On evaluation of permanents

dynamic programming [1,2,7] across column subsets; in commutative semirings, a "transposed" variant is shown to be considerably faster. In arbitrary rings, the starting point is Ryser's classic algorithm [13] that we manage to expedite for rectangular matrices, but that remains the fastest known algorithm for square matrices; again, in commutative rings, a transposed variant is shown to be substantially faster for rectangular matrices.

To state our main results, we take the time requirement of an algorithm as the number of additions and multiplications it performs, while the space requirement is taken as the maximum number of semiring elements that it needs to keep simultaneously in memory at any point in the computation. Also, denote by q ↓r the sum of the binomial coefficients

Theorem 1. The permanent of any m × n matrix, m ≤ n, can be computed All previous works we are aware of on evaluation of permanents assume commutativity, besides perhaps what is implicit in Ryser’s formula, see (1) below. For commutative rings, our bounds improve upon the state-of-theart achieved in a series of works based on arguably more involved techniques:

Using the Binet-Minc formulas [12], Kawabata and Tarui [9] presented an algorithm that runs in time O(n2 m + 3 m ) and space O(n2 m ). Recently, Vassilevska and Williams [14] took a different approach and obtained improved bounds O(mn 3 2 m ) and O(n 2 2 m ), respectively. Finally, by a yet different, algebraic approach, Koutis and Williams [11] further improved these bounds to poly(m, n)2 m and poly(m, n). For commutative semirings, Vassilevska and Williams [14] gave a Gurevich-Shelah [6] type recursive partitioning algorithm running in time poly(m, n)4 m and space poly(m, n). Koutis and Williams [11] presented bounds comparable to Theorem 1(ii) using a dynamic programming algorithm similar to ours but in an algebraic guise.

We begin without any further assumptions about the semiring and adopt the standard dynamic programming treatment of sequencing problems. That is, the algorithm tabulates intermediate results α(i, J) for sets J ⊆ N of size i, given by the recurrence

Here J corresponds to the image σ({1, 2, . . . , i}) of the injection σ, and it is easy to show that the permanent of A is obtained as the sum of the terms α(m, J) over all J ⊆ N of size m. Straightforward analysis proves Theorem 1(i).

In commutative semirings, we may transpose the previous algorithm, as follows. The idea is to go through the column indices j one by one, associating j with either one row index i not already associated with some other column, or associating j with none of the rows. Formally, for all

Here I corresponds to the preimage σ -1 ({1, 2, . . . , j}) of the injection σ. In rings, we start with Ryser’s inclusion-exclusion formula. Denote by a iX the partial row sum of the entries a ij with j ∈ X. Ryser [13] found that

(While Ryser’s original derivation is for fields, it immediately extends to arbitrary rings.) Visiting the sets X, for instance, in the lexicographical order, the terms a iX can be computed in an incremental fashion, each in constant amortized time. Thus the permanent can be evaluated in time O m n ↓m and space O(m). For square matrices this remains the most efficient way to evaluate the permanent.

But, when m is much less than n we can, in fact, do significantly better. For a set family F, denote by ↓F the family of sets in F and their subsets.

Theorem 2 (Björklund et al. [4], Kennes [10]). Let f and g be two functions from the subsets of a finite set U to a ring R. Then, To apply this result, we first note that the cardinality of

. Second, note that the permanent per A KP , for all P ⊆ N of size m/2, can be computed in time O m n ↓m/2 and space O n ↓m/2

; similarly for the permanents per A LQ . Combining these bounds yields Theorem 1(iii). We also note without proof that the space requirement can be reduced to O(m) at the cost of an extra factor of 3 m/2 in the time requirement; the idea is the same as what we have recently used to count paths and packings [4].

Finally, in commutative rings we may transpose Ryser’s formula in analogue to the transposed dynamic programming algorithm for commutative semirings. To this end, denote by a Xj the partial column sum of the entries a ij with i ∈ X. Then we may write

where the inner-most summation is over all binary sequences p = p 1 p 2 To analyze the time and space complexity, we note that, for any fixed X ⊆ M , the summation over the binary sequences p can be performed using simple dynamic programming in time O(n + m(n -m)) and space O(n). 2Here we assume that the sets X are visited in a suitable order such that each partial column sum can be updated in an incremental fashion in constant amortized time. Theorem 1(iv) follows.

We end by discussing the role of commutativity. With the given definition of permanents, Theorem 1 suggests that commutativity is crucial for efficient evaluation of permanents. However, we po

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