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Series: ACO Student Seminar

Thomassen proved that there are only finitely many 6-critical graphs
embeddable on a fixed surface. He also showed that planar graphs are
5-list-colorable. We develop new techniques to prove a general theorem
for 5-list-coloring graphs embedded on a fixed surface. Namely, for every surface S and every integer C > 0, there exists D
such that the following holds: Let G be a graph embedded in a surface S
with edge-width at least D and a list assignment L such that, for every
vertex v in G, L(v) has size at least five. If F is a collection of any
number of facial cycles of length at most C such that every two cycles
in F are distance at least D apart and every cycle in F has a locally
planar neighborhood up to distance D/2, then any proper L-coloring of F
extends to an L-coloring of G.
This theorem implies the following results. In what follows, let S be a
fixed surface, G be a graph embedded in S (except in 4, where G is drawn
in S) and L a list assignment such that, for every vertex v of G, L(v)
has size at least five.
1. If G has large edge-width, then G is 5-list-colorable. (Devos, Kawarabayashi and Mohar)
2. There exists only finitely many 6-list-critical graphs embeddable in
S. (Conjectured by Thomassen, Proof announced by Kawarabayashi and
Mohar) As a corollary, there exists a linear-time algorithm for deciding
5-list-colorability of graphs embeddable on S. Furthermore, we exhibit
an explicit bound on the size of such graphs.
3. There exists D(S) such that the following holds: If X is a subset of
the vertices of G that are pairwise distance at least D(S) apart, then
any L-coloring of X extends to an L-coloring G. For planar graphs, this
was conjectured by Albertson and recently proved by Dvorak, Lidicky,
Mohar, and Postle.
4. There exists D(S) such that the following holds: If G is a graph
drawn in S with face-width at least D(S) such that any pair of crossings
is distance at least D apart, then G is L-colorable. For planar graphs,
this was recently proved by Dvorak, Lidicky and Mohar.
Joint work with Robin Thomas.

Series: ACO Student Seminar

A mixed integer point is a vector in $\mathbb{R}^n$ whose first $n_1$ coordinates are integer. We present necessary and sufficient conditions for the convex hull of mixed integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones and strictly convex sets. Furthermore, by using these results, we show that there exists a polynomial time algorithm to check the closedness of the convex hull of the mixed integer points contained in the feasible region of a second order conic programming problem, for the special case this region is defined by just one Lorentz cone and one rational matrix. This is joint work with Santanu Dey.

Series: ACO Student Seminar

Consider a series of n single-server queues each with unlimited waiting
space and FIFO discipline. At first the system is empty, then m customers
are placed in the first queue. The service times of all the customers at
all the queues
are iid. We are interested in the exit time of the last customer from the
last
server and that's when queues meet random matrices and
GUEs. If the number of customers
is a small fractional power of the number of servers, and as such
customers stay within a tube, that's when queues encounter Tracy and Widom.
This talk will be self contained and accessible to graduate students.

Series: ACO Student Seminar

In a celebrated result, Raghavendra [Rag08] showed that, assuming Unique Game Conjecture, every Max-CSP problem has a sharp approximation threshold that matches the integrality gap of a natural SDP relaxation. Raghavendra and Steurer [RS09] also gave a simple and unified framework for optimally approximating all the Max-CSPs.In this work, we consider the problem of approximating CSPs with global cardinality constraints. For example, Max-Cut is a boolean CSP where the input is a graph $G = (V,E)$ and the goal is to find a cut $S \cup \bar S = V$ that maximizes the number of crossing edges, $|E(S,\bar S)|$. The Max-Bisection problem is a variant of Max-Cut with an additional global constraint that each side of the cut has exactly half the vertices, i.e., $|S| = |V|/2$. Several other natural optimization problems like Small Set Expansion, Min Bisection and approximating Graph Expansion can be formulated as CSPs with global constraints.In this talk, I will introduce a general approach towards approximating CSPs with global constraints using SDP hierarchies. To demonstrate the approach, I will present an improved algorithm for Max-Bisection problem that achieves the following:- Given an instance of Max-Bisection with value $1-\epsilon$, the algorithm finds a bisection with value at least $1-O(\sqrt{\epsilon})$ with running time $O(n^{poly(1/\eps)})$. This approximation is near-optimal (up to constant factors in $O()$) under the Unique Games Conjecture.- Using computer-assisted proof, we show that the same algorithm also achieves a 0.85-approximation for Max-Bisection, improving on the previous bound of 0.70 (note that it is UGC-hard to approximate better than 0.878 factor). As an attempt to prove matching hardness result, we show a generic conversion from SDP integrality gap to dictatorship test for any CSP with global cardinality constraints. The talk is based on joint work with Prasad Raghavendra.

