Popular matchings in the marriage and roommates problems

Popular matchings have recently been a subject of study in the context of the so-called House Allocation Problem, where the objective is to match applicants to houses over which the applicants have preferences. A matching M is called popular if there is no other matching M′ with the property that more applicants prefer their allocation in M′ to their allocation in M. In this paper we study popular matchings in the context of the Roommates Problem, including its special (bipartite) case, the Marriage Problem. We investigate the relationship between popularity and stability, and describe efficient algorithms to test a matching for popularity in these settings. We also show that, when ties are permitted in the preferences, it is NP-hard to determine whether a popular matching exists in both the Roommates and Marriage cases

Intrusion Detection Systems for Community Wireless Mesh Networks

Wireless mesh networks are being increasingly used to provide affordable network connectivity to communities where wired deployment strategies are either not possible or are prohibitively expensive. Unfortunately, computer networks (including mesh networks) are frequently being exploited by increasingly profit-driven and insidious attackers, which can affect their utility for legitimate use. In response to this, a number of countermeasures have been developed, including intrusion detection systems that aim to detect anomalous behaviour caused by attacks. We present a set of socio-technical challenges associated with developing an intrusion detection system for a community wireless mesh network. The attack space on a mesh network is particularly large; we motivate the need for and describe the challenges of adopting an asset-driven approach to managing this space. Finally, we present an initial design of a modular architecture for intrusion detection, highlighting how it addresses the identified challenges

Speeding up shortest path algorithms

Given an arbitrary, non-negatively weighted, directed graph $G=(V,E)$ we present an algorithm that computes all pairs shortest paths in time $\mathcal{O}(m^* n + m \lg n + nT_\psi(m^*, n))$, where $m^*$ is the number of different edges contained in shortest paths and $T_\psi(m^*, n)$ is a running time of an algorithm to solve a single-source shortest path problem (SSSP). This is a substantial improvement over a trivial $n$ times application of $\psi$ that runs in $\mathcal{O}(nT_\psi(m,n))$. In our algorithm we use $\psi$ as a black box and hence any improvement on $\psi$ results also in improvement of our algorithm. Furthermore, a combination of our method, Johnson's reweighting technique and topological sorting results in an $\mathcal{O}(m^*n + m \lg n)$ all-pairs shortest path algorithm for arbitrarily-weighted directed acyclic graphs. In addition, we also point out a connection between the complexity of a certain sorting problem defined on shortest paths and SSSP.Comment: 10 page

Node-balancing by edge-increments

Suppose you are given a graph $G=(V,E)$ with a weight assignment $w:V\rightarrow\mathbb{Z}$ and that your objective is to modify $w$ using legal steps such that all vertices will have the same weight, where in each legal step you are allowed to choose an edge and increment the weights of its end points by $1$. In this paper we study several variants of this problem for graphs and hypergraphs. On the combinatorial side we show connections with fundamental results from matching theory such as Hall's Theorem and Tutte's Theorem. On the algorithmic side we study the computational complexity of associated decision problems. Our main results are a characterization of the graphs for which any initial assignment can be balanced by edge-increments and a strongly polynomial-time algorithm that computes a balancing sequence of increments if one exists.Comment: 10 page

Almost-Tight Distributed Minimum Cut Algorithms

We study the problem of computing the minimum cut in a weighted distributed message-passing networks (the CONGEST model). Let $\lambda$ be the minimum cut, $n$ be the number of nodes in the network, and $D$ be the network diameter. Our algorithm can compute $\lambda$ exactly in $O((\sqrt{n} \log^{*} n+D)\lambda^4 \log^2 n)$ time. To the best of our knowledge, this is the first paper that explicitly studies computing the exact minimum cut in the distributed setting. Previously, non-trivial sublinear time algorithms for this problem are known only for unweighted graphs when $\lambda\leq 3$ due to Pritchard and Thurimella's $O(D)$-time and $O(D+n^{1/2}\log^* n)$-time algorithms for computing $2$-edge-connected and $3$-edge-connected components. By using the edge sampling technique of Karger's, we can convert this algorithm into a $(1+\epsilon)$-approximation $O((\sqrt{n}\log^{*} n+D)\epsilon^{-5}\log^3 n)$-time algorithm for any $\epsilon>0$. This improves over the previous $(2+\epsilon)$-approximation $O((\sqrt{n}\log^{*} n+D)\epsilon^{-5}\log^2 n\log\log n)$-time algorithm and $O(\epsilon^{-1})$-approximation $O(D+n^{\frac{1}{2}+\epsilon} \mathrm{poly}\log n)$-time algorithm of Ghaffari and Kuhn. Due to the lower bound of $\Omega(D+n^{1/2}/\log n)$ by Das Sarma et al. which holds for any approximation algorithm, this running time is tight up to a $\mathrm{poly}\log n$ factor. To get the stated running time, we developed an approximation algorithm which combines the ideas of Thorup's algorithm and Matula's contraction algorithm. It saves an $\epsilon^{-9}\log^{7} n$ factor as compared to applying Thorup's tree packing theorem directly. Then, we combine Kutten and Peleg's tree partitioning algorithm and Karger's dynamic programming to achieve an efficient distributed algorithm that finds the minimum cut when we are given a spanning tree that crosses the minimum cut exactly once

