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Series: ACO Colloquium

Refreshments at 10:30am in the atrium outside Skiles 006

The traveling salesman problem (TSP) is the most famous problem in
discrete optimization. Given $n$ cities and the costs $c_{ij}$ for
traveling from city $i$ to city $j$ for all $i,j$, the goal of the
problem is to find the least expensive tour that visits each city
exactly once and returns to its starting point. We consider cases in
which the costs are symmetric and obey the triangle inequality. In
1954, Dantzig, Fulkerson, and Johnson introduced a linear programming
relaxation of the TSP now known as the subtour LP, and used it to find
the optimal solution to a 48-city instance. Ever since then, the
subtour LP has been used extensively to find optimal solutions to TSP
instances, and it is known to give extremely good lower bounds on the
length of an optimal tour.
Nevertheless, the quality of the subtour LP bound is poorly understood
from a theoretical point of view. For 30 years it has been known that
it is at least 2/3 times the length of an optimal tour for all instances
of the problem, and it is known that there are instances such that it
is at most 3/4 times the length of an optimal tour, but no progress has
been made in 30 years in tightening these bounds.
In this talk we will review some of the results that are known about the
subtour LP, and give some new results that refine our understanding in
some cases. In particular, we resolve a conjecture of Boyd and Carr
about the ratio of an optimal 2-matching to the subtour LP bound in the
worst case. We also begin a study of the subtour LP bound for the
extremely simple case in which all costs $c_{ij}$ are either 1 or 2.
For these instances we can show that the subtour LP is always strictly
better than 3/4 times the length of an optimal tour.
These results are joint work with Jiawei Qian, Frans Schalekamp, and Anke van Zuylen.

Series: ACO Colloquium

Motivated by mass-spectrometry protein sequencing,
we consider the simple problem of reconstructing
a string from its substring compositions. Relating
the question to the long-standing turnpike problem,
polynomial factorization, and cyclotomic polynomials,
we cleanly characterize the lengths of reconstructable
strings and the structure of non-reconstructable ones.
The talk is elementary and self contained and covers
work with Jayadev Acharya, Hirakendu Das, Olgica
Milenkovic, and Shengjun Pan.

Series: ACO Colloquium

***Refreshments at 4PM in Skiles 236.***

Forbidding (undirected or directed) paths in graphs, what can be easier? Yet we show that in the context of coloring problems (CSP) and structural graph theory, this is related to the notions tree depth, (restricted) dualities, bounded expansion and nowhere dense classes with applications both in and out of combinatorics.

Series: ACO Colloquium

We establish the existence of scaling limits for several combinatorial optimization models on Erdos-Renyi and sparse random regular graphs. For a variety of models, including maximum independent sets, MAX-CUT, coloring and K-SAT, we prove that the optimal value appropriately rescaled, converges to a limit with high probability (w.h.p.), as the size of the underlying graph divergesto infinity. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. thus resolving an open problem. Our approach is based on developing a simple combinatorial approach to an interpolation method developed recently in the statistical physics literature. Among other things, theinterpolation method was used to prove the existence of the so-called free energy limits for several spin glass models including Viana-Bray and random K-SAT models. Our simpler combinatorial approach allows us to work with the zero temperature case (optimization) directly and extend the approach to many other models. Additionally, using our approach, we establish the large deviationsprinciple for the satisfiability property for constraint satisfaction problems such as coloring, K-SAT and NAE(Not-All-Equal)-K-SAT. The talk will be completely self-contained. No background on random graph theory/statistical physics is necessary. Joint work with Mohsen Bayati and Prasad Tetali

Series: ACO Colloquium

Refreshments at 4:00PM in Skiles 236

I will talk about new approximation algorithms for the Asymmetric Traveling Salesman Problem (ATSP) when the costs satisfy the triangle inequality. Our approach is based on constructing a "thin" spanning tree from the solution of a classical linear programming relaxation of the problem and augmenting the tree to an Eulerian subgraph. I will talk about Goddyn's conjecture on the existence of such trees and its relations to nowhere-zero flows. I will present an O(log n/log log n) approximation algorithm that uses a new randomized rounding method. Our rounding method is based on sampling from a distribution and could be of independent interest. Also, I will talk about the special case where the underlying undirected graph of the LP relaxation of the problem has bounded genus. This is the case for example, when the distance functions are shortest paths in a city with few bridges and underpasses. We give a constant factor approximation algorithm in that case. The first result is a joint work with A. Asadpour, M. Goemans, A. Madry and S. Oveis Gharan, and the second result is a joint work with S. Oveis Gharan.

Series: ACO Colloquium

In my talk I will give a survey on the rise and early development of Extremal Graph Theory, one of the large areas in Discrete Mathematics.I will give a description of the asymptotic solution of extremal graph problems for ordinary graphs, describe the stability method and expose the difficulties connected to hypergraph extremal problems.I will expose several unsolved problems in the field, and move on to some new results.I will also describe the connection of the field to several other areas of Discrete Mathematics, like to Ramsey Theory,Random graphs, Regularity lemma, Quasi-randomness, etc.I will also mention some applications of extremal graph theory. The lecture will be a non-technical one.***Refreshments at 4PM in Skiles 236.***

Series: ACO Colloquium

The past 10 years have seen a confluence of research in sparse approximation amongst computer science, mathematics, and electrical engineering. Sparse approximation encompasses a large number of mathematical, algorithmic, and signal processing problems which all attempt to balance the size of a (linear) representation of data and the fidelity of that representation. I will discuss several of the basic algorithmic problems and their solutions, focusing on special classes of matrices. I will conclude with an application in biological testing.

Series: ACO Colloquium

One of the most interesting aspects of the Linear Complementarity Problem (LCP) is its range from relatively easy problems such as linear and convex quadratic programming problems to NP-hard problems. A major effort in LCP theory had been the study of the Lemke algorithm, a simplex-like algorithm which is guaranteed to terminate in finite number of iterations but not necessarily with a solution (or a certificate that no solution exists). Over the years, many subclasses of LCP were proven to be workable by the Lemke algorithm. Those subclasses were often characterized as ‘nice’ even when no polynomial upper bound for the algorithm was known to exist. In fact, for most of these classes, instances with exponential number of steps had been discovered. In this talk, I’ll discuss the close connection between these classes and the PPAD (Polynomial-time Parity Argument Directed) complexity class. The discovery that computing Nash equilibrium (which is an LCP) is PPAD complete is particularly significant in analyzing the complexity of LCP. I’ll also discuss the LCP reduction-via-perturbation technique and its relation to the PPAD class and the Lemke Algorithm.
This talk is based on a joint work with Sushil Verma.

Series: ACO Colloquium

We describe recent progress in the study of the binary deletion channel and related channels with synchronization errors, including a clear description of many open problems in this area. As an example, while the capacity of the binary symmetric error channel and the binary erasure channel have been known since Shannon, we still do not have a closed-form description of the capacity of the binary deletion channel. We highlight a recent result that shows that the capacity is at least (1-p)/9 when each bit is deleted independently with fixed probability p.