Abstract. Betweenness is a centrality measure based on shortest paths, widely used in complex network analysis. It is computationally-expensive to exactly determine betweenness; currently the fastest-known algorithm by Brandes requires O(nm) time for unweighted graphs and O(nm + n 2 log n) time for weighted graphs, where n is the number of vertices and m is the number of edges in the network. These are also the worstcase time bounds for computing the betweenness score of a single vertex. In this paper, we present a novel approximation algorithm for computing betweenness centrality of a given vertex, for both weighted and unweighted graphs. Our approximation algorithm is based on an adaptive sampling technique that significantly reduces the number of single-source shortest path computations for vertices with high centrality. We conduct an extensive experimental study on real-world graph instances, and observe that our random sampling algorithm gives very good betweenness approximations for biological networks, road networks and web crawls.

As has often been the case with NP-completeness proofs, PPAD-completeness proofs will be eventually refined to cover simpler and more realistic looking classes of games. And then researchers will strive to identify even simpler classes." -Papadimitriou (chapter 2 of [37])In a landmark paper [39], Papadimitriou introduced a number of syntactic subclasses of TFNP based on proof styles that (unlike TFNP) admit complete problems. A recent series of results [12,19,6,7,8,9] has shown that finding Nash equilibria is complete for PPAD, a particularly notable subclass of TFNP. A major goal of this work is to expand the universe of known PPAD-complete problems. We resolve the computational complexity of a number of outstanding open problems with practical applications.Here is the list of problems we show to be PPAD-complete, along with the domains of Theory. In fact, we show that no fully polynomial-time approximation schemes exist (unless PPAD is in FP).This paper is entirely a series of reductions that build in nontrivial ways on the framework established in previous work. In the course of deriving these reductions, we created two new concepts -preference games and personalized equilibria. The entire set of new reductions can be presented as a lattice with the above problems sandwiched between preference games (at the "easy" end) and personalized equilibria (at the "hard" end). Our completeness results extend to natural approximate versions of most of these problems. On a technical note, we wish to highlight our novel "continuous-to-discrete" reduction from exact personalized equilibria to approximate personalized equilibria using a linear program augmented with an exponential number of "min" constraints of a specific form. In addition to enhancing our repertoire of PPAD-complete problems, we expect the concepts and techniques in this paper to find future use in algorithmic game theory.Intuitively, the notion of stability implies the absence of oscillations over time and encompasses the concepts of fixed points and equilibria. Stability is important in a variety of fields ranging from the practical -the Internet -to the theoretical -combinatorics and game theory. For important practical systems (e.g. Internet), the existence and computational feasibility of stable operating modes is of profound real-world significance. On the more abstract front, the study of stable solutions to combinatorial problems has a distinguished tradition dating back to, at least, the Gale-Shapley algorithm [17]. It is often the case, as with Nash's celebrated theorem [36], that fractional stable points are guaranteed to exist even when integral points don't. In this paper, we focus on fractional stability and resolve the computational complexity of a set of eight problems with applications to a variety of different domains. Six of these are pre-existing problems. Below we provide elaborate motivation for two of the pre-existing problems -Fractional Stable Paths Problem (FSPP) and Core of Balanced Games. The remaining four are: Scarf 's lemma, a...

Abstract"As has often been the case with NP-completeness proofs, PPAD-completeness proofs will be eventually refined to cover simpler and more realistic looking classes of games. And then researchers will strive to identify even simpler classes." -Papadimitriou (chapter 2 of [37])In a landmark paper [39], Papadimitriou introduced a number of syntactic subclasses of TFNP based on proof styles that (unlike TFNP) admit complete problems. A recent series of results [12,19,6,7,8,9] has shown that finding Nash equilibria is complete for PPAD, a particularly notable subclass of TFNP. A major goal of this work is to expand the universe of known PPAD-complete problems. We resolve the computational complexity of a number of outstanding open problems with practical applications.Here is the list of problems we show to be PPAD-complete, This paper is entirely a series of reductions that build in nontrivial ways on the framework established in previous work. In the course of deriving these reductions, we created two new concepts -preference games and personalized equilibria. The entire set of new reductions can be presented as a lattice with the above problems sandwiched between preference games (at the "easy" end) and personalized equilibria (at the "hard" end). Our completeness results extend to natural approximate versions of most of these problems. On a technical note, we wish to highlight our novel "continuous-to-discrete" reduction from exact personalized equilibria to approximate personalized equilibria using a linear program augmented with an exponential number of "min" constraints of a specific form. In addition to enhancing our repertoire of PPAD-complete problems, we expect the concepts and techniques in this paper to find future use in algorithmic game theory.

We prove that the directed treewidth, DAG-width and Kelly-width of a digraph are bounded above by its circumference plus one.

Several problems that are NP-hard on general graphs are efficiently solvable on graphs with bounded treewidth. Efforts have been made to generalize treewidth and the related notion of pathwidth to digraphs. Directed treewidth, DAG-width and Kelly-width are some such notions which generalize treewidth, whereas directed pathwidth generalizes pathwidth. Each of these digraph width measures have an associated decomposition structure.In this paper, we present approximation algorithms for all these digraph width parameters. In particular, we give an O( √ log n)-approximation algorithm for directed treewidth, and an O(log 3/2 n)-approximation algorithm for directed pathwidth, DAG-width and Kelly-width. Our algorithms construct the corresponding decompositions whose widths are within the above mentioned approximation factors.