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Counting Solution Clusters Divide-and-Conquer Recursively: Arbitrarily pick a variable, say x, of formula F Count how many clusters contain solutions with x=0 (ok if the cluster has solutions with both x=0 and x=1) Add number of clusters that contain solutions with x=1 Subtract number of clusters that contain both solutions with x=0 and solutions with x=1 #clusters = #clusters(F)| x=0 + #clusters(F)| x=1 #clusters(F)| x=0 & x=1 Key issues: how can we compute #clusters(F)| x=0 ? (#clusters| x=1 would be similar) how do we compute #clusters(F)| x=0 & x=1 ? (not a problem for SAT) Physics of Algorithms, 2009Bart Selman1 x=1 x=0 (Inclusion - exclusion formula)

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Computing #clusters(F)| x=0 : Fragmentation Algorithmically, easiest way is to “fix” x to 0 in the formula F, compute #clusters in new formula (F| x=0 ) So, use as approximation: #clusters(F)| x=0 #clusters(F| x=0 ) Risk? Potential over-counting: a cluster of F may break/fragment into several smaller, disconnected clusters when x is fixed to 0 Interestingly: Clusters often do not fragment! In particular, provably no fragmentation in 2-SAT and 3-COL* instances! (any instance, i.e., worst-case). Also, empirically holds for almost all clusters in random 3-SAT, logistics, circuits, … 2 x=0x=1 a cluster in F could fragment to 2 clusters in F| x=0

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Apply recursively: Z (-1) #clusters(F) = #clusters(F)| x=0 + #clusters(F)| x=1 #clusters(F)| x=0 & x=1 = #clusters(F| x=0 ) + #clusters(F| x=1 ) #clusters(F| x= ) 3 #clusters(F| x=0,y=0 ) + #clusters(F| x=0,y=1 ) #clusters(F| x=0,y= ) smaller problem instances; solve recursively #clusters(F| x=1,y=0 ) + #clusters(F| x=1,y=1 ) #clusters(F| x=1,y= ) #clusters(F| x= ,y=0 ) + #clusters(F| x= ,y=1 ) #clusters(F| x= ,y= ) Total 3 N terms of the form #clusters(F| x=0,y= ,z=1,w= … p=0,q=1 )

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Putting it all together: Z (-1) Observation: when all N variables have been assigned values, #clusters is either 0 (no solution) or 1 (a full hypercube) #clusters(F| x1=0,x2= ,x3=1,x4= ,…,xN=0 ) = F( x1=0,x2= ,x3=1,x4= ,…,xN=0 ) So, #clusters(F) = v full generalized assignments (-1) no. of in v #clusters(F| v ) = v full generalized assignments (-1) no. of in v F(v) = Z (-1) (F) 4 any full “generalized” assignment

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5 Empirical Results: Z (-1) for SAT Z (-1) is accurate even for many structured formulas (encoding real-world problems): [Kroc, Sabharwal, Selman ‘09]

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6 Checks whether all points in y are good # (y) counts the number of elements of y Leads directly to SP equations! Talk by Lukas Kroc Written in factor graph notation:

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4.2B Geometric Explicit and Recursive Sequences

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