Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1908.03252

On Symbolic Approaches for Computing the Matrix Permanent

Supratik Chakraborty·Aditya A. Shrotri·Moshe Y. Vardi
View original source ↗Read source document ↗Export metadata ↓

Abstract

Counting the number of perfect matchings in bipartite graphs, or equivalently computing the permanent of 0-1 matrices, is an important combinatorial problem that has been extensively studied by theoreticians and practitioners alike. The permanent is #P-Complete; hence it is unlikely that a polynomial-time algorithm exists for the problem. Researchers have therefore focused on finding tractable subclasses of matrices for permanent computation. One such subclass that has received much attention is that of sparse matrices i.e. matrices with few entries set to 1, the rest being 0. For this subclass, improved theoretical upper bounds and practically efficient algorithms have been developed. In this paper, we ask whether it is possible to go beyond sparse matrices in our quest for developing scalable techniques for the permanent, and answer this question affirmatively. Our key insight is to represent permanent computation symbolically using Algebraic Decision Diagrams (ADDs). ADD-based techniques naturally use dynamic programming, and hence avoid redundant computation through memoization. This permits exploiting the hidden structure in a large class of matrices that have so far remained beyond the reach of permanent computation techniques. The availability of sophisticated libraries implementing ADDs also makes the task of engineering practical solutions relatively straightforward. While a complete characterization of matrices admitting a compact ADD representation remains open, we provide strong experimental evidence of the effectiveness of our approach for computing the permanent, not just for sparse matrices, but also for dense matrices and for matrices with "similar" rows.

Explore related subjects

cs.DScs.SC

Keep this discovery

BibTeXRIS

Supratik Chakraborty, Aditya A. Shrotri, Moshe Y. Vardi. 2019-08-08. On Symbolic Approaches for Computing the Matrix Permanent. https://arxiv.org/abs/1908.03252

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