## Polynomial Time Algorithms To Approximate Permanents And Mixed Discriminants Within A Simply Exponential Factor (1999)

Venue: | Random Structures & Algorithms |

Citations: | 30 - 3 self |

### BibTeX

@INPROCEEDINGS{Barvinok99polynomialtime,

author = {Alexander Barvinok},

title = {Polynomial Time Algorithms To Approximate Permanents And Mixed Discriminants Within A Simply Exponential Factor},

booktitle = {Random Structures & Algorithms},

year = {1999},

pages = {29--61}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present real, complex, and quaternionic versions of a simple randomized polynomial time algorithm to approximate the permanent of a non-negative matrix and, more generally, the mixed discriminant of positive semidefinite matrices. The algorithm provides an unbiased estimator, which, with high probability, approximates the true value within a factor of O(c n ), where n is the size of the matrix (matrices) and where c 0:28 for the real version, c 0:56 for the complex version and c 0:76 for the quaternionic version. We discuss possible extensions of our method as well as applications of mixed discriminants to problems of combinatorial counting.

### Citations

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Citation Context ...8 in the real algorithm, c = 0:56 in the complex algorithm, and c = 0:76 in the quaternionic algorithm. The computational model is the RAM (Random Access Machine) with the uniform cost criterion (see =-=[1]-=-), so the algorithms operate with real numbers and allow arithmetic operations (addition, subtraction, multiplication, and division) and comparison of real numbers. A preprocessing requires taking the... |

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Citation Context ... ff is a random variable. Specifically, we use sampling from the Gaussian distribution in R 1 with the density / oe (x) = 1 oe p 2 e \Gamma x 2 2oe 2 . As is known (see, for example, Section 3.4.1 of =-=[20]-=-), sampling from the Gaussian distribution can be efficiently simulated from the standard Bernoulli distribution (sampling a random bit). The output ff is an unbiased estimator for per A and it turns ... |

299 | Approximating the permanent
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Citation Context ...ermanent of a given matrix within a relative error ffl? In other words, does there exist a polynomial time approximation scheme? Polynomial time approximation schemes are known for dense 0-1 matrices =-=[15], for &quo-=-t;almost all" 0-1 matrices (see [15], [12], and [27]) and for some special 0-1 matrices, such as those corresponding to lattice graphs (see [16] for a survey on approximation algorithms). However... |

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Citation Context ...east 0:6. As we noted in Section 1, for any ffl ? 0 we can improve the probability to 1 \Gamma ffl by running the algorithm O(log ffl \Gamma1 ) times and choosing the median of the computed ff's, cf. =-=[17]-=-. We can choose c = 0:28 in the real algorithm, c = 0:56 in the complex algorithm, and c = 0:76 in the quaternionic algorithm. The computational model is the RAM (Random Access Machine) with the unifo... |

235 | The Markov Chain Monte Carlo method: an approach to approximate counting and integration
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Citation Context ...oximation schemes are known for dense 0-1 matrices [15], for "almost all" 0-1 matrices (see [15], [12], and [27]) and for some special 0-1 matrices, such as those corresponding to lattice gr=-=aphs (see [16] for a sur-=-vey on approximation algorithms). However, no polynomial time approximation scheme is known for an arbitrary 0-1 matrix (see [18] for the fastest known "mildly exponential" approximation sch... |

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Citation Context .... This transformation is straightforward; it is sketched in the first version of the paper [7]. The general technique of going from "real randomized" to "binary randomized" algorit=-=hms can be found in [24]-=-. (2.1) The Real Algorithm. Input: A non-negative n \Theta n matrix A. Output: A non-negative number ff approximating per A. Algorithm: Sample independently n 2 numbers u ij : i; j = 1; : : : ; n at r... |

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Citation Context ...ight definition of the "squared norm of the determinant" of a quaternionic matrix H (it is known as the "reduced norm", or the squared norm of the Dieudonn'e determinant, see Chapt=-=er VI, Section 1 of [3]-=-). As an analogue, let us point out that the the squared absolute value of the determinant of an n \Theta n complex matrix can be interpreted as the determinant of a 2n \Theta 2n real matrix: j det(A ... |