Series: ACO Student Seminar

I present a new class of vertex cover and set cover games, with the price of anarchy bounds matching the best known
constant factor approximation guarantees for the centralized optimization problems for linear and also for submodular
costs. In particular, the price of anarchy is 2 for vertex cover. The
basic intuition is that the members of the vertex cover form a Mafia
that has to "protect" the graph, and may ask ransoms from their
neighbors in exchange for the protection. These ransoms turn out to
capture a good dual solution to the linear programming relaxation. For linear costs we also exhibit
linear time best response dynamics that converge that mimic the classical greedy approximation algorithm of Bar-Yehuda
and Even. This is a joint work with Georgios Piliouras and Tomas Valla.

Series: ACO Student Seminar

Recently, there has been great interest in understanding the fundamental
limits of our ability to sample from the independent sets (i.s.) of a
graph. One approach involves the study of the so-called hardcore model,
in which each i.s. is selected with probability proportional to some fixed
activity $\lambda$ raised to the cardinality of the given i.s. It is
well-known that for any fixed degree $\Delta$, there exists a critical
activity $\lambda_{\Delta}$ s.t. for all activities below
$\lambda_{\Delta}$, the sampled i.s. enjoys a long-range independence
(a.k.a. uniqueness) property when implemented on graphs with maximum
degree $\Delta$, while for all activities above $\lambda_{\Delta}$, the
sampled i.s. exhibits long-range dependencies. Such phase transitions are
known to have deep connections to the inherent computational complexity of
the underlying combinatorial problems. In this talk, we study a family of
measures which generalizes the hardcore model by taking more structural
information into account, beyond just the number of nodes belonging to the
i.s., with the hope of further probing the fundamental limits of what we
can learn about the i.s. of a graph using only local information. In our
model, the probability assigned to a given i.s. depends not only on its
cardinality, but also on how many excluded nodes are adjacent to exactly
$k$ nodes belonging to the i.s., for each $k$, resulting in a parameter
for each $k$. We generalize the notion of critical activity to these
``neighborly measures", and give necessary and sufficient conditions for
long-range independence when certain parameters satisfy a
log-convexity(concavity) requirement. To better understand the phase
transitions in this richer model, we view the classical critical activity
as a particular point in the parameter space, and ask which directions can
one move and still maintain long-range independence. We show that the set
of all such ``directions of uniqueness” has a simple polyhedral
description, which we use to study how moving along these directions
changes the probabilities associated with the sampled i.s. We conclude by
discussing implications for choosing how to sample when trying to optimize
a linear function of the underlying probabilities.

Series: ACO Student Seminar

In this seminar, I will talk about a few recent developments in the random colorings, random weighted independent sets and other 2-spin
models on different classes of graphs such as the square lattices and the triangular free graphs. I will focus on the so-called spatial mixing property of these models and discuss about the consequences (e.g., fast mixing of the Markov chains) of the spatial mixing property
as well as the techniques of proving it.

Series: ACO Student Seminar

I will show a new approach based on the discrepancy of the constraint
matrix to verify integer feasibility of polytopes. I will then use this
method to show a threshold phenomenon for integer feasibility of random
polytopes.
The random polytope model that we consider is P(n,m,x0,R) - these are
polytopes in n-dimensional space specified by m "random" tangential
hyperplanes to a ball of radius R
centered around the point x0. We show
that there exist constants c_1 < c_2 such
that with high probability, the random polytope
P(n,m,x0=(0.5,...,0.5),R) is integer infeasible if R is less than
c_1sqrt(log(2m/n))
and the random polytope P(n,m,x0,R) is integer feasible for every center
x0 if the radius R is at least c_2sqrt(log(2m/n)). Thus, a
transition from infeasibility to feasibility happens within a constant
factor
increase in the radius. Moreover, if the polytope contains a ball
of radius Omega(log (2m/n)), then we can find an integer solution with
high probability (over the input) in randomized polynomial time.
This is joint work with Santosh Vempala.

Series: ACO Student Seminar

I will define planted distributions of random structures and give plenty of examples in different contexts: from balls and bins, to random permutations, to random graphs and CSP's. I will
give an idea of how they are used and why they are interesting. Then
I'll focus on one particular problem: under what conditions can you
distinguish a planted distribution from the standard distribution on a random structure and how can you do it?

Series: ACO Student Seminar

I'll give a high-level tour of how lattices are providing a powerful new mathematical foundation for cryptography. Lattices provide simple, fast, and highly parallel cryptoschemes that, in contrast with many of today's popular methods (like RSA and elliptic curves), even appear to remain secure against quantum computers.
No background in lattices, cryptography, or quantum computers will be
necessary -- you only need to know how to add and multiply vectors and
matrices.