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Reducing WCNF-3SAT to WCNF-2SAT - (among other things)

Reducing WCNF-3SAT to WCNF-2SAT - (among other things). A presentation of results in: Rod G. Downey and Michael R. Fellows Fixed-parameter tractability and completeness II: on completeness for W[1], Theoretical Computer Science 141 , pp. 109-131, (1995). Presentation by Nick Neumann.

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Reducing WCNF-3SAT to WCNF-2SAT - (among other things)

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  1. Reducing WCNF-3SAT toWCNF-2SAT-(among other things) A presentation of results in: Rod G. Downey and Michael R. Fellows Fixed-parameter tractability and completeness II: on completeness for W[1], Theoretical Computer Science 141, pp. 109-131, (1995). Presentation by Nick Neumann

  2. Outline of talk • Summary of paper • Necessary definitions • Lemma for proving reduction • Result yielding reduction • Other results in paper/implications • Conclusion

  3. Summary of paper • Defines W[t] differently than in class(makes W[1] much “bigger”) • Shows collapse of W[1] to WCNF-2SAT- . • In process, shows WCNF-3SAT <=fpt WCNF-2SAT- • Result immediately yields several problems to be W[1]-complete

  4. Necessary definitions • Small gates – bounded fan-in • Large gates – unrestricted fan-in • Depth (of circuit) – max # gates on path from literal to output in circuit • Weft (of circuit) – max # LARGE gates on path from literal to output in circuit

  5. More definitions • LF(t,h) = language of circuits of weft <= t, depth <= h, satisfied by weight k assignment • W[t]: LÎW[t] if L <=fpt LF(t,h) for some h • W[1,s]: Weft 1, depth 2 circuits with fan- in for small gates bounded by s • W[t] bounds weft at t;W[1,s] bounds fan-in at s, fixes weft at 1, depth at 2

  6. Result yielding reduction • Paper proves W[1]=W[1,2] • Does this as follows: • Proves W[1,s]=antimonotone W[1,s] • Proves W[1]=Ès=1..∞ W[1,s] • Proves antimonotone W[1,s]=W[1,2]

  7. Lemma for proving reduction • W[1,s]=antimonotone W[1,s] for s>=2 • Proof strategy: • Takes a problem in antimonotone W[1,s] and shows it is hard for W[1,s] • Problem is s-RED/BLUE NONBLOCKER(s-R/B N): • Graph G=(V,E) of max degree s,V = VblueÈVred partition V • Is there a set V’ÍVred of size k s.t. every blue vertex has at least one neighbor not in V’?

  8. Lemma (cont’d) • Showing s-R/B NÎantimonotone W[1,s] • Easy • Õ(u blue)Sx_i a red neighbor of uØx_i • Showing W[1,s] <=fpt s-R/B N • Involved graph construction • Use blue vertices to limit weight of satisfying assignment

  9. Proof ofantimonotone W[1,s]=W[1,2] • Replaces literals with new variables representing subsets of literals of size 2..s • Replace inputs to OR gates with new variables • Adds additional variables and gates to enforce consistency (while keeping fan-in of OR gates bounded by 2) • Parameter k becomes k2k + Si=2..s C(k,i)

  10. Reducing WCNF-3SAT • So W[1,s]=antimonotone W[1,s]=W[1,2]=antimonotone W[1,2], • WCNF-3SATÎW[1,3],WCNF-2SAT- isantimonotone W[1,2]-hard • Paper’s result is much stronger, but: • WCNF-3SAT <=fpt WCNF-2SAT-

  11. Other results/implications • W[t]=antimonotone W[t] for t odd • Result that W[1]=Ès=1..∞ W[1,s] comes from LF(1,h) <=fpt W[1,s], s=f(h)(trade depth for fan-in) • IS, CLIQUE are W[1] – complete • Perfect Code, Weighted Exact CNF-SAT, Sized Subset Sum are W[1]-hard

  12. Conclusions • W[t] formulated uniformly for all t>=1 • W[1] stratification to Ès=1..∞ W[1,s] • W[1] collapses to W[1,2]

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