Collapsing Superstring Conjecture

In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits 2 11/23-approximation in polynomial time (Mucha, SODA\u2713). While this algorithm and its analysis are technically involved, the 30 years old Greedy Conjecture claims that the trivial and efficient Greedy Algorithm gives a 2-approximation for SCS. We develop a graph-theoretic framework for studying approximation algorithms for SCS. The framework is reminiscent of the classical 2-approximation for Traveling Salesman: take two copies of an optimal solution, apply a trivial edge-collapsing procedure, and get an approximate solution. In this framework, we observe two surprising properties of SCS solutions, and we conjecture that they hold for all input instances. The first conjecture, that we call Collapsing Superstring conjecture, claims that there is an elementary way to transform any solution repeated twice into the same graph G. This conjecture would give an elementary 2-approximate algorithm for SCS. The second conjecture claims that not only the resulting graph G is the same for all solutions, but that G can be computed by an elementary greedy procedure called Greedy Hierarchical Algorithm. While the second conjecture clearly implies the first one, perhaps surprisingly we prove their equivalence. We support these equivalent conjectures by giving a proof for the special case where all input strings have length at most 3 (which until recently had been the only case where the Greedy Conjecture was proven). We also tested our conjectures on millions of instances of SCS. We prove that the standard Greedy Conjecture implies Greedy Hierarchical Conjecture, while the latter is sufficient for an efficient greedy 2-approximate approximation of SCS. Except for its (conjectured) good approximation ratio, the Greedy Hierarchical Algorithm provably finds a 3.5-approximation, and finds exact solutions for the special cases where we know polynomial time (not greedy) exact algorithms: (1) when the input strings form a spectrum of a string (2) when all input strings have length at most 2

Fixed-Parameter Algorithms For Protein Similarity Search Under mRNA Structure Constraints

International audienceIn the context of protein engineering, we consider the problem of computing an mRNA sequence of maximal codon-wise similarity to a given mRNA (and consequently, to a given protein) that additionally satisfies some secondary structure constraints, the so-called mRNA Structure Optimization (MRSO) problem. Since MRSO is known to be APX-hard, Bongartz [10] suggested to attack the problem using the approach of parameterized complexity. In this paper we propose three fixed-parameter algorithms that apply for several interesting parameters of MRSO. We believe these algorithms to be relevant for practical applications today, as well as for possible future applications. Furthermore, our results extend the known tractability borderline of MRSO, and provide new research horizons for further improvements of this sort

Complexity of the List Homomorphism Problem in Hereditary Graph Classes

A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). For a fixed graph H, in the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is equipped with a list L(v) ? V(H). We ask if there exists a homomorphism f from G to H, in which f(v) ? L(v) for every v ? V(G). Feder, Hell, and Huang [JGT 2003] proved that LHom(H) is polynomial time-solvable if H is a so-called bi-arc-graph, and NP-complete otherwise. We are interested in the complexity of the LHom(H) problem in F-free graphs, i.e., graphs excluding a copy of some fixed graph F as an induced subgraph. It is known that if F is connected and is not a path nor a subdivided claw, then for every non-bi-arc graph the LHom(H) problem is NP-complete and cannot be solved in subexponential time, unless the ETH fails. We consider the remaining cases for connected graphs F. If F is a path, we exhibit a full dichotomy. We define a class called predacious graphs and show that if H is not predacious, then for every fixed t the LHom(H) problem can be solved in quasi-polynomial time in P_t-free graphs. On the other hand, if H is predacious, then there exists t, such that the existence of a subexponential-time algorithm for LHom(H) in P_t-free graphs would violate the ETH. If F is a subdivided claw, we show a full dichotomy in two important cases: for H being irreflexive (i.e., with no loops), and for H being reflexive (i.e., where every vertex has a loop). Unless the ETH fails, for irreflexive H the LHom(H) problem can be solved in subexponential time in graphs excluding a fixed subdivided claw if and only if H is non-predacious and triangle-free. On the other hand, if H is reflexive, then LHom(H) cannot be solved in subexponential time whenever H is not a bi-arc graph

Routing in packet switched computer communication networks

This thesis concerns the optimization of the routing path in packet-switched computer-communication networks. Computer-communication networks over the past decade are outlined. A glossary of some of the terms used throughout this thesis are introduced. A brief description follows of the advantages of packet switching over the more conventional circuit-switched scheme for information transfer. The important design variables that a network planner is faced with in the design of these networks are discussed. A general design problem is stated and then decomposed into simpler subproblems one of which is the link-capacity assignment problem, which is briefly discussed. The route-assignment problem is identified as being of particular importance and is specified. A network model is introduced and relationships between performance measures, input parameters and constraints that appear in the general design problem are discussed. The routing problem is the formulated and a heuristic routing procedure is suggested as a sub-optimum solution to the problem. Basic routing methods are discussed. The principles of datagram and virtual circuit techniques are explained with reference to the routing of packets throughout the network. The directory routing technique with alternate routing is identified as being a specific requirement and the operation of this technique is explained in more detail. Two basic algorithms are introduced. The first which determines the shortest, second shortest, third shortest, etc., paths between all pairs of nodes in a network. The second which determines from all the paths in the first algorithm, the best alternative paths between all pairs of nodes in a network. A heuristic routing algorithm for establishing routing tables at each of the individual nodes in a packet switched data network is presented. Among the properties of a desirable routing algorithm is that the paths established between all node pairs are such that the average packet delay from source to destination node is minimal. The heuristic-routing algorithm proposed is to-be implemented on a newly proposed SAPONET packet-switching network, with special emphasis on the minimization of the average packet delay of the network. Results are presented and discussed for different combinations of the primary, secondary, tertiary and fourth alternative paths obtained. Finally, results are summarized and areas for further work identified

Analysis of Free Browser-based Accessibility Tools: WCAG 2.1 Evaluation of Mississippi Gulf Coast Public Library Websites

This webometrics study compared free browser-based accessibility tools and determine the WCAG2.1 compliance levels of Mississippi Gulf Coast public library websites based on homepage analysis through free browser-based accessibility tools—ARC Toolkit, Lighthouse, Accessibility Insights for the Web, and Axe Accessibility

Clustering with minimum spanning trees: How good can it be?

Minimum spanning trees (MSTs) provide a convenient representation of datasets in numerous pattern recognition activities. Moreover, they are relatively fast to compute. In this paper, we quantify the extent to which they can be meaningful in data clustering tasks. By identifying the upper bounds for the agreement between the best (oracle) algorithm and the expert labels from a large battery of benchmark data, we discover that MST methods can overall be very competitive. Next, instead of proposing yet another algorithm that performs well on a limited set of examples, we review, study, extend, and generalise existing, the state-of-the-art MST-based partitioning schemes, which leads to a few new and interesting approaches. It turns out that the Genie method and the information-theoretic approaches often outperform the non-MST algorithms such as k-means, Gaussian mixtures, spectral clustering, BIRCH, and classical hierarchical agglomerative procedures