2,432 research outputs found
Streaming Lower Bounds for Approximating MAX-CUT
We consider the problem of estimating the value of max cut in a graph in the
streaming model of computation. At one extreme, there is a trivial
-approximation for this problem that uses only space, namely,
count the number of edges and output half of this value as the estimate for max
cut value. On the other extreme, if one allows space, then a
near-optimal solution to the max cut value can be obtained by storing an
-size sparsifier that essentially preserves the max cut. An
intriguing question is if poly-logarithmic space suffices to obtain a
non-trivial approximation to the max-cut value (that is, beating the factor
). It was recently shown that the problem of estimating the size of a
maximum matching in a graph admits a non-trivial approximation in
poly-logarithmic space.
Our main result is that any streaming algorithm that breaks the
-approximation barrier requires space even if the
edges of the input graph are presented in random order. Our result is obtained
by exhibiting a distribution over graphs which are either bipartite or
-far from being bipartite, and establishing that
space is necessary to differentiate between these
two cases. Thus as a direct corollary we obtain that
space is also necessary to test if a graph is bipartite or -far
from being bipartite.
We also show that for any , any streaming algorithm that
obtains a -approximation to the max cut value when edges arrive
in adversarial order requires space, implying that
space is necessary to obtain an arbitrarily good approximation to
the max cut value
The Quantum and Classical Streaming Complexity of Quantum and Classical Max-Cut
We investigate the space complexity of two graph streaming problems: Max-Cut
and its quantum analogue, Quantum Max-Cut. Previous work by Kapralov and
Krachun [STOC `19] resolved the classical complexity of the \emph{classical}
problem, showing that any -approximation requires
space (a -approximation is trivial with
space). We generalize both of these qualifiers, demonstrating space
lower bounds for -approximating Max-Cut and Quantum Max-Cut,
even if the algorithm is allowed to maintain a quantum state. As the trivial
approximation algorithm for Quantum Max-Cut only gives a -approximation, we
show tightness with an algorithm that returns a -approximation to the Quantum Max-Cut value of a graph in
space. Our work resolves the quantum and classical
approximability of quantum and classical Max-Cut using space.
We prove our lower bounds through the techniques of Boolean Fourier analysis.
We give the first application of these methods to sequential one-way quantum
communication, in which each player receives a quantum message from the
previous player, and can then perform arbitrary quantum operations on it before
sending it to the next. To this end, we show how Fourier-analytic techniques
may be used to understand the application of a quantum channel
Noisy Boolean Hidden Matching with Applications
The Boolean Hidden Matching (BHM) problem, introduced in a seminal paper of Gavinsky et al. [STOC\u2707], has played an important role in lower bounds for graph problems in the streaming model (e.g., subgraph counting, maximum matching, MAX-CUT, Schatten p-norm approximation). The BHM problem typically leads to ?(?n) space lower bounds for constant factor approximations, with the reductions generating graphs that consist of connected components of constant size. The related Boolean Hidden Hypermatching (BHH) problem provides ?(n^{1-1/t}) lower bounds for 1+O(1/t) approximation, for integers t ? 2. The corresponding reductions produce graphs with connected components of diameter about t, and essentially show that long range exploration is hard in the streaming model with an adversarial order of updates.
In this paper we introduce a natural variant of the BHM problem, called noisy BHM (and its natural noisy BHH variant), that we use to obtain stronger than ?(?n) lower bounds for approximating a number of the aforementioned problems in graph streams when the input graphs consist only of components of diameter bounded by a fixed constant.
