Multiple-choice load balancing has been a topic of intense study since the seminal paper of Azar, Broder, Karlin, and Upfal. Questions in this area can be phrased in terms of orientations of a graph, or more generally a k-uniform random hypergraph. A (d,b)-orientation is an assignment of each edge to d of its vertices, such that no vertex has more than b edges assigned to it. Conditions for the existence of such orientations have been completely documented except for the "extreme" case of (k-1,1)-orientations. We consider this remaining case, and establish: - The density threshold below which an orientation exists with high probability, and above which it does not exist with high probability. - An algorithm for finding an orientation that runs in linear time with high probability, with explicit polynomial bounds on the failure probability. Previously, the only known algorithms for constructing (k-1,1)-orientations worked for k<=3, and were only shown to have expected linear running time.
In the off-line setting, the threshold for @math -orientations with @math can be shown (see, e.g. @cite_12 ) to coincide with the appearance of a giant component in the random graph, which is known to happen at @math with high probability @cite_4 . Several groups of researchers @cite_16 @cite_9 @cite_17 independently established the thresholds for @math -orientations for every @math . Generalizing in another direction, Fernholz and Ramachandran @cite_10 and Cain, Sanders, and Wormald @cite_5 showed thresholds for @math -orientations for @math , and gave expected linear time algorithms for computing an orientation. This result was later extended to @math by @cite_18 . Gao and Wormald @cite_14 established thresholds for @math -orientations, given that @math is a sufficiently large constant (depending on @math and @math ). Independently of our work, Lelarge @cite_7 recently developed new technical machinery for this problem, which handles all combinations of the parameters @math , @math , and @math that satisfy @math .
@cite_18 @cite_14 @cite_4 @cite_7 @cite_9 @cite_5 @cite_16 @cite_10 @cite_12 @cite_17
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Given a discrete random walk on a finite graph @math , the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step @math . These sets induce subgraphs of the underlying graph. Let @math be the subgraph of @math induced by the vacant set of the walk at step @math . Similarly, let @math be the subgraph of @math induced by the edges of the vacant net. For random @math -regular graphs @math , it was previously established that for a simple random walk, the graph @math of the vacant set undergoes a phase transition in the sense of the phase transition on Erd os -Renyi graphs @math . Thus, for @math there is an explicit value @math of the walk, such that for @math , @math has a unique giant component, plus components of size @math , whereas for @math all the components of @math are of size @math . We establish the threshold value @math for a phase transition in the graph @math of the vacant net of a simple random walk on a random @math -regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for @math even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random @math -regular graphs. The main findings are the following: As @math increases the threshold for the vacant set converges to @math in all three walks. For the vacant net, the threshold converges to @math for both the simple random walk and non-backtracking random walk. When @math is even, the threshold for the vacant net of the unvisited edge process converges to @math , which is also the vertex cover time of the process. We design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with "nice" threshold behavior and degree almost as low as the probabilistic polynomials, and a new notion of probabilistic PTFs where we combine the above techniques to achieve even lower degree with similar "nice" threshold behavior. Utilizing these polynomial constructions, we design faster algorithms for a variety of problems: @math Offline Hamming Nearest (and Furthest) Neighbors: Given @math red and @math blue points in @math -dimensional Hamming space for @math , we can find an (exact) nearest (or furthest) blue neighbor for every red point in randomized time @math or deterministic time @math . These also lead to faster MAX-SAT algorithms for sparse CNFs. @math Offline Approximate Nearest (and Furthest) Neighbors: Given @math red and @math blue points in @math -dimensional @math or Euclidean space, we can find a @math -approximate nearest (or furthest) blue neighbor for each red point in randomized time near @math . @math SAT Algorithms and Lower Bounds for Circuits With Linear Threshold Functions: We give a satisfiability algorithm for @math circuits with a subquadratic number of linear threshold gates on the bottom layer, and a subexponential number of gates on the other layers, that runs in deterministic @math time. This also implies new circuit lower bounds for threshold circuits. We also give a randomized @math -time SAT algorithm for subexponential-size @math circuits, where the top @math gate and middle @math gates have @math fan-in. In this paper we consider a fundamental problem in the area of viral marketing, called T ARGET S ET S ELECTION problem. In a a viral marketing setting, social networks are modeled by graphs with potential customers of a new product as vertices and friend relationships as edges, where each vertex @math is assigned a threshold value @math . The thresholds represent the different latent tendencies of customers (vertices) to buy the new product when their friend (neighbors) do. Consider a repetitive process on social network @math where each vertex @math is associated with two states, active and inactive, which indicate whether @math is persuaded into buying the new product. Suppose we are given a target set @math . Initially, all vertices in @math are inactive. At time step 0, we choose all vertices in @math to become active. Then, at every time step @math , all vertices that were active in time step @math remain active, and we activate any vertex @math if at least @math of its neighbors were active at time step @math . The activation process terminates when no more vertices can get activated. We are interested in the following optimization problem, called T ARGET S ET S ELECTION : Finding a target set @math of smallest possible size that activates all vertices of @math . There is an important and well-studied threshold called strict majority threshold, where for every vertex @math in @math we have @math and @math is the degree of @math in @math . In this paper, we consider the T ARGET S ET S ELECTION problem under strict majority thresholds and focus on three popular regular network structures: cycle permutation graphs, generalized Petersen graphs and torus cordalis. We prove that, for k ≥ 2, the k-orientability threshold for the random graph Gn, p coincides with the threshold at which the (k + 1)-core has average degree 2k. The proof involves the analysis of a heuristic algorithm that attempts to find a k-orientation of the random graph. The k-orientation threshold has several applications including offline balanced allocation with a limit of k on maximum bin-size, perfect hashing with a limit of k on maximum chain-length, and concurrent access to parallel memories through redundancy,