Showing papers on "Vertex (graph theory) published in 2009"
TL;DR: The geometrical arithmetic index (GA) as discussed by the authors is a topological index based on the end-vertex degrees of edges and its basic features are presented in this paper.
Abstract: Research on the topological indices based on end-vertex degrees of edges has been intensively rising recently. Randic index, one of the best-known topological indices in chemical graph theory, is belonging to this class of indices. In this paper, we introduce a novel topological index based on the end-vertex degrees of edges and its basic features are presented here. We named it as geometrical-arithmetic index (GA).
525 citations
TL;DR: This is the first calculation of complete three-loop vertex corrections within massless perturbative quantum chromodynamics and provides building blocks for many third-order cross sections.
Abstract: We compute the form factors of the photon-quark-anti-quark vertex and the effective vertex of a Higgs-boson and two gluons to three-loop order within massless perturbative quantum chromodynamics. These results provide building blocks for many third-order cross sections. Furthermore, this is the first calculation of complete three-loop vertex corrections.
306 citations
TL;DR: This article presents a worst-case O(n) 3-time algorithm for the problem when the two trees have size n, and proves the optimality of the algorithm among the family of decomposition strategy algorithms—which also includes the previous fastest algorithms—by tightening the known lower bound.
Abstract: The edit distance between two ordered rooted trees with vertex labels is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. In this article, we present a worst-case O(n3)-time algorithm for the problem when the two trees have size n, improving the previous best O(n3 log n)-time algorithm. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems, together with a deeper understanding of the previous algorithms for the problem. We prove the optimality of our algorithm among the family of decomposition strategy algorithms—which also includes the previous fastest algorithms—by tightening the known lower bound of Ω(n2 log2n) to Ω(n3), matching our algorithm's running time. Furthermore, we obtain matching upper and lower bounds for decomposition strategy algorithms of Θ(nm2 (1 + log n/m)) when the two trees have sizes m and n and m
264 citations
TL;DR: In this article, it was shown that for a sequence of such multigraphs with the number of edges, the probability that the multigraph is simple stays away from 0 if and only if.
Abstract: Consider a random multigraph G* with given vertex degrees d1,…,dn, constructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges , the probability that the multigraph is simple stays away from 0 if and only if . This was previously known only under extra assumptions on the maximum degree maxidi. We also give an asymptotic formula for this probability, extending previous results by several authors.
229 citations
TL;DR: The result indicates that dynamical chiral symmetry breaking enhances spin-orbit splitting in the meson spectrum.
Abstract: An exact form is presented for the axial-vector Bethe-Salpeter equation, which is valid when the quark-gluon vertex is fully dressed. A Ward-Takahashi identity for the Bethe-Salpeter kernel is derived therefrom and solved for a class of dressed quark-gluon-vertex models. The solution provides a symmetry-preserving closed system of gap and vertex equations. The analysis can be extended to the vector equation. This enables a comparison between the responses of pseudoscalar and scalar meson masses to nonperturbatively dressing the quark-gluon vertex. The result indicates that dynamical chiral symmetry breaking enhances spin-orbit splitting in the meson spectrum.
225 citations
01 Jan 2009
TL;DR: In this article, an on-the-fly kinetic mechanism reduction scheme, referred to as dynamic adaptive chemistry (DAC), was developed to incorporate detailed chemical kinetics into reactive flow computations with high efficiency and accuracy.
