Reconfiguration of Minimum Steiner Trees via Vertex Exchanges

In this paper, we study the problem of deciding if there is a transformation between two given minimum Steiner trees of an unweighted graph such that each transformation step respects a prescribed reconfiguration rule and results in another minimum Steiner tree of the graph. We consider two reconfiguration rules, both of which exchange a single vertex at a time, and generalize the known reconfiguration problem for shortest paths in an unweighted graph. This generalization implies that our problems under both reconfiguration rules are PSPACE-complete for bipartite graphs. We thus study the problems with respect to graph classes, and give some boundaries between the polynomial-time solvable and PSPACE-complete cases

A Generalized Matching Reconfiguration Problem

The goal in reconfiguration problems is to compute a gradual transformation between two feasible solutions of a problem such that all intermediate solutions are also feasible. In the Matching Reconfiguration Problem (MRP), proposed in a pioneering work by Ito et al. from 2008, we are given a graph G and two matchings M and M\u27, and we are asked whether there is a sequence of matchings in G starting with M and ending at M\u27, each resulting from the previous one by either adding or deleting a single edge in G, without ever going through a matching of size < min{|M|,|M\u27|}-1. Ito et al. gave a polynomial time algorithm for the problem, which uses the Edmonds-Gallai decomposition. In this paper we introduce a natural generalization of the MRP that depends on an integer parameter ? ? 1: here we are allowed to make ? changes to the current solution rather than 1 at each step of the {transformation procedure}. There is always a valid sequence of matchings transforming M to M\u27 if ? is sufficiently large, and naturally we would like to minimize ?. We first devise an optimal transformation procedure for unweighted matching with ? = 3, and then extend it to weighted matchings to achieve asymptotically optimal guarantees. The running time of these procedures is linear. We further demonstrate the applicability of this generalized problem to dynamic graph matchings. In this area, the number of changes to the maintained matching per update step (the recourse bound) is an important quality measure. Nevertheless, the worst-case recourse bounds of almost all known dynamic matching algorithms are prohibitively large, much larger than the corresponding update times. We fill in this gap via a surprisingly simple black-box reduction: Any dynamic algorithm for maintaining a ?-approximate maximum cardinality matching with update time T, for any ? ? 1, T and ? > 0, can be transformed into an algorithm for maintaining a (?(1 +?))-approximate maximum cardinality matching with update time T + O(1/?) and worst-case recourse bound O(1/?). This result generalizes for approximate maximum weight matching, where the update time and worst-case recourse bound grow from T + O(1/?) and O(1/?) to T + O(?/?) and O(?/?), respectively; ? is the graph aspect-ratio. We complement this positive result by showing that, for ? = 1+?, the worst-case recourse bound of any algorithm produced by our reduction is optimal. As a corollary, several key dynamic approximate matching algorithms - with poor worst-case recourse bounds - are strengthened to achieve near-optimal worst-case recourse bounds with no loss in update time

Reconfiguration of Subgraphs Satisfying Specific Properties

Tohoku University周暁課

Reliability and fault tolerance modelling of multiprocessor systems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Reliability evaluation by analytic modelling constitute an important issue of designing a reliable multiprocessor system. In this thesis, a model for reliability and fault tolerance analysis of the interconnection network is presented, based on graph theory. Reliability and fault tolerance are considered as deterministic and probabilistic measures of connectivity. Exact techniques for reliability evaluation fail for large multiprocessor systems because of the enormous computational resources required. Therefore, approximation techniques have to be used. Three approaches are proposed, the first by simplifying the symbolic expression of reliability; the other two by applying a hierarchical decomposition to the system. All these methods give results close to those obtained by exact techniques.Consejo Nacional de Ciencia y Tecnologia" (National Council for Science and Technology of Mexico) and "Instituto de Investigaciones Electricas" (Institute for Electrical Research

Order-Related Problems Parameterized by Width

In the main body of this thesis, we study two different order theoretic problems. The first problem, called Completion of an Ordering, asks to extend a given finite partial order to a complete linear order while respecting some weight constraints. The second problem is an order reconfiguration problem under width constraints. While the Completion of an Ordering problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, and computer memory management. Each application yields a special partial order ρ. We also relate the interval width of ρ to parameterizations for these problems that have been studied earlier in the context of these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems. In our second main result, we combine our parameterized approach with the paradigm of solution diversity. The idea of solution diversity is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. There, we show that the considered diversity version of the Completion of an Ordering problem is fixed-parameter tractable with respect to natural paramaters that capture the notion of diversity and the notion of sufficiently good solutions. We apply this algorithm in the study of the Kemeny Rank Aggregation class of problems, a well-studied class of problems lying in the intersection of order theory and social choice theory. Up to this point, we have been looking at problems where the goal is to find an optimal solution or a diverse set of good solutions. In the last part, we shift our focus from finding solutions to studying the solution space of a problem. There we consider the following order reconfiguration problem: Given a graph G together with linear orders τ and τ ′ of the vertices of G, can one transform τ into τ ′ by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most w? We show that this problem always has an affirmative answer when the input linear orders τ and τ ′ have cutwidth (pathwidth) at most w/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory. In addition to the main part of this work, we present results on two unrelated problems, namely on the Steiner Tree problem and on the Intersection Non-emptiness problem from automata theory.Doktorgradsavhandlin

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum