Extended formulations, nonnegative factorizations, and randomized communication protocols

An extended formulation of a polyhedron P is a linear description of a polyhedron Q together with a linear map π such that π(Q)=P. These objects are of fundamental importance in polyhedral combinatorics and optimization theory, and the subject of a number of studies. Yannakakis’ factorization theorem (Yannakakis in J Comput Syst Sci 43(3):441–466, 1991) provides a surprising connection between extended formulations and communication complexity, showing that the smallest size of an extended formulation of $$P$$P equals the nonnegative rank of its slack matrix S. Moreover, Yannakakis also shows that the nonnegative rank of S is at most 2^c, where c is the complexity of any deterministic protocol computing S. In this paper, we show that the latter result can be strengthened when we allow protocols to be randomized. In particular, we prove that the base-2 logarithm of the nonnegative rank of any nonnegative matrix equals the minimum complexity of a randomized communication protocol computing the matrix in expectation. Using Yannakakis’ factorization theorem, this implies that the base-2 logarithm of the smallest size of an extended formulation of a polytope P equals the minimum complexity of a randomized communication protocol computing the slack matrix of P in expectation. We show that allowing randomization in the protocol can be crucial for obtaining small extended formulations. Specifically, we prove that for the spanning tree and perfect matching polytopes, small variance in the protocol forces large size in the extended formulation.

Approximation limits of linear programs (beyond hierarchies)

We develop a framework for proving approximation limits of polynomial size linear programs (LPs) from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any LP as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n^1/2-ε)-approximations for CLIQUE require LPs of size 2^nΩ(ε). This lower bound applies to LPs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by LPs. Our main technical ingredient is a quantitative improvement of Razborov's [38] rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix.

Lovelock terms and BRST cohomology

Lovelock terms are polynomial scalar densities in the Riemann curvature tensor that have the remarkable property that their Euler-Lagrange derivatives contain derivatives of the metric of an order not higher than 2 (while generic polynomial scalar densities lead to Euler-Lagrange derivatives with derivatives of the metric of order 4). A characteristic feature of Lovelock terms is that their first nonvanishing term in the expansion g λμ = η λμ + h λμ of the metric around flat space is a total derivative. In this paper, we investigate generalized Lovelock terms defined as polynomial scalar densities in the Riemann curvature tensor and its covariant derivatives (of arbitrarily high but finite order) such that their first nonvanishing term in the expansion of the metric around flat space is a total derivative. This is done by reformulating the problem as a BRST cohomological one and by using cohomological tools. We determine all the generalized Lovelock terms. We find, in fact, that the class of nontrivial generalized Lovelock terms contains only the usual ones. Allowing covariant derivatives of the Riemann tensor does not lead to a new structure. Our work provides a novel algebraic understanding of the Lovelock terms in the context of BRST cohomology. © 2005 IOP Publishing Ltd.

The Vietnam's Transition Economy and Its Fledgling Financial Markets: 1986-2003

In this paper, we analyze the context of Vietnam’s economic standings in the reform period. The first section embarks on most remarkable factors, which promote the development of financial markets are: (i) Doi Moi policies in 1986 unleash ‘productive powers’. Real GDP growth, and key economic indicators improve. The economy truly departs from the old-style command economy; (ii) FDI component is present in the economy as sine qua non; a crucial growth engine, forming part of the financial markets, planting the ‘seeds’ for its growth; and (iii) the private economy is both the result and cause of the reform. Its growth is steady. Today, it represents a powerhouse, and helps form part of the genuine financial economy. A few noteworthy points found in the next section are: (i) No evidence of financial markets existence was found before Doi Moi. The reform has generated a bulk of private-sector financial companies. New developments have roots in the 1992-amended constitution (x3.2); (ii) The need to reform the financial started with the domino collapse of credit cooperatives in early 1990s. More stress is caused by the ‘blow’ of banking deficiency in late 1990s; and (iii) Laws on SBV and credit institutions, and the launch of the stock market are bold steps. Besides, the Asian financial turmoil forces the economy to reaffirm its reform agenda. Our findings also indicate, through empirical evidences, that economic conditions have stabilized throughout the reform, thanks to the contributions of the FDI and private economic sector. Private investment flows continue to be an eminent factor that drives the economy growth.

Sorting under partial information (without the ellipsoid algorithm)

We revisit the well-known problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set P, and solving the problem amounts to discovering an unknown linear extension of P, using pairwise comparisons. The information-theoretic lower bound on the number of comparisons needed in the worst case is log e(P), the binary logarithm of the number of linear extensions of P. In a breakthrough paper, Jeff Kahn and Jeong Han Kim (STOC 1992) showed that there exists a polynomial-time algorithm for the problem achieving this bound up to a constant factor. Their algorithm invokes the ellipsoid algorithm at each iteration for determining the next comparison, making it impractical. We develop efficient algorithms for sorting under partial information. Like Kahn and Kim, our approach relies on graph entropy. However, our algorithms differ in essential ways from theirs. Rather than resorting to convex programming for computing the entropy, we approximate the entropy, or make sure it is computed only once in a restricted class of graphs, permitting the use of a simpler algorithm. Specifically, we present: an O(n2) algorithm performing O(log n·log e(P)) comparisons; an O(n2.5) algorithm performing at most (1+ε) log e(P) + Oε(n) comparisons; an O(n2.5) algorithm performing O(log e(P)) comparisons. All our algorithms are simple to implement. © 2010 ACM.