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Register a new userSpeeding Up Graph Algorithms via Switching Classes
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Added by
Nathan Lindzey
Publication date
2014
fields
Informatics Engineering
and research's language is
English
Authors
Nathan Lindzey
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Given a graph $G$, a vertex switch of $v in V(G)$ results in a new graph where neighbors of $v$ become nonneighbors and vice versa. This operation gives rise to an equivalence relation over the set of labeled digraphs on $n$ vertices. The equivalence class of $G$ with respect to the switching operation is commonly referred to as $G$s switching class. The algebraic and combinatorial properties of switching classes have been studied in depth; however, they have not been studied as thoroughly from an algorithmic point of view. The intent of this work is to further investigate the algorithmic properties of switching classes. In particular, we show that switching classes can be used to asymptotically speed up several super-linear unweighted graph algorithms. The current techniques for speeding up graph algorithms are all somewhat involved insofar that they employ sophisticated pre-processing, data-structures, or use word tricks on the RAM model to achieve at most a $O(log(n))$ speed up for sufficiently dense graphs. Our methods are much simpler and can result in super-polylogarithmic speedups. In particular, we achieve better bounds for diameter, transitive closure, bipartite maximum matching, and general maximum matching.
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Many algorithms use data structures that maintain properties of matrices undergoing some changes. The applications are wide-ranging and include for example matchings, shortest paths, linear programming, semi-definite programming, convex hull and volume computation. Given the wide range of applications, the exact property these data structures must maintain varies from one application to another, forcing algorithm designers to invent them from scratch or modify existing ones. Thus it is not surprising that these data structures and their proofs are usually tailor-made for their specific application and that maintaining more complicated properties results in more complicated proofs. In this paper we present a unifying framework that captures a wide range of these data structures. The simplicity of this framework allows us to give short proofs for many existing data structures regardless of how complicated the to be maintained property is. We also show how the framework can be used to speed up existing iterative algorithms, such as the simplex algorithm. More formally, consider any rational function $f(A_1,...,A_d)$ with input matrices $A_1,...,A_d$. We show that the task of maintaining $f(A_1,...,A_d)$ under updates to $A_1,...,A_d$ can be reduced to the much simpler problem of maintaining some matrix inverse $M^{-1}$ under updates to $M$. The latter is a well studied problem called dynamic matrix inverse. By applying our reduction and using known algorithms for dynamic matrix inverse we can obtain fast data structures and iterative algorithms for much more general problems.
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We study the application of a counter-diabatic driving (CD) technique to enhance the thermodynamic efficiency and power of a quantum Otto refrigerator based on a superconducting qubit coupled to two resonant circuits. Although the CD technique is originally designed to counteract non-adiabatic coherent excitations in isolated systems, we find that it also works effectively in the open system dynamics, improving the coherence-induced losses of efficiency and power. We compare the CD dynamics with its classical counterpart, and find a deviation that arises because the CD is designed to follow the energy eigenbasis of the original Hamiltonian, but the heat baths thermalize the system in a different basis. We also discuss possible experimental realizations of our model.
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Pavel Dvov{r}ak
, Duv{s}an Knop (Department ofn Applied Mathematics Charles University Prague
2016
We show that it is possible to use Bondy-Chvatal closure to design an FPT algorithm that decides whether or not it is possible to cover vertices of an input graph by at most k vertex disjoint paths in the complement of the input graph. More precisely, we show that if a graph has tree-width at most w and its complement is closed under Bondy-Chvatal closure, then it is possible to bound neighborhood diversity of the complement by a function of w only. A simpler proof where tree-depth is used instead of tree-width is also presented.
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