We next introduce and study the graph classification problem, where the task is to test whether the input graph is isomorphic to a given graph. As a first step, we use the noisy BHM problem to show that the problem of classifying whether an underlying graph is isomorphic to a complete binary tree in insertion-only streams requires ?(n) space, which seems challenging to show using either BHM or BHH
Sketching Cuts in Graphs and Hypergraphs
Sketching and streaming algorithms are in the forefront of current research
directions for cut problems in graphs. In the streaming model, we show that
-approximation for Max-Cut must use space;
moreover, beating -approximation requires polynomial space. For the
sketching model, we show that -uniform hypergraphs admit a
-cut-sparsifier (i.e., a weighted subhypergraph that
approximately preserves all the cuts) with
edges. We also make first steps towards sketching general CSPs (Constraint
Satisfaction Problems)
On streaming approximation algorithms for constraint satisfaction problems
In this thesis, we explore streaming algorithms for approximating constraint
satisfaction problems (CSPs). The setup is roughly the following: A computer
has limited memory space, sees a long "stream" of local constraints on a set of
variables, and tries to estimate how many of the constraints may be
simultaneously satisfied. The past ten years have seen a number of works in
this area, and this thesis includes both expository material and novel
contributions. Throughout, we emphasize connections to the broader theories of
CSPs, approximability, and streaming models, and highlight interesting open
problems.
The first part of our thesis is expository: We present aspects of previous
works that completely characterize the approximability of specific CSPs like
Max-Cut and Max-Dicut with -space streaming algorithm (on
-variable instances), while characterizing the approximability of all CSPs
in space in the special case of "composable" (i.e., sketching)
algorithms, and of a particular subclass of CSPs with linear-space streaming
algorithms.
In the second part of the thesis, we present two of our own joint works. We
begin with a work with Madhu Sudan and Santhoshini Velusamy in which we prove
linear-space streaming approximation-resistance for all ordering CSPs (OCSPs),
which are "CSP-like" problems maximizing over sets of permutations. Next, we
present joint work with Joanna Boyland, Michael Hwang, Tarun Prasad, and
Santhoshini Velusamy in which we investigate the -space streaming
approximability of symmetric Boolean CSPs with negations. We give explicit
-space sketching approximability ratios for several families of CSPs,
including Max-AND; develop simpler optimal sketching approximation
algorithms for threshold predicates; and show that previous lower bounds fail
to characterize the -space streaming approximability of Max-AND.Comment: Harvard College senior thesis; 119 pages plus references; abstract
shortened for arXiv; formatted with Dissertate template (feel free to copy!);
exposits papers arXiv:2105.01782 (APPROX 2021) and arXiv:2112.06319 (APPROX
2022
Streaming Hardness of Unique Games
We study the problem of approximating the value of a Unique Game instance in the streaming model. A simple count of the number of constraints divided by p, the alphabet size of the Unique Game, gives a trivial p-approximation that can be computed in O(log n) space. Meanwhile, with high probability, a sample of O~(n) constraints suffices to estimate the optimal value to (1+epsilon) accuracy. We prove that any single-pass streaming algorithm that achieves a (p-epsilon)-approximation requires Omega_epsilon(sqrt n) space. Our proof is via a reduction from lower bounds for a communication problem that is a p-ary variant of the Boolean Hidden Matching problem studied in the literature. Given the utility of Unique Games as a starting point for reduction to other optimization problems, our strong hardness for approximating Unique Games could lead to downstream hardness results for streaming approximability for other CSP-like problems
Dynamic Graph Stream Algorithms in Space
In this paper we study graph problems in dynamic streaming model, where the
input is defined by a sequence of edge insertions and deletions. As many
natural problems require space, where is the number of
vertices, existing works mainly focused on designing space
algorithms. Although sublinear in the number of edges for dense graphs, it
could still be too large for many applications (e.g. is huge or the graph
is sparse). In this work, we give single-pass algorithms beating this space
barrier for two classes of problems.
We present space algorithms for estimating the number of connected
components with additive error and
-approximating the weight of minimum spanning tree, for any
small constant . The latter improves previous
space algorithm given by Ahn et al. (SODA 2012) for connected graphs with
bounded edge weights.
We initiate the study of approximate graph property testing in the dynamic
streaming model, where we want to distinguish graphs satisfying the property
from graphs that are -far from having the property. We consider
the problem of testing -edge connectivity, -vertex connectivity,
cycle-freeness and bipartiteness (of planar graphs), for which, we provide
algorithms using roughly space, which is
for any constant .
To complement our algorithms, we present space
lower bounds for these problems, which show that such a dependence on
is necessary.Comment: ICALP 201
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