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Register a new userThe Power of Self-Reducibility: Selectivity, Information, and Approximation
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Added by
Lane A. Hemaspaandra
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Authors
Lane A. Hemaspaandra
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This chapter provides a hands-on tutorial on the important technique known as self-reducibility. Through a series of Challenge Problems that are theorems that the reader will---after being given definitions and tools---try to prove, the tutorial will ask the reader not to read proofs that use self-reducibility, but rather to discover proofs that use self-reducibility. In particular, the chapter will seek to guide the reader to the discovery of proofs of four interesting theorems---whose focus areas range from selectivity to information to approximation---from the literature, whose proofs draw on self-reducibility. The chapters goal is to allow interested readers to add self-reducibility to their collection of proof tools. The chapter simultaneously has a related but different goal, namely, to provide a lesson plan (and a coordinated set of slides is available online to support this use [Hem19]) for a lecture to a two-lecture series that can be given to undergraduate students---even those with no background other than basic discrete mathematics and an understanding of what polynomial-time computation is---to immerse them in hands-on proving, and by doing that, to serve as an invitation to them to take courses on Models of Computation or Complexity Theory.
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The Stabbing Planes proof system was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas -- certain unsatisfiable systems of linear equations mod 2 -- which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes. In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients SP* can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas.
One of the strongest techniques available for showing lower bounds on quantum communication complexity is the logarithm of the approximation rank of the communication matrix--the minimum rank of a matrix which is entrywise close to the communication matrix. This technique has two main drawbacks: it is difficult to compute, and it is not known to lower bound quantum communication complexity with entanglement. Linial and Shraibman recently introduced a norm, called gamma_2^{alpha}, to quantum communication complexity, showing that it can be used to lower bound communication with entanglement. Here the parameter alpha is a measure of approximation which is related to the allowable error probability of the protocol. This bound can be written as a semidefinite program and gives bounds at least as large as many techniques in the literature, although it is smaller than the corresponding alpha-approximation rank, rk_alpha. We show that in fact log gamma_2^{alpha}(A)$ and log rk_{alpha}(A)$ agree up to small factors. As corollaries we obtain a constant factor polynomial time approximation algorithm to the logarithm of approximate rank, and that the logarithm of approximation rank is a lower bound for quantum communication complexity with entanglement.
We introduce a uniform representation of general objects that captures the regularities with respect to their structure. It allows a representation of a general class of objects including geometric patterns and images in a sparse, modular, hierarchical, and recursive manner. The representation can exploit any computable regularity in objects to compactly describe them, while also being capable of representing random objects as raw data. A set of rules uniformly dictates the interpretation of the representation into raw signal, which makes it possible to ask what pattern a given raw signal contains. Also, it allows simple separation of the information that we wish to ignore from that which we measure, by using a set of maps to delineate the a priori parts of the objects, leaving only the information in the structure. Using the representation, we introduce a measure of information in general objects relative to structures defined by the set of maps. We point out that the common prescription of encoding objects by strings to use Kolmogorov complexity is meaningless when, as often is the case, the encoding is not specified in any way other than that it exists. Noting this, we define the measure directly in terms of the structures of the spaces in which the objects reside. As a result, the measure is defined relative to a set of maps that characterize the structures. It turns out that the measure is equivalent to Kolmogorov complexity when it is defined relative to the maps characterizing the structure of natural numbers. Thus, the formulation gives the larger class of objects a meaningful measure of information that generalizes Kolmogorov complexity.
A predicate f:{-1,1}^k -> {0,1} with rho(f) = frac{|f^{-1}(1)|}{2^k} is called {it approximation resistant} if given a near-satisfiable instance of CSP(f), it is computationally hard to find an assignment that satisfies at least rho(f)+Omega(1) fraction of the constraints. We present a complete characterization of approximation resistant predicates under the Unique Games Conjecture. We also present characterizations in the {it mixed} linear and semi-definite programming hierarchy and the Sherali-Adams linear programming hierarchy. In the former case, the characterization coincides with the one based on UGC. Each of the two characterizations is in terms of existence of a probability measure with certain symmetry properties on a natural convex polytope associated with the predicate.
In these notes we discuss the self-reducibility property of the Weil representation. We explain how to use this property to obtain sharp estimates of certain higher-dimensional exponential sums which originate from the theory of quantum chaos. As a result, we obtain the Hecke quantum unique ergodicity theorem for generic linear symplectomorphism $A$ of the torus $T^{2N}=R^{2N}/Z^{2N}.
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