Orienting (hyper)graphs under explorable stochastic uncertainty

Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2022

We study the minimum vertex cover problem in the following stochastic setting. Let G be an arbitrary given graph, p ∈ (0, 1] a parameter of the problem, and let G p be a random subgraph that includes each edge of G independently with probability p. We are unaware of the realization G p , but can learn if an edge e exists in G p by querying it. The goal is to find an approximate minimum vertex cover (MVC) of G p by querying few edges of G non-adaptively. This stochastic setting has been studied extensively for various problems such as minimum spanning trees, matroids, shortest paths, and matchings. To our knowledge, however, no nontrivial bound was known for MVC prior to our work. In this work, we present a: • (2 + ε)-approximation for general graphs which queries O(1 ε 3 p) edges per vertex, and a • 1.367-approximation for bipartite graphs which queries poly(1/p) edges per vertex. Additionally, we show that at the expense of a triple-exponential dependence on p −1 in the number of queries, the approximation ratio can be improved down to (1+ε) for bipartite graphs. Our techniques also lead to improved bounds for bipartite stochastic matching. We obtain a 0.731-approximation with nearly-linear in 1/p per-vertex queries. This is the first result to break the prevalent (2/3 ∼ 0.66)-approximation barrier in the poly(1/p) query regime, improving algorithms of [Behnezhad et al., SODA'19] and [Assadi and Bernstein, SOSA'19].

ArXiv, 2019

Given a weighted hypergraph $\mathcal{H}(V, \mathcal{E} \subseteq 2^V, w)$, the approximate $k$-cover problem seeks for a size-$k$ subset of $V$ that has the maximum weighted coverage by \emph{sampling only a few hyperedges} in $\mathcal{E}$. The problem has emerged from several network analysis applications including viral marketing, centrality maximization, and landmark selection. Despite many efforts, even the best approaches require $O(k n \log n)$ space complexities, thus, cannot scale to, nowadays, humongous networks without sacrificing formal guarantees. In this paper, we propose BCA, a family of algorithms for approximate $k$-cover that can find $(1-\frac{1}{e} -\epsilon)$-approximation solutions within an \emph{$O(\epsilon^{-2}n \log n)$ space}. That is a factor $k$ reduction on space comparing to the state-of-the-art approaches with the same guarantee. We further make BCA more efficient and robust on real-world instances by introducing a novel adaptive sampling scheme, ter...

Algorithmic Operations Research, 2010

We study a probabilistic optimization model for graph-problems under vertex-uncertainty. We assume that any vertex vi of the input-graph G(V, E) has only a probability pi to be present in the final graph to be optimized (i.e., the final instance for the problem tackled will be only a sub-graph of the initial graph). Under this model, the original "deterministic" problem gives rise to a new (deterministic) problem on the same input-graph G, having the same set of feasible solutions as the former one, but its objective function can be very different from the original one, the set of its optimal solutions too. Moreover, this objective function is a sum of 2 |V | terms; hence, its computation is not immediately polynomial. We give sufficient conditions for large classes of graph-problems under which objective functions of their probabilistic counterparts are polynomially computable and optimal solutions are well-characterized. Finally, we apply these general results to natural and well-known combinatorial problems that belong to the classes considered.

Journal of Discrete Algorithms, 2012

We consider the minimum vertex cover problem in hypergraphs in which every hyperedge has size k (also known as minimum hitting set problem, or minimum set cover with element frequency k). Simple algorithms exist that provide k-approximations, and this is believed to be the best possible approximation achievable in polynomial time. We show how to exploit density and regularity properties of the input hypergraph to break this barrier. In particular, we provide a randomized polynomial-time algorithm with approximation factor k/(1 + (k − 1)d k∆), whered and ∆ are the average and maximum degree, respectively, and ∆ must be Ω(n k−1 / log n). The proposed algorithm generalizes the recursive sampling technique of Imamura and Iwama (SODA'05) for vertex cover in dense graphs. As a corollary, we obtain an approximation factor k/(2 − 1/k) for subdense regular hypergraphs, which is shown to be the best possible under the unique games conjecture.

