Algorithms for Multityped Branching Processes

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Algorithms compute the extinction probabilities of multitype branching processes by solving the monotone system of probabilistic polynomial fixed‑point equations and by handling decision extensions that add max/min operators. Key claims: simple iteration can require exponentially many steps, but Newton with rounding applied after preprocessing converges in linear iterations and yields arbitrary bits of precision; preprocessing identify 0/1 variables via reachability and eigenvalue tests and SCC decomposition; complexity is polynomial time in the unit‑cost arithmetic model and practical with careful bit rounding.

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1. 0:00Intro
2. 0:34Branching Processes
3. 2:18Multi-type Branching Process
4. 4:56Extinction Probabilities
5. 6:52Probabilistic Polynomial System of Equations (PPS) System x-Pixi of nequations in n variables of the form
6. 8:37Properties of PPS, X=P(x)
7. 9:41Value Iteration
8. 14:02Newton on Monotone Polynomial Systems - Given a monotone polynomial system (MPS), we can first
9. 18:01P-time Approximation Algorithm for Probabilistic Polynomial Systems
10. 20:57Algorithm Outline ctd. 2. Apply Rounded down Newton with rounding parameter
11. 26:53Proof sketch for step 2: Key properties
12. 29:30Comparison/Decision Questions are Hard
13. 32:44Branching Markov Decision Process
14. 41:38Maximizing Branching MDP Equations
15. 43:18Max (Min) Probabilistic Polynomial Systems (Max PPS / Min PPS)
16. 43:54Solving Max (Min) PPS
17. 46:14Generalized Newton's Method
18. 47:10Algorithm for max (min) PPS
19. 48:21Computing (near)-optimal Policies
20. 51:45Conclusions