Abstract: An on-the-fly kinetic mechanism reduction scheme, referred to as dynamic adaptive chemistry (DAC), has been developed to incorporate detailed chemical kinetics into reactive flow computations with high efficiency and accuracy. The procedure entails reducing a detailed mechanism to locally and instantaneously accurate sub-mechanisms at each hydrodynamic time step of the calculation, and consequently no a priori information regarding simulation conditions is needed. The reduction utilizes an extended version of the directed relation graph (DRG) method in which the edges are weighted by a value that measures the dependence of the tail species (vertex) on the head species. An R-value is then defined at each vertex as the maximum of the products of these weights along all paths to that vertex from an initiating species. Active species are identified by their R-values exceeding a threshold value, eR, using a modified breadth-first search (BFS) that starts from a pre-defined set of initiating species. Chemical kinetics equations are then formulated with respect to the active species, with the inactive species considered only as third body collision partners. The DAC method is implemented into CHEMKIN and tested by simulating homogeneous charge compression ignition (HCCI) combustion using detailed and pre-reduced n-heptane mechanisms (578 species and 178 species, respectively) as the full mechanisms. The DAC scheme reproduces with high accuracy the pressure curves and species mass fractions obtained using the full mechanisms. The on-the-fly mechanism reduction scheme introduces minimal computational overhead and achieves more than 30-fold time reduction in calculations using the 578-species mechanism.
178 citations
TL;DR: In this article, a refined version of the topological vertex (arXiv:hep-th/0502061) is presented as a building block of Nekrasov's partition function with two equivariant parameters.
Abstract: It has been argued that Nekrasov's partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi–Yau spaces. We show that a refined version of the topological vertex we previously proposed (arXiv:hep-th/0502061) is a building block of Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal, Kozcaz and Vafa (arXiv:hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on ℂ2. We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang–Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions.
160 citations
TL;DR: This paper establishes properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing or by annihilating each other, and establishes results for random regular graphs.
Abstract: We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the analysis for random regular graphs. The cover time of a random walk on a random $r$-regular graph was studied in [C. Cooper and A. Frieze, SIAM J. Discrete Math., 18 (2005), pp. 728-740], where it was shown with high probability (whp) that for $r\geq3$ the cover time is asymptotic to $\theta_r n\ln n$, where $\theta_r=(r-1)/(r-2)$. In this paper we prove the following (whp) results, arising from the study of multiple random walks on a random regular graph $G$. For $k$ independent walks on $G$, the cover time $C_G(k)$ is asymptotic to $C_G/k$, where $C_G$ is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is $\theta_r n/\binom{k}{2}$. If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for $k$ walks to broadcast a single piece of information to each other is asymptotic to $\frac{2\ln k}{k}\theta_r n$ as $k,n\rightarrow\infty$. We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing or by annihilating each other. For example, the expected extinction time of $k$ explosive particles ($k$ even) tends to $(2\ln2)\theta_r n$ as $k\rightarrow\infty$. The case of $n$ coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. The expected time for a unique opinion to emerge is the same as the expected time for all the particles to coalesce, which is asymptotic to $2\theta_r n$. Combining results from the predator-prey and multiple random walk models allows us to compare expected detection times of all prey in the following scenarios: Both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, and the predators move randomly and the prey stays fixed. In all cases, with $k$ predators and $\ell$ prey the expected detection time is $\theta_r H_{\ell}n/k$, where $H_{\ell}$ is the $\ell$th harmonic number.
142 citations
01 Aug 2009
TL;DR: This paper presents a minimum spanning tree algorithm on Nvidia GPUs under CUDA, as a recursive formulation of Borůvka's approach for undirected graphs, implemented using scalable primitives such as scan, segmented scan and split.
Abstract: Graphics Processor Units are used for many general purpose processing due to high compute power available on them. Regular, data-parallel algorithms map well to the SIMD architecture of current GPU. Irregular algorithms on discrete structures like graphs are harder to map to them. Efficient data-mapping primitives can play crucial role in mapping such algorithms onto the GPU. In this paper, we present a minimum spanning tree algorithm on Nvidia GPUs under CUDA, as a recursive formulation of Boruvka's approach for undirected graphs. We implement it using scalable primitives such as scan, segmented scan and split. The irregular steps of supervertex formation and recursive graph construction are mapped to primitives like split to categories involving vertex ids and edge weights. We obtain 30 to 50 times speedup over the CPU implementation on most graphs and 3 to 10 times speedup over our previous GPU implementation. We construct the minimum spanning tree on a 5 million node and 30 million edge graph in under 1 second on one quarter of the Tesla S1070 GPU.