Proceedings of the 21st ACM international conference on Information and knowledge management - CIKM '12, 2012

This demo presents a framework for running probabilistic graph queries on uncertain graphs and visualizing their results. The framework supports the most common uncertainty model for uncertain graphs, i.e. existential uncertainty for the edges of the graph. A large variety of meaningful graph queries are supported, such as shortest path, range, kNN, reverse kNN, reachability and various aggregation queries. Since the problem of exact probability computation according to possible world semantics is in #P-Time for many combinations of model and query, and since ignoring uncertainty (e.g. by using expectations only) will yield counterintuitive and hard to interpret results, our framework uses an optimized version of Monte-Carlo sampling to estimate the results which allows us not only to perform queries that conform to possible world semantics but also to sample only parts of a graph relevant for a given query. The main strength of this framework is the visualization combined with statistic hypothesis tests, which gives the user not only the estimated result of a query, but also an indication of how significant and reliable these results are. The aim of this demonstration is to give an intuition that a sampling based approach to probabilistic graphs is viable, and that the estimated results quickly converge even for very large graphs.

2011

We study approximation complexity of the Vertex Cover problem restricted to dense and subdense balanced k-partite k-uniform hypergraphs. The best known approximation algorithm for the general k-partite case achieves an approximation ratio of k2 which is the best possible assuming the Unique Game Conjecture. In this paper, we present approximation algorithms for the dense and the subdense nearly regular instances both with an approximation factor strictly better than k2 . On the other hand, we show that the latter approximation upper bound is almost tight under the Unique Games Conjecture.

2010

We consider the problem of finding an unknown graph by using queries with an additive property. This problem was partially motivated by DNA shotgun sequencing and linkage discovery problems of artificial intelligence. Given a graph, an additive query asks the number of edges in a set of vertices while a cross-additive query asks the number of edges crossing between two disjoint sets of vertices. The queries ask the sum of weights for weighted graphs. For a graph G with n vertices and at most m edges, we prove that there exists an algorithm to find the edges of G using O(m log n 2 m log(m+1)) queries of both types for all m. The bound is best possible up to a constant factor. For a weighted graph with a mild condition on weights, it is shown that O(m log n log m) queries are enough provided m (log n) α for a sufficiently large constant α, which is best possible up to a constant factor if m n 2−ε for any constant ε > 0.

2002

This short paper surveys recent results on the gen- eration of implicitly given hypergraphs, as well as their applications in data mining, reliability theory, integer programming and combinatorics. More precisely, we consider a monotone property over the subsets of a finite set V , the correspond- ing family S of subsets satisfying property , and the problem of generating (sequentially) the family F of all minimal subsets in S , when only V is given explic- itly, and is represented by an oracle O. We show that for a number of interesting monotone properties, the family F is uniformly dual-bounded allowing for the incrementally ecient generation of the members of F. Important applications include the ecient gener- ation of minimal infrequent sets of a database (data mining), minimal connectivity ensuring collections of subgraphs from a given list (reliability theory), mini- mal feasible solutions to a system of monotone inequal- ities in integer variables (integer programming), mini-...

Lecture Notes in Computer Science, 2015

How efficiently can we find an unknown graph using distance or shortest path queries between its vertices? Let G = (V, E) be an unweighted, connected graph of bounded degree. The edge set E is initially unknown, and the graph can be accessed using a distance oracle, which receives a pair of vertices (u, v) and returns the distance between u and v. In the verification problem, we are given a hypothetical graphĜ = (V,Ê) and want to check whether G is equal toĜ. We analyze a natural greedy algorithm and prove that it uses n 1+o(1) distance queries. In the more difficult reconstruction problem,Ĝ is not given, and the goal is to find the graph G. If the graph can be accessed using a shortest path oracle, which returns not just the distance but an actual shortest path between u and v, we show that extending the idea of greedy gives a reconstruction algorithm that uses n 1+o(1) shortest path queries. When the graph has bounded treewidth, we further bound the query complexity of the greedy algorithms for both problems byÕ(n). When the graph is chordal, we provide a randomized algorithm for reconstruction usingÕ(n) distance queries.

SIAM Journal on Optimization, 2004

We address the problem of evaluating the expected optimal objective value of a 0-1 optimization problem under uncertainty in the objective coefficients. The probabilistic model we consider prescribes limited marginal distribution information for the objective coefficients in the form of moments. We show that for a fairly general class of marginal information, a tight upper (lower) bound on the expected optimal objective value of a 0-1 maximization (minimization) problem can be computed in polynomial time if the corresponding deterministic problem is solvable in polynomial time. We provide an efficiently solvable semidefinite programming formulation to compute this tight bound. We also analyze the asymptotic behavior of a general class of combinatorial problems that includes the linear assignment, spanning tree, and traveling salesman problems, under knowledge of complete marginal distributions, with and without independence. We calculate the limiting constants exactly.