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 , (1995). Presentation by Nick Neumann

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

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

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

More definitions l L F(t,h) = language of circuits of weft <= t, depth <= h, satisfied by weight k assignment W[t]: L  W[t] if L <= fpt L F(t,h) for some h l 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

Result yielding reduction l Paper proves W[1]=W[1,2] l Does this as follows: –Proves W[1,s]=antimonotone W[1,s] –Proves W[1]=  s=1..∞ W[1,s] –Proves antimonotone W[1,s]=W[1,2]

Lemma for proving reduction l W[1,s]=antimonotone W[1,s] for s>=2 l 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 = V blue  V red partition V Is there a set V’  V red of size k s.t. every blue vertex has at least one neighbor not in V’?

Lemma (cont’d) Showing s-R/B N  antimonotone W[1,s] –Easy –  (u blue)  x_i a red neighbor of u  x_i l Showing W[1,s] <= fpt s-R/B N –Involved graph construction –Use blue vertices to limit weight of satisfying assignment

Proof of antimonotone W[1,s]=W[1,2] l Replaces literals with new variables representing subsets of literals of size 2..s l Replace inputs to OR gates with new variables l Adds additional variables and gates to enforce consistency (while keeping fan-in of OR gates bounded by 2) Parameter k becomes k2 k +  i=2..s C(k,i)

Reducing WCNF-3SAT l So W[1,s]=antimonotone W[1,s] =W[1,2]=antimonotone W[1,2], WCNF-3SAT  W[1,3], WCNF-2SAT - is antimonotone W[1,2]-hard l Paper’s result is much stronger, but: –WCNF-3SAT <= fpt WCNF-2SAT -

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

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