A sample of FOCS19 presentations
As usual,
I am posting brief comments on a few presentations that I attended,
with the usual caveats asserting that some omission
reflect my own limitations rather than anything else.
Here is a list of the presentation I will comment on.

Derandomization from Algebraic Hardness: Treading the Borders
by Zeyu Guo, Mrinal Kumar, Ramprasad Saptharishi, and Noam Solomon

Sampling Graphs without Forbidden Subgraphs
and Unbalanced Expanders with Negligible Error
by Benny Applebaum and Eliran Kachlon

ReedMuller codes polarize by Emmanuel Abbe and Min Ye

Lower bounds for maximal matchings and maximal independent sets
by Alkida Balliu, Sebastian Brandt, Juho Hirvonen, Dennis Olivetti,
Mikal Rabie, and Jukka Suomela

New Notions and Constructions of Sparsification for Graphs and Hypergraphs
by Nikhil Bansal, Ola Svensson, and Luca Trevisan

Efficient Construction of Rigid Matrices Using an NP Oracle
by Josh Alman and Lijie Chen

Truly Optimal Euclidean Spanners
by Hung Le and Shay Solomon

Hardness Magnification for all Sparse NP Languages
by Lijie Chen, Ce Jin, and Ryan Williams

The AverageCase Complexity of Counting Cliques in ErdosRenyi
Hypergraphs by Enric BoixAdsera, Matthew Brennan, and Guy Bresler

Nondeterministic QuasiPolynomial Time is Averagecase Hard
for ACC Circuits by Lijie Chen

Finding monotone patterns in sublinear time
by Omri BenEliezer, Clement L. Canonne, Shoham Letzter, and Erik Waingarten.
(No comments here; see link).

Agreement testing theorems on layered set systems
by Yotam Dikstein and Irit Dinur

A characterization of graph properties testable for general planar graphs
with onesided error (It's all about forbidden subgraphs)
by Artur Czumaj and Christian Sohler