126 citations
TL;DR: In this paper, it was shown that a simple vertex operator algebra generated by two weight 2 vectors is either a vertex algebra associated to an irreducible highest weight W(2, 2)- module or a tensor product of two simple Virasoro vertex operator algebras.
Abstract: In this paper the W-algebra W(2, 2) and its representation theory are studied. It is proved that a simple vertex operator algebra generated by two weight 2 vectors is either a vertex operator algebra associated to an irreducible highest weight W(2, 2)- module or a tensor product of two simple Virasoro vertex operator algebras. Furthermore, we show that any rational, C
2-cofinite and simple vertex operator algebra whose weight 1 subspace is zero, weight 2 subspace is 2-dimensional and with central charge c = 1 is isomorphic to $${L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}$$
.
120 citations
01 Oct 2009
TL;DR: In this paper, the authors used vertex decomposability from geometric combinatorics to show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay.
Abstract: Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.
TL;DR: In this paper, the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs was studied and a sufficient condition for non-uniqueness was given. But the uniqueness criterion was not defined.
Abstract: We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex. A sufficient condition for non-uniqueness is also presented. Furthermore, we give a lower bound on the bottom of the spectrum of the discrete Laplacian and use this bound to give a condition ensuring that the essential spectrum of the Laplacian is empty.
TL;DR: It is proved that κ 2 ( S n ) = 6 ( n − 3 ) for n ≥ 4 , where S n is the n -dimensional star graph.
TL;DR: In this paper, the independence complex of a bipartite graph with edge ideal I(G) was shown to be vertex decomposable, and the Castelnuovo-Mumford regularity of R/I (G) can be determined from the invariants of G.
Abstract: Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen–Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo–Mumford regularity of R/I(G) can be determined from the invariants of G.
TL;DR: In this paper, it was shown that the double Grothendieck polynomials have vertex decompositions of simplicial com- plexes, whose ideals are also known as (one-sided) ladder determinantal ideals.
Abstract: We relate a classic algebro-geometric degeneration technique, dating at least to (Hodge 1941), to the notion of vertex decompositions of simplicial com- plexes. The good case is when the degeneration is reduced, and we call this a geometric vertex decomposition. Our main example in this paper is the family of vexillary matrix Schubert vari- eties, whose ideals are also known as (one-sided) ladder determinantal ideals. Us- ing a diagonal term order to specify the (Grobner) degeneration, we show that these have geometric vertex decompositions into simpler varieties of the same type. From this, together with the combinatorics of the pipe dreams of (Fomin- Kirillov 1996), we derive a new formula for the numerators of their multigraded Hilbert series, the double Grothendieck polynomials, in terms of flagged set-valued tableaux. This unifies work of (Wachs 1985) on flagged tableaux, and (Buch 2002) on set-valued tableaux, giving geometric meaning to both. This work focuses on diagonal term orders, giving results complementary to those of (Knutson-Miller 2004), where it was shown that the generating minors form a Grobner basis for any antidiagonal term order and any matrix Schubert va- riety. We show here that under a diagonal term order, the only matrix Schubert varieties for which these minors form Grobner bases are the vexillary ones, reach- ing an end toward which the ladder determinantal literature had been building. CONTENTS
01 Jan 2009
TL;DR: The setting of metapopulation models, which consist of graphs, with systems of differential equations at each vertex, are presented, and some of the mathematical techniques that can be used to study them are presented.
Abstract: Metapopulation models consist of graphs, with systems of differential equations at each vertex This modeling paradigm is appropriate for the description of the spatio-temporal spread of infectious diseases In this document, I present the setting of these models, and some of the mathematical techniques that can be used to study them I conclude with a brief review of some models using this approach
TL;DR: In this paper, a general method for constructing logarithmic modules in vertex operator algebra theory is presented, and explicit vertex operator construction of certain indecomposable modules for the triplet vertex algebra \({\mathcal{W}(p)}\) and for other subalgebras of lattice vertex algebra and their N = 1 super extensions.
