A Survey of Counting Dichotomies and Holographic Algorithms

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Survey reports that reductions to weighted perfect matching and matchgates produce unexpected polynomial-time algorithms for many #P problems on planar inputs, while a complete dichotomy classifies each fixed constraint set as either polynomial-time or #P-hard. Key tools are tensor representations, a 2×2 basis T (using a third root of unity) and holographic transformations that turn Boolean signatures into matchgate signatures. Surprising arithmetic constraints arise: solvability of the realization equations depends on moduli (examples mention mod 7, mod 5 and mod 15), explaining when the matchgate approach works.

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" chapters (17)

1. 0:00Holographic algorithms overview
2. 1:46Holographic algorithms basics
3. 4:54Match Gates applications
4. 12:08Match Gates limitations
5. 17:48Match Gates definition
6. 20:14Holographic transformations
7. 28:57Basis transformations
8. 36:40Match Gates signatures theory
9. 37:01Dichotomy theorems
10. 40:18Graph homomorphism
11. 42:48CSP dichotomy criteria
12. 47:21Dichotomy theorems decidability
13. 49:06Holant frameworks
14. 50:13Tractable matrices
15. 53:48Dichotomy versus modular counting
16. 54:30Boolean CSPs transformations
17. 55:55Summary and open problems