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

Covering Pairs in Directed Acyclic Graphs

The Minimum Path Cover problem on directed acyclic graphs (DAGs) is a classical problem that provides a clear and simple mathematical formulation for several applications in different areas and that has an efficient algorithmic solution. In this paper, we study the computational complexity of two constrained variants of Minimum Path Cover motivated by the recent introduction of next-generation sequencing technologies in bioinformatics. The first problem (MinPCRP), given a DAG and a set of pairs of vertices, asks for a minimum cardinality set of paths "covering" all the vertices such that both vertices of each pair belong to the same path. For this problem, we show that, while it is NP-hard to compute if there exists a solution consisting of at most three paths, it is possible to decide in polynomial time whether a solution consisting of at most two paths exists. The second problem (MaxRPSP), given a DAG and a set of pairs of vertices, asks for a path containing the maximum number of the given pairs of vertices. We show its NP-hardness and also its W[1]-hardness when parametrized by the number of covered pairs. On the positive side, we give a fixed-parameter algorithm when the parameter is the maximum overlapping degree, a natural parameter in the bioinformatics applications of the problem

Maximum flow and topological structure of complex networks

The problem of sending the maximum amount of flow $q$ between two arbitrary nodes $s$ and $t$ of complex networks along links with unit capacity is studied, which is equivalent to determining the number of link-disjoint paths between $s$ and $t$. The average of $q$ over all node pairs with smaller degree $k_{\rm min}$ is $_{k_{\rm min}} \simeq c k_{\rm min}$ for large $k_{\rm min}$ with $c$ a constant implying that the statistics of $q$ is related to the degree distribution of the network. The disjoint paths between hub nodes are found to be distributed among the links belonging to the same edge-biconnected component, and $q$ can be estimated by the number of pairs of edge-biconnected links incident to the start and terminal node. The relative size of the giant edge-biconnected component of a network approximates to the coefficient $c$. The applicability of our results to real world networks is tested for the Internet at the autonomous system level.Comment: 7 pages, 4 figure

Balancing Minimum Spanning and Shortest Path Trees

This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the two trees and epsilon > 0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+epsilon times the shortest-path distance, and yet the total weight of the tree is at most 1+2/epsilon times the weight of a minimum spanning tree. This is the best tradeoff possible. The paper also describes a fast parallel implementation.Comment: conference version: ACM-SIAM Symposium on Discrete Algorithms (1993

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

Polynomial algorithms for the Maximal Pairing Problem: efficient phylogenetic targeting on arbitrary trees

Background: The Maximal Pairing Problem (MPP) is the prototype of a class of combinatorial optimization problems that are of considerable interest in bioinformatics: Given an arbitrary phylogenetic tree T and weights ωxy for the paths between any two pairs of leaves (x, y), what is the collection of edge-disjoint paths between pairs of leaves that maximizes the total weight? Special cases of the MPP for binary trees and equal weights have been described previously; algorithms to solve the general MPP are still missing, however. Results: We describe a relatively simple dynamic programming algorithm for the special case of binary trees. We then show that the general case of multifurcating trees can be treated by interleaving solutions to certain auxiliary Maximum Weighted Matching problems with an extension of this dynamic programming approach, resulting in an overall polynomial-time solution of complexity (n^4 log n) w.r.t. the number n of leaves. The source code of a C implementation can be obtained under the GNU Public License from http://www.bioinf.uni-leipzig.de/Software/Targeting. For binary trees, we furthermore discuss several constrained variants of the MPP as well as a partition function approach to the probabilistic version of the MPP. Conclusions: The algorithms introduced here make it possible to solve the MPP also for large trees with high-degree vertices. This has practical relevance in the field of comparative phylogenetics and, for example, in the context of phylogenetic targeting, i.e., data collection with resource limitations.Human Evolutionary Biolog

Diagnosis of autosomal dominant polycystic kidney disease in utero and in the young infant.

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135563/1/jum198765249.pd

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

A time study of physicians' work in a German university eye hospital to estimate unit costs.

BACKGROUND: Technical efficiency of hospital services is debated since performance has been heterogeneous. Staff time represents the main resource in patient care and its inappropriate allocation has been identified as a key factor of inefficiency. The aim of this study was to analyse the utilisation of physicians' work time stratified by staff groups, tasks and places of work. A further aim was to use these data to estimate resource use per unit of output. METHODS: A self-reporting work-sampling study was carried during 14-days at a University Eye Hospital. Staff costs of physicians per unit of output were calculated at the wards, the operating rooms and the outpatient unit. RESULTS: Forty per cent of total work time was spent in contact with the patient. Thirty per cent was spent with documentation tasks. Time spent with documentation tasks declined monotonically with increasing seniority of staff. Unit costs were 56 € per patient day at the wards, 77 € and 20 € per intervention at the operating rooms for inpatients and outpatients, respectively, and 33 € per contact at the outpatient unit. Substantial differences in resources directly dedicated to the patient were found between these locations. CONCLUSION: The presented data provide unprecedented units costs in inpatient Ophthalmology. Future research should focus on analysing factors that influence differences in time allocation, such as types of patients, organisation of care processes and composition of staff