19 research outputs found
Optimal Query Complexity for Reconstructing Hypergraphs
In this paper we consider the problem of reconstructing a hidden weighted
hypergraph of constant rank using additive queries. We prove the following: Let
be a weighted hidden hypergraph of constant rank with n vertices and
hyperedges. For any there exists a non-adaptive algorithm that finds the
edges of the graph and their weights using
additive queries. This solves the open problem in [S. Choi, J. H. Kim. Optimal
Query Complexity Bounds for Finding Graphs. {\em STOC}, 749--758,~2008].
When the weights of the hypergraph are integers that are less than
where is the rank of the hypergraph (and therefore for
unweighted hypergraphs) there exists a non-adaptive algorithm that finds the
edges of the graph and their weights using additive queries.
Using the information theoretic bound the above query complexities are tight
Graph Reconstruction with a Betweenness Oracle
Graph reconstruction algorithms seek to learn a hidden graph by repeatedly querying a black-box oracle for information about the graph structure. Perhaps the most well studied and applied version of the problem uses a distance oracle, which can report the shortest path distance between any pair of nodes.
We introduce and study the betweenness oracle, where bet(a, m, z) is true iff m lies on a shortest path between a and z. This oracle is strictly weaker than a distance oracle, in the sense that a betweenness query can be simulated by a constant number of distance queries, but not vice versa. Despite this, we are able to develop betweenness reconstruction algorithms that match the current state of the art for distance reconstruction, and even improve it for certain types of graphs. We obtain the following algorithms: (1) Reconstruction of general graphs in O(n^2) queries, (2) Reconstruction of degree-bounded graphs in ~O(n^{3/2}) queries, (3) Reconstruction of geodetic degree-bounded graphs in ~O(n) queries
In addition to being a fundamental graph theoretic problem with some natural applications, our new results shed light on some avenues for progress in the distance reconstruction problem
Computing Exact Minimum Cuts Without Knowing the Graph
We give query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs. Our oracle model is inspired by the submodular function minimization problem:
on query S subset V, the oracle returns the size of the cut between S and V S.
We provide algorithms computing an exact minimum - cut in with ~{O}(n^{5/3}) queries, and computing an exact global minimum cut of G with only ~{O}(n) queries (while learning the graph requires ~{Theta}(n^2) queries)
Topology Discovery of Sparse Random Graphs With Few Participants
We consider the task of topology discovery of sparse random graphs using
end-to-end random measurements (e.g., delay) between a subset of nodes,
referred to as the participants. The rest of the nodes are hidden, and do not
provide any information for topology discovery. We consider topology discovery
under two routing models: (a) the participants exchange messages along the
shortest paths and obtain end-to-end measurements, and (b) additionally, the
participants exchange messages along the second shortest path. For scenario
(a), our proposed algorithm results in a sub-linear edit-distance guarantee
using a sub-linear number of uniformly selected participants. For scenario (b),
we obtain a much stronger result, and show that we can achieve consistent
reconstruction when a sub-linear number of uniformly selected nodes
participate. This implies that accurate discovery of sparse random graphs is
tractable using an extremely small number of participants. We finally obtain a
lower bound on the number of participants required by any algorithm to
reconstruct the original random graph up to a given edit distance. We also
demonstrate that while consistent discovery is tractable for sparse random
graphs using a small number of participants, in general, there are graphs which
cannot be discovered by any algorithm even with a significant number of
participants, and with the availability of end-to-end information along all the
paths between the participants.Comment: A shorter version appears in ACM SIGMETRICS 2011. This version is
scheduled to appear in J. on Random Structures and Algorithm
Learning Spanning Forests Optimally using CUT Queries in Weighted Undirected Graphs
In this paper we describe a randomized algorithm which returns a maximal
spanning forest of an unknown {\em weighted} undirected graph making
queries in expectation. For weighted graphs, this is optimal due
to a result in [Auza and Lee, 2021] which shows an lower bound for
zero-error randomized algorithms. %To our knowledge, it is the only regime of
this problem where we have upper and lower bounds tight up to constants. These
questions have been extensively studied in the past few years, especially due
to the problem's connections to symmetric submodular function minimization. We
also describe a simple polynomial time deterministic algorithm that makes
queries on undirected unweighted graphs and
returns a maximal spanning forest, thereby (slightly) improving upon the
state-of-the-art
Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty
We study how to utilize (possibly erroneous) predictions in a model for computing under uncertainty in which an algorithm can query unknown data. Our aim is to minimize the number of queries needed to solve the minimum spanning tree problem, a fundamental combinatorial optimization problem that has been central also to the research area of explorable uncertainty. For all integral ? ? 2, we present algorithms that are ?-robust and (1+1/?)-consistent, meaning that they use at most ?OPT queries if the predictions are arbitrarily wrong and at most (1+1/?)OPT queries if the predictions are correct, where OPT is the optimal number of queries for the given instance. Moreover, we show that this trade-off is best possible. Furthermore, we argue that a suitably defined hop distance is a useful measure for the amount of prediction error and design algorithms with performance guarantees that degrade smoothly with the hop distance. We also show that the predictions are PAC-learnable in our model. Our results demonstrate that untrusted predictions can circumvent the known lower bound of 2, without any degradation of the worst-case ratio. To obtain our results, we provide new structural insights for the minimum spanning tree problem that might be useful in the context of query-based algorithms regardless of predictions. In particular, we generalize the concept of witness sets - the key to lower-bounding the optimum - by proposing novel global witness set structures and completely new ways of adaptively using those
Graph Connectivity and Single Element Recovery via Linear and OR Queries
We study the problem of finding a spanning forest in an undirected,
-vertex multi-graph under two basic query models. One is the Linear query
model which are linear measurements on the incidence vector induced by the
edges; the other is the weaker OR query model which only reveals whether a
given subset of plausible edges is empty or not. At the heart of our study lies
a fundamental problem which we call the {\em single element recovery} problem:
given a non-negative real vector in dimension, return a single element
from the support. Queries can be made in rounds, and our goals is to
understand the trade-offs between the query complexity and the rounds of
adaptivity needed to solve these problems, for both deterministic and
randomized algorithms. These questions have connections and ramifications to
multiple areas such as sketching, streaming, graph reconstruction, and
compressed sensing. Our main results are:
* For the single element recovery problem, it is easy to obtain a
deterministic, -round algorithm which makes -queries per-round.
We prove that this is tight: any -round deterministic algorithm must make
linear queries in some round. In contrast, a -round
-query randomized algorithm which succeeds 99% of the time is
known to exist.
* We design a deterministic -round, -OR query
algorithm for graph connectivity. We complement this with an
-lower bound for any -round deterministic
algorithm in the OR-model.
* We design a randomized, -round algorithm for the graph connectivity
problem which makes -OR queries. In contrast, we prove that any
-round algorithm (possibly randomized) requires -OR
queries
Graph Reconstruction via Distance Oracles
We study the problem of reconstructing a hidden graph given access to a
distance oracle. We design randomized algorithms for the following problems:
reconstruction of a degree bounded graph with query complexity
; reconstruction of a degree bounded outerplanar graph with
query complexity ; and near-optimal approximate reconstruction of
a general graph