Sharp Thresholds in Adaptive Random Graph Processes

The $\mathcal{D}$-process is a single player game in which the player is initially presented the empty graph on $n$ vertices. In each step, a subset of edges $X$ is independently sampled according to a distribution $\mathcal{D}$. The player then selects one edge $e$ from $X$, and adds $e$ to its current graph. For a fixed monotone increasing graph property $\mathcal{P}$, the objective of the player is to force the graph to satisfy $\mathcal{P}$ in as few steps as possible. Through appropriate choices of $\mathcal{D}$, the $\mathcal{D}$-process generalizes well-studied adaptive random graph processes, such as the Achlioptas process and the semi-random graph process We prove a sufficient condition for the existence of a sharp threshold for $\mathcal{P}$ in the $\mathcal{D}$-process. For the semi-random process, we use this condition to prove the existence of a sharp threshold when $\mathcal{P}$ corresponds to being Hamiltonian or to containing a perfect matching. These are the first results for the semi-random graph process which show the existence of a sharp threshold when $\mathcal{P}$ corresponds to containing a sparse spanning graph. Using a separate analytic argument, we show that each sharp threshold is of the form $C_{\mathcal{P}}n$ for some fixed constant $C_{\mathcal{P}}>0$. This answers two of the open problems proposed by Ben-Eliezer et al. (SODA 2020) in the affirmative. Unlike similar results which establish sharp thresholds for certain distributions and properties, we establish the existence of sharp thresholds without explicitly identifying asymptotically optimal strategies.Comment: Accepted to Random Structures and Algorithms (RSA). Minor corrections made to Section 3, and the exposition of Section 4 was improved from the previous arXiv versio

Secretary Matching Meets Probing with Commitment

We consider the online bipartite matching problem within the context of stochastic probing with commitment. This is the one-sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist based on edge probabilities that become known when an online vertex arrives. If a probed edge exists, it must be used in the matching. We consider the competitiveness of online algorithms in the adversarial order model (AOM) and the secretary/random order model (ROM). More specifically, we consider an unknown bipartite stochastic graph G = (U,V,E) where U is the known set of offline vertices, V is the set of online vertices, G has edge probabilities (p_{e})_{e ? E}, and G has edge weights (w_{e})_{e ? E} or vertex weights (w_u)_{u ? U}. Additionally, G has a downward-closed set of probing constraints (?_{v})_{v ? V}, where ?_v indicates which sequences of edges adjacent to an online vertex v can be probed. This model generalizes the various settings of the classical bipartite matching problem (i.e. with and without probing). Our contributions include the introduction and analysis of probing within the random order model, and our generalization of probing constraints which includes budget (i.e. knapsack) constraints. Our algorithms run in polynomial time assuming access to a membership oracle for each ?_v. In the vertex weighted setting, for adversarial order arrivals, we generalize the known 1/2 competitive ratio to our setting of ?_v constraints. For random order arrivals, we show that the same algorithm attains an asymptotic competitive ratio of 1-1/e, provided the edge probabilities vanish to 0 sufficiently fast. We also obtain a strict competitive ratio for non-vanishing edge probabilities when the probing constraints are sufficiently simple. For example, if each ?_v corresponds to a patience constraint ?_v (i.e., ?_v is the maximum number of probes of edges adjacent to v), and any one of following three conditions is satisfied (each studied in previous papers), then there is a conceptually simple greedy algorithm whose competitive ratio is 1-1/e. - When the offline vertices are unweighted. - When the online vertex probabilities are "vertex uniform"; i.e., p_{u,v} = p_v for all (u,v) ? E. - When the patience constraint ?_v satisfies ?_v ? {[1,|U|} for every online vertex; i.e., every online vertex either has unit or full patience. Finally, in the edge weighted case, we match the known optimal 1/e asymptotic competitive ratio for the classic (i.e. without probing) secretary matching problem

A Fully Adaptive Strategy for Hamiltonian Cycles in the Semi-Random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on n vertices. In each round, a vertex u is presented to the player independently and uniformly at random. The player then adaptively selects a vertex v, and adds the edge uv to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a Hamiltonian cycle in as few rounds as possible. In particular, we present an adaptive strategy for the player which achieves it in ? n rounds, where ? 1.26575. These results improve the previously best known bounds and, as a result, the gap between the upper and lower bounds is decreased from 1.39162 to 0.75102