Juntacorrelation is testable
by Anindya De, Elchanan Mossel, and Joe Neeman.
(No comments here; see link).
See also my report on the event
Why I Am Excited About This Research Direction:
A Tribute to Shafi Goldwasser.
Derandomization from Algebraic Hardness: Treading the Borders
by Zeyu Guo, Mrinal Kumar, Ramprasad Saptharishi, and Noam Solomon
Eventually, it turned out (in a subsequent work by some of the authors)
that there is no need to tread the borders.
Anyhow, the result is a tight conversion of hardness
into pseudorandomness in the form of a hitting set generator.
It also turns out that any construction of a nontrivial hitting set
(i.e., one that is smaller than the trivial by one unit)
yields a hitting set of optimal size (up to a polynomial relation).
Sampling Graphs without Forbidden Subgraphs
and Unbalanced Expanders with Negligible Error
by Benny Applebaum and Eliran Kachlon
I have already recommended this interesting work
(see 259),
but wish to reiterate the main question addressed here.
For a fixed graph $H$ and parameters $n,m$ (and $k$),
we wish to generate an $n$vertex graph with $m$ edges
that is $H$free. Furthermore, we want a distribution
of $k$wise independent graphs such that with overwhelming
probability the graph is $H$free (i.e., the graph is not $H$free
with negligible probability). Note that, for $m=Theta(n)$,
a truely random graph will not be $H$free with noticeable probability.
This computational problem is solved by employing a pseudorandom
construction that supports superfast checking of being $H$free
(via a computation on the seed of the generator rather than
by running a search on the resulting graph).
ReedMuller codes polarize by Emmanuel Abbe and Min Ye
My takeaway from this presentation was a clear description
of the polarization theory. For an $n$by$n$ matrix $M$,
it considers the process of applying $M$ to a random vector $u$,
subjecting the result to noise, yielding a vector $uM+e$,
and iteratively decoding the bits of $u$
by scanning the bits $(uM+e)M^{1}$.
On the average, in the $i$th iteration, we have $I_i=1H_2(\eta)$ units
of information regarding $u_i$, and that point is that for suitable
matrices almost each of the $i$'s is either close to zero or to one.
If that is the case, then a good encoding is obtained by placing
the plaintext at the $i$ for which $I_i$ is close to 1.
(Warning: Finding this set of indices may not be easy,
but a polynomialtime algorithm is known.)
Lower bounds for maximal matchings and maximal independent sets
by Alkida Balliu, Sebastian Brandt, Juho Hirvonen, Dennis Olivetti,
Mikal Rabie, and Jukka Suomela
I have already recommended this interesting line of works:
see my comments on Jukka Suomela's spotlight presentation
at WOLA'19.
The current work is arguablly the crown jewel of that line, so far.
New Notions and Constructions of Sparsification for Graphs and Hypergraphs
by Nikhil Bansal, Ola Svensson, and Luca Trevisan
I vaguely recall thinking of this notion of a cutsparsifier two decades ago,
but I definitely got nowhere.
The point is that this work (like my own thinking)
did not allow for weighted subgraphs and aimed only
at graphs that admit an unweighted cutsparsifier.
The work suggests a weaker notion that in some cases coincides
with the natural notion of an unweighted cutsparsifier,
but can be achieved for all graphs.
Efficient Construction of Rigid Matrices Using an NP Oracle
by Josh Alman and Lijie Chen
Its not that I care so much about this specific problem
(of obtaining rigidity at this range of parameters),
let alone the weak notion of explicitness (mod an NP oracle),
but I was stunned by the insights that underlie this work.
The authors envision and materialize Ryan Williams'
(of deriving lower bounds by algorithmic improvements)
in a domain that seems totally unrelated. Of course,
the construction of rigid matrices is a derandomization problem,
but it is not one that refers to a computational model.
Still, they manage to derive a corresponding computational model
and apply Ryan's method to it, which is indeed highly nontrivial
given that this computational model is quite weird and restricted
(so one has to transform all ingrediants of the method accordingly).
Truly Optimal Euclidean Spanners by Hung Le and Shay Solomon
Given $n$ points in the plane
(or in a higher dimensional Euclidean space),
the problem is to construct an $n$vertex weighted graph
that approximately represents the Euclidean distances.
The first result in this work is that the sparsity parameter
achieved by the greedy spanner is optimal, and that its
lightness parameter can be improved to the optimal.
The second result is that one can improve over the
foregoing bounds by allowing Steiner points.
Hardness Magnification for all Sparse NP Languages
by Lijie Chen, Ce Jin, and Ryan Williams
Hardness magnification means showing that seemingly weak
lower bounds imply very strong lower bounds.
Several results of this type have been proved in recent years
for various models and computational problems
(including works on quantified derandomization
such as Roei Tell's these this).
Most relevant to the current work is the recent work
Weak Lower Bounds on ResourceBounded Compression Imply
Strong Separations of Complexity Classes (by McKay, Murray, and Williams).
What the current work shows is that the results obtained in the latter
work using the Min Circuit Size Problem and its variants
are actually an artifact of these problems referring to sparse sets.
Specifically, they show results of the following form:
If a (subexp) sparse NPset has no $n^{1.01}$size circuits,
then it has no $n^{100}$size circuits.
The AverageCase Complexity of Counting Cliques in ErdosRenyi Hypergraphs
by Enric BoixAdsera, Matthew Brennan, and Guy Bresler
My own focus is on the case of graphs. For this case, this work
provides a worstcase to averagecase reduction of the problem
of counting kcliques, where the averagecase result refers to
the ErdosRenyi distribution (with various edge densities).
This result should be compared to my own work
with Guy Rothblum. On the one hand, the distribution used
in the current work is far simpler (than the sampleable distribution
used in the prior work). On the other hand, the current work refers
to algorithms that err with low probability tending to zero,
whereas the prior work refers to algorithms that err with high
probability tending to one. (In both cases, the rate of convergence
is the reciprocal of a polylogarithmic function).
Nondeterministic QuasiPolynomial Time is Averagecase Hard
for ACC Circuits by Lijie Chen
Prior works have shown worstcase ACChardness for NQP
(see this),
and averagecase ACChardness for NEXP.
The current work obtains the best (or worst) of these results,
proving averagecase ACChardness for NQP.
The proof strategy follows Williams' celebrated method
(of deriving lower bounds by algorithmic improvements)
but runs against many difficulties that arise from the
averagecase setting (most notoriously in proving
an adequate easy witness lemma).
Agreement testing theorems on layered set systems
by Yotam Dikstein and Irit Dinur
An agreement test is specified by a distribution on pairs
of sets in a set system (or by a graph in which these sets are
vertices and considering the uniform distribution over edges).
Such a test is intended to test whether partial assignments
(i.e., local views of the assignments to sets)
given to the individual sets fits some global assignment.
The test picks a pair of sets according to this distribution
and accepts if and only if the assignments to elements in
the intersection is the same in both sets.
The performance of the test is the function relating
its rejection probability to the disagreement rate
with the best global function; that is, in the current regime,
the aim is showing that if the test rejects with probability $\eps$,
then there exist a global assignment that agrees with $1O(\eps)$
of the local views (i.e., the assignments to sets).
This works shows that a sufficient condition for such a good performance.
A characterization of graph properties testable for general planar graphs
with onesided error (It's all about forbidden subgraphs)
by Artur Czumaj and Christian Sohler
One commendable aspect of this work is that it studies the
general graph model, which is aimed at dealing with sparse graphs
that lack a feasible degree bound.
The main result is a (onesided error) tester of sizeoblivious complexity
for subgraphfreeness under the promise that the tested graph is planar.
The point being that the straightforward tester (which takes a BFS
of depth bounded by the radius of the forbidden subgraph)
that works in the boundeddegree model cannot be used here.
Instead one takes a ``randomly trimmed'' BFS such that
for each vertex in the frontier we only extend a bounded
number of randomly selected vertices.
Of course, the issue is proving that this works,
and planarity is used in that analysis,
which begs the question of what is actually needed.
(Of course, some promise is required, since no sizeoblivious
algorithm can distinguish a $n$cycle from an $(nsqrt(n))$cycle
that is connected to a sqrt(n)sized clique.)
See 60th FOCS.
Back to
list of Oded's choices.