Networks of Complements

We consider a network of sellers, each selling a single product, where the graph structure represents pair-wise complementarities between products. We study how the network structure affects revenue and social welfare of equilibria of the pricing game between the sellers. We prove positive and negative results, both of "Price of Anarchy" and of "Price of Stability" type, for special families of graphs (paths, cycles) as well as more general ones (trees, graphs). We describe best-reply dynamics that converge to non-trivial equilibrium in several families of graphs, and we use these dynamics to prove the existence of approximately-efficient equilibria.Comment: An extended abstract will appear in ICALP 201

Arboricity, h-Index, and Dynamic Algorithms

In this paper we present a modification of a technique by Chiba and Nishizeki [Chiba and Nishizeki: Arboricity and Subgraph Listing Algorithms, SIAM J. Comput. 14(1), pp. 210--223 (1985)]. Based on it, we design a data structure suitable for dynamic graph algorithms. We employ the data structure to formulate new algorithms for several problems, including counting subgraphs of four vertices, recognition of diamond-free graphs, cop-win graphs and strongly chordal graphs, among others. We improve the time complexity for graphs with low arboricity or h-index.Comment: 19 pages, no figure

Improved Dynamic Graph Coloring

This paper studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within n^{1-epsilon} for any epsilon > 0, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring, or alternatively, study restricted families of graphs. Towards understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm for C-coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. Optimizing the number of recolorings, sometimes referred to as the recourse bound, is important for various practical applications. In WADS\u2717, Barba et al. devised two complementary algorithms: For any beta > 0, the first (respectively, second) maintains an O(C beta n^{1/beta}) (resp., O(C beta))-coloring while recoloring O(beta) (resp., O(beta n^{1/beta})) vertices per update. Barba et al. also showed that the second trade-off appears to exhibit the right behavior, at least for beta = O(1): Any algorithm that maintains a c-coloring of an n-vertex dynamic forest must recolor Omega(n^{2/(c(c-1))}) vertices per update, for any constant c >= 2. Our contribution is two-fold: - We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: For any beta > 0, we get a O~(C/(beta)log^2 n)-coloring with O(beta) recolorings per update, where the O~ notation supresses polyloglog(n) factors. In particular, for beta = O(1) we get constant recolorings with polylog(n) colors; not only is this an exponential improvement over the previous bound, but it also unveils a rather surprising phenomenon: The trade-off between the number of colors and recolorings is highly non-symmetric. - For uniformly sparse graphs, we use low out-degree orientations to strengthen the above result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest

A Simple Greedy Algorithm for Dynamic Graph Orientation

Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between out-degree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in O(log n) flips) for general arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case O(1) and O(sqrt(log n)) flips nearly matching degree bounds to their respective amortized solutions