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Citation Context ...be improved to 1 \Gamma ffl by running the algorithm independently O(log ffl \Gamma1 ) times and choosing ff to be median of the computed ff's. Recently, N. Linial, A. Samorodnitsky, and A. Wigderson =-=[22]-=- constructed a polynomial time deterministic algorithm, which achieves (1.1.1) with C = 1 and c = 1=es0:37. The algorithm uses scaling of a given non-negative matrix to a doubly stochastic matrix. In ... |

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Citation Context ...d volumes ([2], see also [21]). The relation between the mixed discriminant and the permanent was used in the proof of the van der Waerden conjecture for permanents of doubly stochastic matrices (see =-=[9]-=-). It is known that D(Q 1 ; : : : ; Q n )s0 provided Q 1 ; : : : ; Q n are positive semidefinite (see [21]). Just as it is natural to restrict the permanent to non-negative matrices, it is natural to ... |

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Citation Context ...o be difficult. Polynomial time algorithms for computing per A are known when A has some special structure, for example, when A has a small rank [5], or when A is a 0-1 matrix and per A is small (see =-=[14]-=- and Section 7.3 of [25]). Since the exact computation is difficult, a natural question is how well one can approximate the permanent in polynomial time. In particular, is it true that for any ffl ? 0... |

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Citation Context ...sults of the paper. (5.1) Conditional expectations. In this subsection, we summarize some general results on measures and integration, which we exploit in Lemma 5.2 below. As a general source, we use =-=[4]-=-. Let us fix a probability measureson the Euclidean space R m . Suppose thatsis absolutely continuous with respect to the Lebesgue measure and let /(x) be the density of . Suppose that we have k copie... |

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Citation Context ... #P-complete problem and even to estimate per A seems to be difficult. Polynomial time algorithms for computing per A are known when A has some special structure, for example, when A has a small rank =-=[5]-=-, or when A is a 0-1 matrix and per A is small (see [14] and Section 7.3 of [25]). Since the exact computation is difficult, a natural question is how well one can approximate the permanent in polynom... |

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Citation Context ...matrices, such as those corresponding to lattice graphs (see [16] for a survey on approximation algorithms). However, no polynomial time approximation scheme is known for an arbitrary 0-1 matrix (see =-=[18] for the f-=-astest known "mildly exponential" approximation scheme). In [6], the author suggested a polynomial time randomized algorithm, which, given an n \Theta n non-negative matrix A, outputs a non-... |

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Citation Context ...a real matrix and its transpose. This is a standard procedure of linear algebra, which requires O(n 3 ) arithmetic operations and n square root extractions (see, for example, Chapter 2, Section 10 of =-=[11]-=-). 11 (3.1) The Real Algorithm. The algorithm requires sampling vectors x 2 R n , x = ( 1 ; : : : ;sn ) from the standard Gaussian distribution in R n with the density /R n(x) = (2) \Gamman=2 exp \Phi... |

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Citation Context ...: : ; a in g, we have D(Q 1 ; : : : ; Q n ) = per A; where A = (a ij ): Mixed discriminants were introduced by A.D. Aleksandrov in his proof of the Aleksandrov - Fenchel inequality for mixed volumes (=-=[2]-=-, see also [21]). The relation between the mixed discriminant and the permanent was used in the proof of the van der Waerden conjecture for permanents of doubly stochastic matrices (see [9]). It is kn... |

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Citation Context ...heta 8n matrix and per A n = (40; 320) n . The described above binary version of the Godsil-Gutman estimator outputs 0 with the probability at least 1 \Gamma (0:8) n . The complex discrete version of =-=[19]-=- (see also Remark in Section 2.2) outputs 0 with the probability at least 1 \Gamma (0:99) n . Algorithm 2.1 outputs at least (1:52) n with the probability approaching 1 as n \Gamma! +1. Similarly, the... |

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Citation Context ...where c 1s0:28 is the constant defined in Section 1.3. Remark. Relation to the Godsil-Gutman estimator. It is immediately seen that Algorithm 2.1 is a modification of the Godsil-Gutman estimator (see =-=[13]-=-) and Chapter 8 of [23]). Indeed, in the Godsil-Gutman estimator we sample u ij from the binary distribution: u ij = ae 1 with probability 1=2 \Gamma1 with probability 1=2: Furthermore, Part 1 and Par... |