Chinese and African migrant entrepreneur's articulation shaped by African agency

Much has been written on the relationship of China and Africa in the past decade. However, the subject of Chinese migrant entrepreneurs in Africa and their articulation with African counterparts was little explored up to the early 2010s. Apparently, this research gap has been closed, as shown by four publications in recent years: three edited volumes and one monography, focusing on this subject. In view of early prejudices on the passive or even disapproving reception of Chinese migrants by Africans, the state of the art underlines the importance and scope of African agency vis à vis Chinese migrant entrepreneurs in Africa. This has been underlined unison in the four books under review. Book review of:(1) Giese, Karsten, Marfaing, Laurence (eds) (2016): Entrepreneurs Africains et Chinois. Les impacts sociaux d’une rencontre particulière. Paris: Karthala. (2) Gadzala, Aleksandra W. (ed) (2015): Africa and China: how Africans and their governments are shaping relations with China. Lanham/Md.: Rowman & Littlefield. (3) Mohan, Giles, Lampert, Ben et al (eds) (2014): Chinese Migrants and Africa's Development: new imperialists or agents of change? London: Zed books. (4) French, Howard W. (2015): China's Second Continent: how a million migrants are building a new empire in Africa. London: Penguin Random House / New York: Knopf.On a beaucoup écrit sur la relation entre la Chine et l'Afrique dans la dernière décennie. Cependant, le sujet des entrepreneurs migrants chinois en Afrique et leur articulation avec leurs homologues africains a été peu exploré jusqu'au début des années 2010. Apparemment, cet écart de recherche a été fermé, comme indiqué par quatre publications au cours des dernières années: trois volumes édités et une monographie, en se concentrant sur ce sujet. Compte tenu des premiers préjugés sur la réception passive ou même désapprobateur des migrants chinois par les Africains, l'état de l'art souligne l'importance et la portée de l'agence africaine vis-à-vis des entrepreneurs migrants chinois en Afrique

On rr-Simple kk-Path

An $r$-simple $k$-path is a {path} in the graph of length $k$ that passes through each vertex at most $r$ times. The $r$-SIMPLE $k$-PATH problem, given a graph $G$ as input, asks whether there exists an $r$-simple $k$-path in $G$. We first show that this problem is NP-Complete. We then show that there is a graph $G$ that contains an $r$-simple $k$-path and no simple path of length greater than $4\log k/\log r$. So this, in a sense, motivates this problem especially when one's goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex. We then give a randomized algorithm that runs in time $$\mathrm{poly}(n)\cdot 2^{O( k\cdot \log r/r)}$$ that solves the $r$-SIMPLE $k$-PATH on a graph with $n$ vertices with one-sided error. We also show that a randomized algorithm with running time $\mathrm{poly}(n)\cdot 2^{(c/2)k/ r}$ with $c<1$ gives a randomized algorithm with running time \poly(n)\cdot 2^{cn} for the Hamiltonian path problem in a directed graph - an outstanding open problem. So in a sense our algorithm is optimal up to an $O(\log r)$ factor

Lombardi Drawings of Graphs

We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure

On the complexity of strongly connected components in directed hypergraphs

We study the complexity of some algorithmic problems on directed hypergraphs and their strongly connected components (SCCs). The main contribution is an almost linear time algorithm computing the terminal strongly connected components (i.e. SCCs which do not reach any components but themselves). "Almost linear" here means that the complexity of the algorithm is linear in the size of the hypergraph up to a factor alpha(n), where alpha is the inverse of Ackermann function, and n is the number of vertices. Our motivation to study this problem arises from a recent application of directed hypergraphs to computational tropical geometry. We also discuss the problem of computing all SCCs. We establish a superlinear lower bound on the size of the transitive reduction of the reachability relation in directed hypergraphs, showing that it is combinatorially more complex than in directed graphs. Besides, we prove a linear time reduction from the well-studied problem of finding all minimal sets among a given family to the problem of computing the SCCs. Only subquadratic time algorithms are known for the former problem. These results strongly suggest that the problem of computing the SCCs is harder in directed hypergraphs than in directed graphs.Comment: v1: 32 pages, 7 figures; v2: revised version, 34 pages, 7 figure

Verified Efficient Implementation of Gabow’s Strongly Connected Component Algorithm

Abstract. We present an Isabelle/HOL formalization of Gabow’s al-gorithm for finding the strongly connected components of a directed graph. Using data refinement techniques, we extract efficient code that performs comparable to a reference implementation in Java. Our style of formalization allows for reusing large parts of the proofs when defining variants of the algorithm. We demonstrate this by verifying an algorithm for the emptiness check of generalized Büchi automata, reusing most of the existing proofs.