Abstract: A general method for constructing logarithmic modules in vertex operator algebra theory is presented. By utilizing this approach, we give explicit vertex operator construction of certain indecomposable and logarithmic modules for the triplet vertex algebra \({\mathcal{W}(p)}\) and for other subalgebras of lattice vertex algebras and their N = 1 super extensions. We analyze in detail indecomposable modules obtained in this way, giving further evidence for the conjectural equivalence between the category of \({\mathcal{W}(p)}\)-modules and the category of modules for the restricted quantum group \({\overline{\mathcal{U}}_q(sl_2)}\) , q = eπi/p. We also construct logarithmic representations for a certain affine vertex operator algebra at admissible level realized in Adamovic (J. Pure Appl. Algebra 196:119–134, 2005). In this way we prove the existence of the logarithmic representations predicted in Gaberdiel (Int. J. Modern Phys. A 18, 4593–4638, 2003). Our approach enlightens related logarithmic intertwining operators among indecomposable modules, which we also construct in the paper.
TL;DR: The unitary Cayley graph associated to an arbitrary finite ring is studied, determining precisely its diameter, girth, eigenvalues, vertex and edge connectivity, andvertex and edge chromatic number.
Abstract: We study the unitary Cayley graph associated to an arbitrary finite ring, determining precisely its diameter, girth, eigenvalues, vertex and edge connectivity, and vertex and edge chromatic number. We also compute its automorphism group, settling a question of Klotz and Sander. In addition, we classify all planar graphs and perfect graphs within this class.
TL;DR: This paper presents a (10+ε)-approximation algorithm to compute minimum-weight connected dominating set (MWCDS) in unit disk graph, which computes a dominating set which has approximation ratio 6+ε (ε is an arbitrary positive number).
Abstract: This paper presents a (10+e)-approximation algorithm to compute minimum-weight connected dominating set (MWCDS) in unit disk graph. MWCDS is to select a vertex subset with minimum weight for a given unit disk graph, such that each vertex of the graph is contained in this subset or has a neighbor in this subset. Besides, the subgraph induced by this vertex subset is connected. Our algorithm is composed of two phases: the first phase computes a dominating set, which has approximation ratio 6+e (e is an arbitrary positive number), while the second phase connects the dominating sets computed in the first phase, which has approximation ratio 4.
TL;DR: In this paper, the MHV vertex expansion for all tree amplitudes of = 4 SYM theory is shown to fall off as 1/zk, or faster, for large z under a shift acting on all external momenta.
Abstract: We prove the MHV vertex expansion for all tree amplitudes of = 4 SYM theory. The proof uses a shift acting on all external momenta, and we show that every NkMHV tree amplitude falls off as 1/zk, or faster, for large z under this shift. The MHV vertex expansion allows us to derive compact and efficient generating functions for all NkMHV tree amplitudes of the theory. We also derive an improved form of the anti-NMHV generating function. The proof leads to a curious set of sum rules for the diagrams of the MHV vertex expansion.
01 Jan 2009
TL;DR: The rule, or update rule, of a cellular automaton describes how any given state is transformed into its successor state, and a rule table, which defines a local neighborhood mapping, or equivalently as a global update mapping, is described.
