Random graph generation is an important tool for studying large complex networks. Despite abundance of random graph models, constructing models with application-driven constraints is poorly understood. In order to advance state-of-the-art in this are
a, we focus on random graphs without short cycles as a stylized family of graphs, and propose the RandGraph algorithm for randomly generating them. For any constant k, when m=O(n^{1+1/[2k(k+3)]}), RandGraph generates an asymptotically uniform random graph with n vertices, m edges, and no cycle of length at most k using O(n^2m) operations. We also characterize the approximation error for finite values of n. To the best of our knowledge, this is the first polynomial-time algorithm for the problem. RandGraph works by sequentially adding $m$ edges to an empty graph with n vertices. Recently, such sequential algorithms have been successful for random sampling problems. Our main contributions to this line of research includes introducing a new approach for sequentially approximating edge-specific probabilities at each step of the algorithm, and providing a new method for analyzing such algorithms.
We study the evolution of cooperation in populations where individuals play prisoners dilemma on a network. Every node of the network corresponds on an individual choosing whether to cooperate or defect in a repeated game. The players revise their ac
tions by imitating those neighbors who have higher payoffs. We show that when the interactions take place on graphs with large girth, cooperation is more likely to emerge. On the flip side, in graphs with many cycles of length 3 and 4, defection spreads more rapidly. One of the key ideas of our analysis is that our dynamics can be seen as a perturbation of the voter model. We write the transition kernel of the corresponding Markov chain in terms of the pairwise correlations in the voter model. We analyze the pairwise correlation and show that in graphs with relatively large girth, cooperators cluster and help each other.
We consider the online stochastic matching problem proposed by Feldman et al. [FMMM09] as a model of display ad allocation. We are given a bipartite graph; one side of the graph corresponds to a fixed set of bins and the other side represents the set
of possible ball types. At each time step, a ball is sampled independently from the given distribution and it needs to be matched upon its arrival to an empty bin. The goal is to maximize the number of allocations. We present an online algorithm for this problem with a competitive ratio of 0.702. Before our result, algorithms with a competitive ratio better than $1-1/e$ were known under the assumption that the expected number of arriving balls of each type is integral. A key idea of the algorithm is to collect statistics about the decisions of the optimum offline solution using Monte Carlo sampling and use those statistics to guide the decisions of the online algorithm. We also show that our algorithm achieves a competitive ratio of 0.705 when the rates are integral. On the hardness side, we prove that no online algorithm can have a competitive ratio better than 0.823 under the known distribution model (and henceforth under the permutation model). This improves upon the 5/6 hardness result proved by Goel and Mehta cite{GM08} for the permutation model.
We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs $G$ and $W$ and an integer $k$, we are asked to find a $k$-edge weighted gra
ph $W_k$ such that $G+W_k$ is a good spectral sparsifer of $G+W$. We will refer to this problem as the subgraph (spectral) sparsification. We present a nontrivial condition on $G$ and $W$ such that a good sparsifier exists and give a polynomial time algorithm to find the sparsifer. %$O(frac{n}{k})log n tilde{O}(log log n)$ As a significant application of our technique, we show that for each positive integer $k$, every $n$-vertex weighted graph has an $(n-1+k)$-edge spectral sparsifier with relative condition number at most $frac{n}{k} log n tilde{O}(loglog n)$ where $tilde{O}()$ hides lower order terms. Our bound is within a factor of $tilde{O}(log log n)$ from optimal. This nearly settles a question left open by Spielman and Teng about ultrasparsifiers, which is a key component in their nearly linear-time algorithms for solving diagonally dominant symmetric linear systems. We also present another application of our technique to spectral optimization in which the goal is to maximize the algebraic connectivity of a graph (e.g. turn it into an expander) with a limited number of edges.
We give a constant factor approximation algorithm for the asymmetric traveling salesman problem when the support graph of the solution of the Held-Karp linear programming relaxation has bounded orientable genus.
Network alignment generalizes and unifies several approaches for forming a matching or alignment between the vertices of two graphs. We study a mathematical programming framework for network alignment problem and a sparse variation of it where only a
small number of matches between the vertices of the two graphs are possible. We propose a new message passing algorithm that allows us to compute, very efficiently, approximate solutions to the sparse network alignment problems with graph sizes as large as hundreds of thousands of vertices. We also provide extensive simulations comparing our algorithms with two of the best solvers for network alignment problems on two synthetic matching problems, two bioinformatics problems, and three large ontology alignment problems including a multilingual problem with a known labeled alignment.
We study the envy-free cake-cutting problem for $d+1$ players with $d$ cuts, for both the oracle function model and the polynomial time function model. For the former, we derive a $theta(({1overepsilon})^{d-1})$ time matching bound for the query comp
lexity of $d+1$ player cake cutting with Lipschitz utilities for any $d> 1$. When the utility functions are given by a polynomial time algorithm, we prove the problem to be PPAD-complete. For measurable utility functions, we find a fully polynomial-time algorithm for finding an approximate envy-free allocation of a cake among three people using two cuts.
We consider a robust model proposed by Scarf, 1958, for stochastic optimization when only the marginal probabilities of (binary) random variables are given, and the correlation between the random variables is unknown. In the robust model, the objecti
ve is to minimize expected cost against worst possible joint distribution with those marginals. We introduce the concept of correlation gap to compare this model to the stochastic optimization model that ignores correlations and minimizes expected cost under independent Bernoulli distribution. We identify a class of functions, using concepts of summable cost sharing schemes from game theory, for which the correlation gap is well-bounded and the robust model can be approximated closely by the independent distribution model. As a result, we derive efficient approximation factors for many popular cost functions, like submodular functions, facility location, and Steiner tree. As a byproduct, our analysis also yields some new results in the areas of social welfare maximization and existence of Walrasian equilibria, which may be of independent interest.
Coordination games describe social or economic interactions in which the adoption of a common strategy has a higher payoff. They are classically used to model the spread of conventions, behaviors, and technologies in societies. Here we consider a two
-strategies coordination game played asynchronously between the nodes of a network. Agents behave according to a noisy best-response dynamics. It is known that noise removes the degeneracy among equilibria: In the long run, the ``risk-dominant behavior spreads throughout the network. Here we consider the problem of computing the typical time scale for the spread of this behavior. In particular, we study its dependence on the network structure and derive a dichotomy between highly-connected, non-local graphs that show slow convergence, and poorly connected, low dimensional graphs that show fast convergence.
In this paper, we present approximation algorithms for combinatorial optimization problems under probabilistic constraints. Specifically, we focus on stochastic variants of two important combinatorial optimization problems: the k-center problem and t
he set cover problem, with uncertainty characterized by a probability distribution over set of points or elements to be covered. We consider these problems under adaptive and non-adaptive settings, and present efficient approximation algorithms for the case when underlying distribution is a product distribution. In contrast to the expected cost model prevalent in stochastic optimization literature, our problem definitions support restrictions on the probability distributions of the total costs, via incorporating constraints that bound the probability with which the incurred costs may exceed a given threshold.
