# Holographic Algorithms Beyond Matchgates

Jin-Yi Cai, Heng Guo, Tyson Williams

Research output: Chapter in Book/Report/Conference proceedingConference contribution

## Abstract

Holographic algorithms were first introduced by Valiant as a new methodology to derive polynomial time algorithms. The algorithms introduced by Valiant are based on matchgates, which are intrinsically for problems over planar structures. In this paper we introduce two new families of holographic algorithms. These algorithms work over general, i.e., not necessarily
planar, graphs. Instead of matchgates, the two underlying families of constraint functions are of the affine type and of the product type. These play the role of Kasteleyn's algorithm for counting planar perfect matchings. The new algorithms are obtained by transforming a problem to one of these two families by holographic reductions.
The tractability of affine and product-type constraint functions is known. The real challenge is to determine when some concrete problem, expressed by its constraint functions, has such a holographic reduction. We present a polynomial time algorithm to decide if a given counting problem has a holographic algorithm using the affine or product-type constraint functions. Our
algorithm also finds a holographic transformation when one exists. We exhibit concrete problems that can be solved by the new holographic algorithms. When the constraint functions are symmetric, we further present a polynomial time algorithm for the same decision and search problems, where the complexity is measured in terms of the (exponentially more) succinct presentation of symmetric constraint functions. The algorithm for the symmetric case also
shows that the recent dichotomy theorem for Holant problems with symmetric constraints is efficiently decidable. Our proof techniques are mainly algebraic, e.g., stabilizers and orbits of group actions.
Original language English Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I Springer 271-282 12 https://doi.org/10.1007/978-3-662-43948-7_23 Published - 2014 ICALP 2014 - Copenhagen, DenmarkDuration: 7 Jul 2014 → 11 Jul 2014http://icalp2014.itu.dk/

### Conference

Conference ICALP 2014 Denmark Copenhagen 7/07/14 → 11/07/14 http://icalp2014.itu.dk/

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