Abstract: Cellular automata Cellular automata are dynamical systems that are discrete in space, time, and value. A state of a cellular automaton is a spatial array of discrete cells, each containing a value chosen from a finite alphabet. The state space for a cellular automaton is the set of all such configurations. Alphabet of a cellular automaton The alphabet of a cellular automaton is the set of symbols or values that can appear in each cell. The alphabet contains a distinguished symbol called the null or quiescent symbol, usually indicated by 0, which satisfies the condition of an additive identity: 0C x D x. Cellular automata rule The rule, or update rule of a cellular automaton describes how any given state is transformed into its successor state. The update rule of a cellular automaton is described by a rule table, which defines a local neighborhood mapping, or equivalently as a global update mapping. Additive cellular automata An additive cellular automaton is a cellular automaton whose update rule satisfies the condition that its action on the sum of two states is equal to the sum of its actions on the two states separately. Linear cellular automata A linear cellular automaton is a cellular automaton whose update rule satisfies the condition that its action on the sum of two states separately equals action on the sum of the two states plus its action on the state in which all cells contain the quiescent symbol. Note that some researchers reverse the definitions of additivity and linearity. Neighborhood The neighborhood of a given cell is the set of cells that contribute to the update of value in that cell under the specified update rule. Rule table The rule table of a cellular automaton is a listing of all neighborhoods together with the symbol that each neighborhood maps to under the local update rule. Local maps of a cellular automaton The local mapping for a cellular automaton is a map from the set of all neighborhoods of a cell to the automaton alphabet. State transition diagram The state transition diagram (STD) of a cellular automaton is a directed graph with each vertex labeled by a possible state and an edge directed from a vertex x to a vertex y if and only if the state labeling vertex xmaps to the state labeling vertex y under application of the automaton update rule. Transient states A transient state of a cellular automaton is a state that can at most appear only once in the evolution of the automaton rule. Cyclic states A cyclic state of a cellular automaton is a state lying on a cycle of the automaton update rule, hence it is periodically revisited in the evolution of the rule. Basins of attraction The basins of attraction of a cellular automaton are the equivalences classes of cyclic states together with their associated transient states, with two states being equivalent if they lie on the same cycle of the update rule. Predecessor state A state x is the predecessor of a state y if and only if xmaps to y under application of the cellular automaton update rule.More specifically, a state x is an nth order predecessor of a state y if it maps to y under n applications of the update rule. Garden-of-Eden A Garden-of-Eden state is a state that has no predecessor. It can be present only as an initial condition. Surjectivity A mapping is surjective (or onto) if every state has a predecessor. Injectivity A mapping is injective (one-to-one) if every state in its domain maps to a unique state in its range. That is, if states x and y both map to a state z then x D y. Reversibility A mapping X is reversible if and only if a second mapping X 1 exists such that if X(x) D y then X 1(y) D x. For finite state spaces reversibility and injectivity are identical.
TL;DR: It is proved the asymptotically sharp upper bound D^'(G)@?14nd(n-d)^2+O(n^7^/^2) for graphs of order n and diameter d is correct.
TL;DR: This work examines the computational complexity of calculating two prominent power indices, the Banzhaf index and the Shapley-Shubik index, in a particular multiagent domain, a threshold network flow game, and shows that for some restricted network flow domains there exists a polynomial algorithm for calculating agents’Banzhaf power indices.
Abstract: Preference aggregation is used in a variety of multiagent applications, and as a result, voting theory has become an important topic in multiagent system research. However, power indices (which reflect how much "real power" a voter has in a weighted voting system) have received relatively little attention, although they have long been studied in political science and economics. We consider a particular multiagent domain, a threshold network flow game. Agents control the edges of a graph; a coalition wins if it can send a flow that exceeds a given threshold from a source vertex to a target vertex. The relative power of each edge/agent reflects its significance in enabling such a flow, and in real-world networks could be used, for example, to allocate resources for maintaining parts of the network. We examine the computational complexity of calculating two prominent power indices, the Banzhaf index and the Shapley-Shubik index, in this network flow domain. We also consider the complexity of calculating the core in this domain. The core can be used to allocate, in a stable manner, the gains of the coalition that is established. We show that calculating the Shapley-Shubik index in this network flow domain is NP-hard, and that calculating the Banzhaf index is #P-complete. Despite these negative results, we show that for some restricted network flow domains there exists a polynomial algorithm for calculating agents' Banzhaf power indices. We also show that computing the core in this game can be performed in polynomial time.
