- Meeting 01 : Wed, Jan 17, 03:30 pm-04:45 pm
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| Exercises | |
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Administrative Announcements. Overview of the course. Counting Complexity. Computational Model. The class FP. Clique vs #Clique Problem. Cycle vs #Cycle Problem. Decision vs Counting Problems. Counting the number of spanning trees in a given undirected graph. Kirchoff's Matrix Tree Theorem. The need of a theory.
- Meeting 02 : Thu, Jan 18, 10:00 am-10:50 am
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| Exercises | |
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The Class #P. If #CYCLE is in FP, then P = NP. Discussion on #BPM problem.
- Meeting 03 : Fri, Jan 19, 02:00 pm-03:15 pm
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| Exercises | |
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The bits and value of the #P function. Counting as a generalization of NP, BPP, RP, PP and Parity P. NP is contained in PP.
#P = FP if and only if PP = P.
- Meeting 04 : Wed, Jan 24, 03:30 pm-04:45 pm
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| Exercises | |
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Notion of #P-completeness. Attempt 1 : Parsimonious reductions. #SAT is #P-complete. #IND-SET is #P-complete. Shortcomings of the theory. General oracle query model of #P-completeness. Valiants theorem. Algebraic interpretation of the number of perfect matchings in bipartite graphs. The permanent and the determinant functions. Discussion on algorithmic approaches.
- Meeting 05 : Thu, Jan 25, 10:00 am-10:50 am
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| Exercises | |
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Permanent over integers. Roadmap for Valiants proof. A combinatorial interpretation of the permanent over integers using cycle covers. Example tricks using cycle covers.
- Meeting 06 : Mon, Jan 29, 12:00 pm-12:50 pm
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| Exercises | |
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Valiant Gadgets. Computation of Cycle covers in various cases. Every satisfying assignment corresponds to a cycle cover of weight 4^m where m is the number of clauses in the 3SAT instance.
- Meeting 07 : Wed, Jan 31, 03:30 pm-04:45 pm
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| Exercises | |
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Reduction to (-1,0,1)-permanent. Reduction to 0-1 permanent.
