We consider two optimization problems related to finding dense subgraphs. The densest at-least-k-subgraph problem (DalkS) is to find an induced subgraph of highest average degree among all subgraphs with at least k vertices, and the densest at-most-k-subgraph problem (DamkS) is defined similarly. These problems are related to the well-known densest k-subgraph problem (DkS), which is to find the densest subgraph on exactly k vertices. We show that DalkS can be approximated efficiently, while DamkS is nearly as hard to approximate as the densest k-subgraph problem.
Arora, Karger, and Karpinski @cite_11 gave a PTAS for the special case @math and @math . Asahiro, Hassin, and Iwama @cite_12 showed that the problem is still @math -complete in very sparse graphs.
@cite_12 @cite_11
2140566134 2399682168 2889042315 2147668104
We consider (uniform) sparsest cut, optimal linear arrangement and the precedence constrained scheduling problem 1 |prec| SigmawjCj-So far, these three notorious NP-hard problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme (PTAS), unless NP-complete pmblems can be solved in randomized subexponential time. Furthermore, we prove that the scheduling problem is as-hard to approximate as vertex cover when the so-called fixed cost, that is present in all feasible solutions, is subtracted from the objective function. We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an @math lower bound for computing the diameter in sparse networks, which was previously known only for dense networks [, SODA 2012]. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define. Moreover, our technique allows us to show @math lower bounds for computing @math -approximations of the diameter or the radius, and for computing a @math -approximation of all eccentricities. For radius, we are unaware of any previous lower bounds. For diameter, these greatly improve upon previous lower bounds and are tight up to polylogarithmic factors [, SODA 2012], and for eccentricities the improvement is both in the lower bound and in the approximation factor [Holzer and Wattenhofer, PODC 2012]. Interestingly, our technique also allows showing an almost-linear lower bound for the verification of @math -spanners, for @math . Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap @math . This is achieved by exhibiting @math -point metric spaces of negative type whose @math distortion is @math . Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to @math when restricted to cosets of the center.