

We generalise a multiple string pattern matching algorithm, recently proposed by Fredriksson and Grabowski [J. Discr. Alg. 7, 2009], to deal with arbitrary dictionaries on an alphabet of size $s$. If $r_m$ is the number of words of length $m$ in the dictionary, and $ϕ(r) = \max_m \ln(s\, m\, r_m)/m$, the complexity rate for the string characters to be read by this algorithm is at most $κ_{{}_\textrm{UB}}\, ϕ(r)$ for some constant $κ_{{}_\textrm{UB}}$. On the other side, we generalise the classical lower bound of Yao [SIAM J. Comput. 8, 1979], for the problem with a single pattern, to deal with…

单遍半流匹配 II：贪心算法是最优的

Semi-Streaming Matching in a Single Pass II: Greedy is Optimal

研究最大匹配问题的单遍半流算法近似率，证明无算法能超二分之一近似，表明朴素贪心算法最优，还通过“蓝图框架”给出构造及结果，解决了带抢占在线匹配最优竞争率的问题。

Cut-matching Games for Generalized Hypergraph Ratio Cuts

Hypergraph clustering is a basic algorithmic primitive for analyzing complex datasets and systems characterized by multiway interactions, such as group email conversations, groups of co-purchased retail products, and co-authorship data. This paper presents a practical $O(\log n)$-approximation algorithm for a broad class of hypergraph ratio cut clustering objectives. This includes objectives involving generalized hypergraph cut functions, which allow a user to penalize cut hyperedges differently depending on the number of nodes in each cluster. Our method is a generalization of the cut-matchin…

Pattern Matching and Consensus Problems on Weighted Sequences and Profiles

We study pattern matching problems on two major representations of uncertain sequences used in molecular biology: weighted sequences (also known as position weight matrices, PWM) and profiles (i.e., scoring matrices). In the simple version, in which only the pattern or only the text is uncertain, we obtain efficient algorithms with theoretically-provable running times using a variation of the lookahead scoring technique. We also consider a general variant of the pattern matching problems in which both the pattern and the text are uncertain. Central to our solution is a special case where the s…

A matching-based heuristic algorithm for school bus routing problems

School bus planning problem (SBPP) has drawn much research attention due to the huge costs of school transportation. In the literature, the SBPP is usually decomposed into the routing and scheduling subproblems due to its complexity. Because of the nature of the decomposition, even if all the subproblems are solved to optimality, the final solution may not be as good as the solution from the integrated model. In this paper, we present a new approach that incorporates the scheduling information (namely the trip compatibility) into the routing stage such that the interrelationship between the su…