Monday, Jan 5, 2015 | |||
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10:00 - 10:10 | Opening remarks | ||

10:10 - 11:00 | Neeraj Kayal | A survey of recent results on depth reduction and lower bounds in arithmetic complexity. | |

11:00 - 11:30 | Coffee Break | ||

11:30 - 11:55 | Srikanth Srinivasan | Lower bounds for non-commutative skew circuits. | |

12:00 - 12:25 | Nutan Limaye | The shifted partial derivative complexity of elementary symmetric polynomials. | |

12:30 - 14:00 | Lunch | ||

14:00 - 14:50 | Chandan Saha | Lower bounds for small depth arithmetic circuits. | |

14:50 - 15:30 | Coffee Break | ||

15:30 - 16:00 | Sagnik Mukhopadhyay | Tribes is hard in the message passing model | |

16:00 - 16:30 | Swagato Sanyal | Fourier dimension and sparsity | |

16:30 - 17:00 | Coffee |

Tuesday, Jan 6, 2015 | ||
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10:00 - 10:50 | Leslie Goldberg | Phase transitions and the complexity of approximating partition functions. |

10:50 - 11:30 | Coffee Break | |

11:30 - 12:25 | Stanislav Zivny | The complexity of Valued CSPs. |

12:30 - 14:00 | Lunch | |

14:00 - 14:50 | Discussion | |

14:50 - 15:30 | Coffee Break | |

15:30 - 16:20 | Iain Stewart | A survey of logical reductions between problems. |

16:30 - 17:30 | Open Problems Session | |

17:30 - 18:00 | Coffee | |

19:00 - 21:30 | Workshop Dinner |

Wednesday, Jan 7, 2015 | ||
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10:00 - 10:50 | Kousha Etessami | Fixed Point Computation Problems for Algebraically-Defined Functions, and their Computational Complexity. |

10:50 - 11:30 | Coffee Break | |

11:30 - 12:20 | Samir Datta | The Dynamic Complexity of Reachability and related problems. |

12:30 - 12:55 | Rahul Savani | The Complexity of the Simplex Method. |

13:00 - 14:00 | Lunch |

Thursday, Jan 8, 2015 | |||
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10:00 - 10:50 | Olaf Beyersdorff | Proof complexity. | |

10:50 - 11:30 | Coffee Break | ||

11:30 - 11:55 | Leroy Chew | Proof complexity of resolution-based QBF calculi. | |

12:00 - 12:25 | Ruiwen Chen | Satisfiability algorithms and lower bounds for boolean formulas. | |

12:30 - 14:00 | Lunch | ||

14:00 - 14:50 | Anuj Dawar | Symmetric Circuits. | |

14:50 - 15:30 | Coffee Break | ||

15:30 - 16:00 | Balagopal Komarath | Comparator Circuits over Lattices | |

16:00 - 16:30 | Arpita Korwar | Polynomial Identity Testing for a sum of ROABPs | |

16:30 - 17:00 | Coffee |

Friday, Jan 9, 2015 | ||
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10:00 - 10:50 | Prahladh Harsha | Role of composition in PCP constructions. |

10:50 - 11:30 | Coffee Break | |

11:30 - 12:00 | Arkadev Chattopadhyay | The power of super-logarithmic number of players. |

12:10 - 12:55 | Jayalal Sarma | Weighing Schemes and NL vs UL problem. |

12:55 - 13:00 | Closing Remarks | |

13:00 | Lunch |

Some of the open slots will be dynamically allotted for talks by graduate students.

In the `Number-on-Forehead' (NOF) model of multiparty communication,
the input is a k times m boolean matrix A (where k is the number of
players) and Player i sees all bits except those in the i-th row, and
the players communicate by broadcast in order to evaluate a specified
function f at A. We discover new computational power when k exceeds
log m. We give a protocol with communication cost poly-logarithmic in
m, for block composed functions with limited block width. These are
functions of the form f o g where f is a symmetric b-variate function,
and g is a (kr)-variate function and (f o g)(A) is defined, for a k
times br matrix to be f(g(A^1),\ldots,g(A^b)) where A^i is the i-th k
times r block of A. Our protocol works provided that k > 1+ \ln b +
2^r. Ada et.al previously obtained *simultaneous* and deterministic
efficient protocols forcomposed functions of block-width r=1. The new
protocol is the first to work for block composed functions with
r>1. Moreover, it is simultaneous, with vanishingly small error
probability, if public coin randomness is allowed. The deterministic
and zero-error version barely uses interaction.

This is joint work with Michael Saks.

The random restriction approach was initially used to prove formula size lower bounds, where it was shown that formulas shrink on average under random restrictions. In several recent work, this approach was revived and improved to the concentrated version where formulas shrink with high probability. This concentrated shrinkage property leads to nontrivial satisfiability algorithms, average-case lower bounds, and pseudorandom generators. In this talk, we will survey techniques in proving concentrated shrinkage, and its applications in designing algorithms and proving lower bounds.

