34 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Improved Hardness of Approximating k-Clique under ETH
In this paper, we prove that assuming the exponential time hypothesis (ETH),
there is no -time algorithm that can decide
whether an -vertex graph contains a clique of size or contains no clique
of size , and no FPT algorithm can decide whether an input graph has a
clique of size or no clique of size , where is some function
in . Our results significantly improve the previous works [Lin21,
LRSW22]. The crux of our proof is a framework to construct gap-producing
reductions for the -Clique problem. More precisely, we show that given an
error-correcting code that is locally testable
and smooth locally decodable in the parallel setting, one can construct a
reduction which on input a graph outputs a graph in time such that:
If has a clique of size , then has a clique of size
, where .
If has no clique of size , then has no clique of size
for some constant .
We then construct such a code with and
, establishing the hardness results above.
Our code generalizes the derivative code [WY07] into the case with a super
constant order of derivatives.Comment: 48 page
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
Sketching Approximability of (Weak) Monarchy Predicates
We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first variable (the president) and the sum of the votes of all remaining variables. The pure version of this function is when the president can only be overruled by when all remaining variables agree. For every k ? 5, we show that CSPs where the underlying predicate is a pure monarchy function on k variables have no non-trivial sketching approximation algorithm in o(?n) space. We also show infinitely many weaker monarchy functions for which CSPs using such constraints are non-trivially approximable by O(log(n)) space sketching algorithms. Moreover, we give the first example of sketching approximable asymmetric Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan, and Velusamy (FOCS 2021) that characterizes the sketching approximability of all CSPs. Their framework can be applied naturally to get a computer-aided analysis of the approximability of any specific constraint satisfaction problem. The novelty of our work is in using their work to get an analysis that applies to infinitely many problems simultaneously
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
Streaming complexity of CSPs with randomly ordered constraints
We initiate a study of the streaming complexity of constraint satisfaction
problems (CSPs) when the constraints arrive in a random order. We show that
there exists a CSP, namely , for which random ordering
makes a provable difference. Whereas a approximation of
requires space with adversarial ordering,
we show that with random ordering of constraints there exists a
-approximation algorithm that only needs space. We also give
new algorithms for in variants of the adversarial ordering
setting. Specifically, we give a two-pass space
-approximation algorithm for general graphs and a single-pass
space -approximation algorithm for bounded degree
graphs.
On the negative side, we prove that CSPs where the satisfying assignments of
the constraints support a one-wise independent distribution require
-space for any non-trivial approximation, even when the
constraints are randomly ordered. This was previously known only for
adversarially ordered constraints. Extending the results to randomly ordered
constraints requires switching the hard instances from a union of random
matchings to simple Erd\"os-Renyi random (hyper)graphs and extending tools that
can perform Fourier analysis on such instances.
The only CSP to have been considered previously with random ordering is
where the ordering is not known to change the
approximability. Specifically it is known to be as hard to approximate with
random ordering as with adversarial ordering, for space
algorithms. Our results show a richer variety of possibilities and motivate
further study of CSPs with randomly ordered constraints
Streaming beyond sketching for Maximum Directed Cut
We give an -space single-pass -approximation
streaming algorithm for estimating the maximum directed cut size
() in a directed graph on vertices. This improves over
an -space approximation algorithm due to Chou,
Golovnev, Velusamy (FOCS 2020), which was known to be optimal for
-space algorithms.
is a special case of a constraint satisfaction problem
(CSP). In this broader context, our work gives the first CSP for which
algorithms with space can provably outperform
-space algorithms on general instances. Previously, this was shown
in the restricted case of bounded-degree graphs in a previous work of the
authors (SODA 2023). Prior to that work, the only algorithms for any CSP were
based on generalizations of the -space algorithm for
, and were in particular so-called "sketching" algorithms.
In this work, we demonstrate that more sophisticated streaming algorithms can
outperform these algorithms even on general instances.
Our algorithm constructs a "snapshot" of the graph and then applies a result
of Feige and Jozeph (Algorithmica, 2015) to approximately estimate the
value from this snapshot. Constructing this snapshot is
easy for bounded-degree graphs and the main contribution of our work is to
construct this snapshot in the general setting. This involves some delicate
sampling methods as well as a host of "continuity" results on the
behaviour in graphs.Comment: 57 pages, 2 figure
Sketching Approximability of (Weak) Monarchy Predicates
We analyze the sketching approximability of constraint satisfaction problems
on Boolean domains, where the constraints are balanced linear threshold
functions applied to literals. In~particular, we explore the approximability of
monarchy-like functions where the value of the function is determined by a
weighted combination of the vote of the first variable (the president) and the
sum of the votes of all remaining variables. The pure version of this function
is when the president can only be overruled by when all remaining variables
agree. For every , we show that CSPs where the underlying predicate
is a pure monarchy function on variables have no non-trivial sketching
approximation algorithm in space. We also show infinitely many
weaker monarchy functions for which CSPs using such constraints are
non-trivially approximable by space sketching algorithms.
Moreover, we give the first example of sketching approximable asymmetric
Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan,
and Velusamy (FOCS 2021) that characterizes the sketching approximability of
all CSPs. Their framework can be applied naturally to get a computer-aided
analysis of the approximability of any specific constraint satisfaction
problem. The novelty of our work is in using their work to get an analysis that
applies to infinitely many problems simultaneously