549 research outputs found
Unconstraining Graph-Constrained Group Testing
In network tomography, one goal is to identify a small set of failed links in a network using as little information as possible. One way of setting up this problem is called graph-constrained group testing. Graph-constrained group testing is a variant of the classical combinatorial group testing problem, where the tests that one is allowed are additionally constrained by a graph. In this case, the graph is given by the underlying network topology.
The main contribution of this work is to show that for most graphs, the constraints imposed by the graph are no constraint at all. That is, the number of tests required to identify the failed links in graph-constrained group testing is near-optimal even for the corresponding group testing problem with no graph constraints. Our approach is based on a simple randomized construction of tests. To analyze our construction, we prove new results about the size of giant components in randomly sparsified graphs.
Finally, we provide empirical results which suggest that our connected-subgraph tests perform better not just in theory but also in practice, and in particular perform better on a real-world network topology
Optimal Lower Bounds for Universal and Differentially Private Steiner Tree and TSP
Given a metric space on n points, an {\alpha}-approximate universal algorithm
for the Steiner tree problem outputs a distribution over rooted spanning trees
such that for any subset X of vertices containing the root, the expected cost
of the induced subtree is within an {\alpha} factor of the optimal Steiner tree
cost for X. An {\alpha}-approximate differentially private algorithm for the
Steiner tree problem takes as input a subset X of vertices, and outputs a tree
distribution that induces a solution within an {\alpha} factor of the optimal
as before, and satisfies the additional property that for any set X' that
differs in a single vertex from X, the tree distributions for X and X' are
"close" to each other. Universal and differentially private algorithms for TSP
are defined similarly. An {\alpha}-approximate universal algorithm for the
Steiner tree problem or TSP is also an {\alpha}-approximate differentially
private algorithm. It is known that both problems admit O(logn)-approximate
universal algorithms, and hence O(log n)-approximate differentially private
algorithms as well. We prove an {\Omega}(logn) lower bound on the approximation
ratio achievable for the universal Steiner tree problem and the universal TSP,
matching the known upper bounds. Our lower bound for the Steiner tree problem
holds even when the algorithm is allowed to output a more general solution of a
distribution on paths to the root.Comment: 14 page
Fast Generation of Random Spanning Trees and the Effective Resistance Metric
We present a new algorithm for generating a uniformly random spanning tree in
an undirected graph. Our algorithm samples such a tree in expected
time. This improves over the best previously known bound
of -- that follows from the work of
Kelner and M\k{a}dry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] --
whenever the input graph is sufficiently sparse.
At a high level, our result stems from carefully exploiting the interplay of
random spanning trees, random walks, and the notion of effective resistance, as
well as from devising a way to algorithmically relate these concepts to the
combinatorial structure of the graph. This involves, in particular,
establishing a new connection between the effective resistance metric and the
cut structure of the underlying graph
Expanders Are Universal for the Class of All Spanning Trees
Given a class of graphs F, we say that a graph G is universal for F, or
F-universal, if every H in F is contained in G as a subgraph. The construction
of sparse universal graphs for various families F has received a considerable
amount of attention. One is particularly interested in tight F-universal
graphs, i.e., graphs whose number of vertices is equal to the largest number of
vertices in a graph from F. Arguably, the most studied case is that when F is
some class of trees.
Given integers n and \Delta, we denote by T(n,\Delta) the class of all
n-vertex trees with maximum degree at most \Delta. In this work, we show that
every n-vertex graph satisfying certain natural expansion properties is
T(n,\Delta)-universal or, in other words, contains every spanning tree of
maximum degree at most \Delta. Our methods also apply to the case when \Delta
is some function of n. The result has a few very interesting implications. Most
importantly, we obtain that the random graph G(n,p) is asymptotically almost
surely (a.a.s.) universal for the class of all bounded degree spanning (i.e.,
n-vertex) trees provided that p \geq c n^{-1/3} \log^2n where c > 0 is a
constant. Moreover, a corresponding result holds for the random regular graph
of degree pn. In fact, we show that if \Delta satisfies \log n \leq \Delta \leq
n^{1/3}, then the random graph G(n,p) with p \geq c \Delta n^{-1/3} \log n and
the random r-regular n-vertex graph with r \geq c\Delta n^{2/3} \log n are
a.a.s. T(n,\Delta)-universal. Another interesting consequence is the existence
of locally sparse n-vertex T(n,\Delta)-universal graphs. For constant \Delta,
we show that one can (randomly) construct n-vertex T(n,\Delta)-universal graphs
with clique number at most five. Finally, we show robustness of random graphs
with respect to being universal for T(n,\Delta) in the context of the
Maker-Breaker tree-universality game.Comment: 25 page
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