Deterministic and Probabilistic Binary Search in Graphs

We consider the following natural generalization of Binary Search: in a given undirected, positively weighted graph, one vertex is a target. The algorithm's task is to identify the target by adaptively querying vertices. In response to querying a node $q$, the algorithm learns either that $q$ is the target, or is given an edge out of $q$ that lies on a shortest path from $q$ to the target. We study this problem in a general noisy model in which each query independently receives a correct answer with probability $p > \frac{1}{2}$ (a known constant), and an (adversarial) incorrect one with probability $1-p$. Our main positive result is that when $p = 1$ (i.e., all answers are correct), $\log_2 n$ queries are always sufficient. For general $p$, we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than $(1 - \delta)\frac{\log_2 n}{1 - H(p)} + o(\log n) + O(\log^2 (1/\delta))$ queries, and identifies the target correctly with probability at leas $1-\delta$. Here, $H(p) = -(p \log p + (1-p) \log(1-p))$ denotes the entropy. The first bound is achieved by the algorithm that iteratively queries a 1-median of the nodes not ruled out yet; the second bound by careful repeated invocations of a multiplicative weights algorithm. Even for $p = 1$, we show several hardness results for the problem of determining whether a target can be found using $K$ queries. Our upper bound of $\log_2 n$ implies a quasipolynomial-time algorithm for undirected connected graphs; we show that this is best-possible under the Strong Exponential Time Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs with non-uniform node querying costs, the problem is PSPACE-complete. For a semi-adaptive version, in which one may query $r$ nodes each in $k$ rounds, we show membership in $\Sigma_{2k-1}$ in the polynomial hierarchy, and hardness for $\Sigma_{2k-5}$

Correlation Clustering with Same-Cluster Queries Bounded by Optimal Cost

Several clustering frameworks with interactive (semi-supervised) queries have been studied in the past. Recently, clustering with same-cluster queries has become popular. An algorithm in this setting has access to an oracle with full knowledge of an optimal clustering, and the algorithm can ask the oracle queries of the form, "Does the optimal clustering put vertices u and v in the same cluster?" Due to its simplicity, this querying model can easily be implemented in real crowd-sourcing platforms and has attracted a lot of recent work. In this paper, we study the popular correlation clustering problem (Bansal et al., 2002) under the same-cluster querying framework. Given a complete graph G=(V,E) with positive and negative edge labels, correlation clustering objective aims to compute a graph clustering that minimizes the total number of disagreements, that is the negative intra-cluster edges and positive inter-cluster edges. In a recent work, Ailon et al. (2018b) provided an approximation algorithm for correlation clustering that approximates the correlation clustering objective within (1+epsilon) with O((k^{14} log{n} log{k})/epsilon^6) queries when the number of clusters, k, is fixed. For many applications, k is not fixed and can grow with |V|. Moreover, the dependency of k^14 on query complexity renders the algorithm impractical even for datasets with small values of k. In this paper, we take a different approach. Let C_{OPT} be the number of disagreements made by the optimal clustering. We present algorithms for correlation clustering whose error and query bounds are parameterized by C_{OPT} rather than by the number of clusters. Indeed, a good clustering must have small C_{OPT}. Specifically, we present an efficient algorithm that recovers an exact optimal clustering using at most 2C_{OPT} queries and an efficient algorithm that outputs a 2-approximation using at most C_{OPT} queries. In addition, we show under a plausible complexity assumption, there does not exist any polynomial time algorithm that has an approximation ratio better than 1+alpha for an absolute constant alpha > 0 with o(C_{OPT}) queries. Therefore, our first algorithm achieves the optimal query bound within a factor of 2. We extensively evaluate our methods on several synthetic and real-world datasets using real crowd-sourced oracles. Moreover, we compare our approach against known correlation clustering algorithms that do not perform querying. In all cases, our algorithms exhibit superior performance

The Query-commit Problem

In the query-commit problem we are given a graph where edges have distinct probabilities of existing. It is possible to query the edges of the graph, and if the queried edge exists then its endpoints are irrevocably matched. The goal is to find a querying strategy which maximizes the expected size of the matching obtained. This stochastic matching setup is motivated by applications in kidney exchanges and online dating. In this paper we address the query-commit problem from both theoretical and experimental perspectives. First, we show that a simple class of edges can be queried without compromising the optimality of the strategy. This property is then used to obtain in polynomial time an optimal querying strategy when the input graph is sparse. Next we turn our attentions to the kidney exchange application, focusing on instances modeled over real data from existing exchange programs. We prove that, as the number of nodes grows, almost every instance admits a strategy which matches almost all nodes. This result supports the intuition that more exchanges are possible on a larger pool of patient/donors and gives theoretical justification for unifying the existing exchange programs. Finally, we evaluate experimentally different querying strategies over kidney exchange instances. We show that even very simple heuristics perform fairly well, being within 1.5% of an optimal clairvoyant strategy, that knows in advance the edges in the graph. In such a time-sensitive application, this result motivates the use of committing strategies

Lower Bounds in the Preprocessing and Query Phases of Routing Algorithms

In the last decade, there has been a substantial amount of research in finding routing algorithms designed specifically to run on real-world graphs. In 2010, Abraham et al. showed upper bounds on the query time in terms of a graph's highway dimension and diameter for the current fastest routing algorithms, including contraction hierarchies, transit node routing, and hub labeling. In this paper, we show corresponding lower bounds for the same three algorithms. We also show how to improve a result by Milosavljevic which lower bounds the number of shortcuts added in the preprocessing stage for contraction hierarchies. We relax the assumption of an optimal contraction order (which is NP-hard to compute), allowing the result to be applicable to real-world instances. Finally, we give a proof that optimal preprocessing for hub labeling is NP-hard. Hardness of optimal preprocessing is known for most routing algorithms, and was suspected to be true for hub labeling

Improved Parallel Algorithms for Spanners and Hopsets

We use exponential start time clustering to design faster and more work-efficient parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of the decomposed pieces. We weaken these bounds in favor of stronger local probabilistic guarantees. This allows more direct analyses of the overall process, giving: * Linear work parallel algorithms that construct spanners with $O(k)$ stretch and size $O(n^{1+1/k})$ in unweighted graphs, and size $O(n^{1+1/k} \log k)$ in weighted graphs. * Hopsets that lead to the first parallel algorithm for approximating shortest paths in undirected graphs with $O(m\;\mathrm{polylog}\;n)$ work

Complexity of coalition structure generation

We revisit the coalition structure generation problem in which the goal is to partition the players into exhaustive and disjoint coalitions so as to maximize the social welfare. One of our key results is a general polynomial-time algorithm to solve the problem for all coalitional games provided that player types are known and the number of player types is bounded by a constant. As a corollary, we obtain a polynomial-time algorithm to compute an optimal partition for weighted voting games with a constant number of weight values and for coalitional skill games with a constant number of skills. We also consider well-studied and well-motivated coalitional games defined compactly on combinatorial domains. For these games, we characterize the complexity of computing an optimal coalition structure by presenting polynomial-time algorithms, approximation algorithms, or NP-hardness and inapproximability lower bounds.Comment: 17 page