We are interested in Boolean circuits that decide properties of graphs
(or similar relational structures). In particular, the output of the
circuit is invariant under permutations of the inputs corresponding to
permutations of the vertices. We consider symmetric circuits -
i.e.those where this invariance is witnessed by automorphisms of the
circuit, and show a close connection between these and definability in
logic. This provides a means of establishing lower bounds for
symmetric circuits and, e.g. separating Boolean symmetric circuits
from those with threshold gates.

This is joint work with Matthew Anderson.

A variety of fundamental computational problems in many fields (in
game theory, economics, stochastic processes, verification, and other
areas), which on the surface may look quite different, can all be
boiled down to computing or approximating a fixed point for an
algebraically-defined function which maps a subset of R^n to itself,
and which is specified by an algebraic circuit (or formula) using
standard gates from among { + , *, - , / , max , min } and using
rational coefficients and constants.

The problems in question share the property that the *existence* of a
fixed point is not in doubt: it is guaranteed by some classic fixed
point theorem (e.g., Brouwer's, Banach's, Tarski's, etc.). But the
proofs of existence do not in general yield an efficient way to find a
fixed point. For many of these problems, we neither have efficient
(P-time) algorithms, nor do we know them to be NP-hard. Indeed, total
search complexity classes such as Papadimitriou's PPAD (=
piecewise-linear-FIXP) and FIXP have been developed to characterize
their complexity.

For some problems in FIXP, by now we know that any non-trivial
approximation of an actual fixed point, even in NP, would resolve long
standing open problems in arithmetic-vs.-Turing complexity, in
particular the PosSLP problem and the square-root-sum problem. For
other such problems, by now we have P-time approximation algorithms,
using a variety of methods (including variants of Newton's method
combined with linear programming).

What makes such a fixed point problem "hard" or "easy"?

In this talk, I will survey a number of such problems, and I will
delineate a taxonomy of the (relative) complexity of such problems,
based on both their combinatorial and their numerical "difficulty",
and based on the nature of their defining algebraic circuits.

This talk is based on joint works in a series of papers with Mihalis Yannakakis and with Alistair Stewart.

A "partition function" is a generating function for counting combinatorial objects. For example, the partition function of the so-called "hard-core lattice gas model" counts the weighted independent sets of a graph, where the weight of an independent set is an (exponential) function of its size. Recent results have shown deep connections between the complexity of approximating such partition functions and phase transitions which arise when the relevant models are viewed on infinite trees. I will tell you about some of the results that are known, focussing especially on the basic connections between phase transitions and inapproximability.

Proof composition is an essential ingredient of all known constructions of probabilistically checkable proofs (PCPs), first introduced by Arora and Safra in the proof of the PCP Theorem. Informally speaking, proof composition is a recursive procedure applied to PCP constructions to reduce the alphabet size. Proof composition is applied (possibly several times over) to PCPs over the large alphabet to obtain PCPs over a small (even binary) alphabet. In the last decade, our understanding of proof composition has considerably improved. (1) Composition of PCPs with high soundness error (greater than 1/2) is by now well understood using the notion of PCPs of proximity. These allow for modular composition which in turn led to Dinur's alternate proof of the PCP Theorem and constructions of shorter PCPs. (2) Compositions of PCPs with low soundness error (less than 1/2) is obtained using the notion of decodable PCPs. These gave rise to PCPs with sub-constant error and nearly linear size improving over parameters previously obtained via parallel repetition theorem. (3) Recently, alternate composition techniques have been proposed towards resolving the sliding scale conjecture.

In this talk, I'll survey these composition techniques and indicate how several improvements in PCP constructions have in fact arisen from a better understanding of proof composition.

Two basic questions in arithmetic complexity are the following:

(i) Can explicit polynomials be efficiently computed?

(ii) Can computation be efficiently parallelized?

We will first motivate these two questions and then survey some
recent results that give some
partial answers.

The shifted partial derivative measure was introduced by Kayal (ECCC
2012) and has been used to prove many strong depth-4 circuit lower
bounds starting from the works of Kayal and Gupta et al (CCC 2013). We
show a strong lower bound on the dimension of the shifted partial
derivative space of the Elementary Symmetric Polynomials (under upper
bounds on their degree). This extends the work of Nisan and Wigderson
(Computational Complexity 1997), who studied the partial derivative
space of these polynomials. Prior to our work, there have been no
results on the shifted partial derivative measure of these
polynomials.

This is a joint work with Hervé Fournier, Meena Mahajan, and Srikanth Srinivasan.