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Citation Context ...rvey on approximation algorithms). However, no polynomial time approximation scheme is known for an arbitrary 0-1 matrix (see [18] for the fastest known "mildly exponential" approximation sc=-=heme). In [6]-=-, the author suggested a polynomial time randomized algorithm, which, given an n \Theta n non-negative matrix A, outputs a non-negative number ff approximating per A within a simply exponential in n f... |

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Citation Context ...n other words, does there exist a polynomial time approximation scheme? Polynomial time approximation schemes are known for dense 0-1 matrices [15], for "almost all" 0-1 matrices (see [15], =-=[12], and [27]-=-) and for some special 0-1 matrices, such as those corresponding to lattice graphs (see [16] for a survey on approximation algorithms). However, no polynomial time approximation scheme is known for an... |

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Citation Context ...ror ffl? In other words, does there exist a polynomial time approximation scheme? Polynomial time approximation schemes are known for dense 0-1 matrices [15], for "almost all" 0-1 matrices (=-=see [15], [12]-=-, and [27]) and for some special 0-1 matrices, such as those corresponding to lattice graphs (see [16] for a survey on approximation algorithms). However, no polynomial time approximation scheme is kn... |

4 |
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Citation Context ...onstant defined in Section 1.3. Remark. Relation to the Godsil-Gutman estimator. It is immediately seen that Algorithm 2.1 is a modification of the Godsil-Gutman estimator (see [13]) and Chapter 8 of =-=[23]-=-). Indeed, in the Godsil-Gutman estimator we sample u ij from the binary distribution: u ij = ae 1 with probability 1=2 \Gamma1 with probability 1=2: Furthermore, Part 1 and Part 2 of Theorem 2.1.1 re... |

4 |
Two combinatorial applications of the Alexandrov-Fenchel inequalities
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Citation Context ...ll n-subsets I, having precisely one element of each color, and the proof follows. R. Stanley obtained a similar formula which involves the mixed volume of zonotopes instead of the mixed discriminant =-=[28]-=-. Example. Trees in a graph. Suppose we have a connected graph G with n vertices and m edges. Suppose further that the edges of G are colored with n \Gamma 1 different colors. We are interested in spa... |

4 |
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Citation Context ...rix A I = [u i 1 ; : : : ; u i n ] is either 0, \Gamma1 or 1. Such an A represents a unimodular matroid on the set f1; : : : ; mg and a subset I with det A I 6= 0 is called a base of the matroid (see =-=[29]-=-). Suppose now that the columns of A are colored with n different colors. The coloring induces a partition f1; : : : ; mg = J 1 [ : : : [ J n . We are interested in the 31 number of bases that have pr... |

2 |
Convexity and Differential Geometry, in: Handbook of Convex Geometry
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Citation Context ...he number D(Q 1 ; : : : ; Q n ) = @ n @t 1 : : : @t n det \Gamma t 1 Q 1 + : : : + t n Q n ) is called the mixed discriminant of Q 1 ; : : : ; Q n . Sometimes the normalizing factor 1=n! is used (cf. =-=[21]-=-). The mixed discriminant D(Q 1 ; : : : ; Q n ) is a polynomial in the entries of Q 1 ; : : : ; Q n : for Q k = (q ij;k ): i; j = 1; : : : ; n, k = 1; : : : ; n, we have (1:2:1) D(Q 1 ; : : : ; Q n ) ... |

1 |
personal communication, Schlo Dagstuhl Meeting on Approximation Algorithms
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Citation Context ...e easier to implement than the algorithm of [6]. The first version (see [7]) of the paper contained the real algorithm only. The complex algorithm was suggested to the author by M. Dyer and M. Jerrum =-=[8]-=-. Building on the complex version, the author constructed the quaternionic version. 2 (1.2) Mixed Discriminant. Let Q 1 ; : : : ; Q n be n \Theta n real symmetric matrices and let t 1 ; : : : ; t n be... |