TL;DR: In this article, the Schrodinger operator was used to approximate vertex couplings of quantum graphs using families of thin branched manifolds, and it was shown that if a Neumann-type Laplacian on such manifolds is amended by suitable potentials, the resulting Schrodger operators can approximate non-trivial vertex Couplings.
Abstract: We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann-type Laplacian on such manifolds is amended by suitable potentials, the resulting Schrodinger operators can approximate non-trivial vertex couplings. The latter include not only the δ-couplings but also those with wavefunctions discontinuous at the vertex. We work out the example of the symmetric δ'-couplings and make a conjecture that the same method can be applied to all couplings invariant with respect to the time reversal. We conclude with a result that certain vertex couplings cannot be approximated by a pure Laplacian.
TL;DR: In this paper, it was shown that the asymptotic analysis of the new vertex amplitude implies that the vertex has precisely the same structure that was indicated as the key necessary condition for overcoming the difficulty in the first article of this series.
Abstract: In the first article of this series, we pointed out a difficulty in the attempt to derive the low-energy behavior of the graviton two-point function, from the loop-quantum-gravity dynamics defined by the Barrett-Crane vertex amplitude. Here we show that this difficulty disappears when using the corrected vertex amplitude recently introduced in the literature. In particular, we show that the asymptotic analysis of the new vertex amplitude recently performed by Barrett, Fairbairn and others, implies that the vertex has precisely the asymptotic structure that, in the second article of this series, was indicated as the key necessary condition for overcoming the difficulty.
TL;DR: A fast new algorithm is proposed for numerical computation of (approximate) D-optimal designs that extends the well-known vertex direction method and the multiplicative algorithm, and shares their simplicity and monotonic convergence properties.
Abstract: A fast new algorithm is proposed for numerical computation of (approximate) D-optimal designs. This "cocktail algorithm" extends the well-known vertex direction method (VDM; Fedorov 1972) and the multiplicative algorithm (Silvey, Titterington and Torsney, 1978), and shares their simplicity and monotonic convergence properties. Numerical examples show that the cocktail algorithm can lead to dramatically improved speed, sometimes by orders of magnitude, relative to either the multiplicative algorithm or the vertex exchange method (a variant of VDM). Key to the improved speed is a new nearest neighbor exchange strategy, which acts locally and complements the global effect of the multiplicative algorithm. Possible extensions to related problems such as nonparametric maximum likelihood estimation are mentioned.
TL;DR: For various values of t, NP-completeness and approximability results (both upper and lower bounds) and FPT algorithms for problems concerned with finding the minimum size of a t-total vertex cover, t- total edge cover and connected vertex cover are presented, in particular improving on a previous FPT algorithm for the latter problem.
TL;DR: It is proved that tvs(G)@[email protected]?n/@ [email protected]?+1 for each graph of order n and with minimum degree @d>0.
01 Oct 2009
TL;DR: The presented algorithm for finding maximum weight matchings in bipartite graphs with nonnegative integer weights works in [email protected]?(Wn^@w) time, where @w is the matrix multiplication exponent, and W is the highest edge weight in the graph.
Abstract: In this paper we consider the problem of finding maximum weight matchings in bipartite graphs with nonnegative integer weights. The presented algorithm for this problem works in [email protected]?(Wn^@w) time, where @w is the matrix multiplication exponent, and W is the highest edge weight in the graph. As a consequence of this result we obtain [email protected]?(Wn^@w) time algorithms for computing: minimum weight bipartite vertex cover, single source shortest paths and minimum weight vertex disjoint s-t paths. All of the presented algorithms are randomized and with small probability can return suboptimal solutions.
TL;DR: An O(logn/loglogn) upper bound on price of stability for the single sink case is presented, i.e., t"i=t for all i.i in the Shapley network design game on undirected networks.