An approach to proving a super-polynomial lower bound for arithmetic
circuits reduces the problem to proving `strong enough' lower bounds for
small depth circuits, in particular (nonhomogeneous) depth-3 circuits and
(homogeneous) depth-4 circuits. Depth of a circuit is the number of layers
of gates in it.

In this talk, we plan to discuss an exponential lower bound for
(homogeneous) depth-4 circuits that comes close to being `strong enough'.
The techniques also yield exponential lower bounds for certain
(nonhomogeneous) depth-3 circuits, in particular depth-3 circuits with low
bottom fanin which also answers a question posed by Shpilka and Wigderson.

Based on joint works with Neeraj Kayal, Srikanth Srinivasan and Nutan Limaye.

For a graph G(V,E) with |V|=n, and a vertex s in V, a weighting scheme
(w : E -> N) is called a min-unique (resp. max-unique) weighting
scheme, if for any vertex v of the graph G, there is a unique path of
minimum(resp.~maximum) weight from s to v. Instead, if the number of
paths of minimum(resp. maximum) weight is bounded by n^c for some
constant c, then the weighting scheme is called a min-poly
(resp. max-poly) weighting scheme.

In this work, we show an unambiguous non-deterministic log-space (UL)
algorithm for the problem of testing reachability in layered directed
acyclic graphs (DAGs) augmented with a min-poly weighting scheme. This
improves the result due to Allender and Reinhardt (2000) where a UL
algorithm was given for the case when the weighting scheme is
min-unique. At the other end of the spectrum, we propose a UL
algorithm for testing reachability in layered DAGs augmented with
max-poly weighting schemes. To achieve this, we first reduce
reachability in DAGs to the longest path problem for DAGs with a
unique source, such that the reduction also preserves the max-poly
property of the graph. We then design a UL algorithm for the longest
path problem on DAGs with a unique source and augmented with a
max-poly weighting scheme.

An important consequence of our results is that, to show NL = UL, it
suffices to design log-space computable min-poly (or max-poly)
weighing schemes for DAGs.

This is joint work with Anant Dhayal (IITM) and Saurabh Sawlani (IITM)

The simplex method is a well-studied and widely-used pivoting method
for solving linear programs. When Dantzig originally formulated the
simplex method, he gave a natural pivot rule that pivots into the
basis a variable with the most violated reduced cost. In their
seminal work, Klee and Minty showed that this pivot rule takes
exponential time in the worst case. We prove two main results on the
simplex method. Firstly, we show that it is PSPACE-complete to find
the solution that is computed by the simplex method using Dantzig's
pivot rule. Secondly, we prove that deciding whether Dantzig's rule
ever chooses a specific variable to enter the basis is
PSPACE-complete. We use the known connection between Markov decision
processes (MDPs) and linear programming, and an equivalence between
Dantzig's pivot rule and a natural variant of policy iteration for
average-reward MDPs. We construct MDPs and show PSPACE-completeness
results for single-switch policy iteration, which in turn imply our
main results for the simplex method.

Joint work with John Fearnley

Abstract
We consider lower bounds for non-commutative arithmetic circuits, i.e.
arithmetic circuits that compute polynomials in the restrictive model
where variables do not commute. Contrary to the more popular
commutative setting, we have strong explicit lower bounds for
non-commutative formulas and more generally, for non-commutative
algebraic branching programs due to Nisan (STOC 1991). However,
proving general non-commutative circuit lower bounds remains open.

It has been observed that Nisan’s hard polynomial is in fact
computable by linear sized "skew" circuits (skew circuits are circuits
where every multiplication gate has the property that all but one of
its children is an input variable or a scalar) and hence, proving
lower bounds for such circuits is a natural question to now consider.
We resolve this question by showing that any non-commutative skew
circuit which computes the square of the polynomial defined by Nisan
must have exponential size.

As a further step towards proving lower bounds for general
non-commutative circuits, we also extend our techniques to prove a
superpolynomial lower bound for any class of circuits that has a
smaller than maximal "non-skew-depth". As far as we know, this is the
strongest model of computation for which we have superpolynomial lower
bounds.

Joint work with Nutan Limaye and Guillaume Malod.

In this survey, I will explain and motivate reductions like first-order translations and, in particular, first-order projections. Taking a logical view of complexity theory, where instead of classifying problems according to the resources required for their solution, we classify according to how difficult it is to specify the problems in logic, we obtain a very different perspective on complexity theory. Amazingly, this perspective dovetails beautifully with more traditional complexity theory. We can associate complexity classes with logics and we can obtain alternative notions of completeness where the reductions involve logical translations.

In this talk, we survey recent results on the computational complexity of optimisation problems that can be cast as Valued Constraint Satisfaction Problems (VCSPs). We will focus on problems that can or provably cannot, assuming standard complexity-theoretic assumptions, be solved optimally in polynomial time.