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I am Patrick Prosser I am a senior lecturer at Glasgow I teach AF2 & CP4 I am a member of the algorithms group the apes (distributed, not disbanded) I am a Glaswegian This is all that I am allowed to tell you (baby) Bio

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10 years of conflict-directed backjumping

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CI 9(4) Hybrid Algorithms for the Constraint Satisfaction Problem Still using that old greasy stuff? Who cares? So? Patrick Prosser

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Plan of the talk Whats a csp? The simplest algorithm (BT) and its behaviour Some improvements (BJ, GBJ) A better way (CBJ) Improvements to CBJ Strange things about CBJ k-inconsistencies the bridge and the long jump value ordering and insolubility So, what about CBJ? some say its good some say its a waste of time whos using it whos not using Wheres it going QBF?

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Whats a csp? a set of variables each with a domain of values a collection of constraints (Im going to assume binary) assign each variable a value from its domain to satisfy the constraint

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Example of a csp C E D B F A G H 3 colour me!

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Example of a csp C E D B F A G H

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C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back Why did it do that? That was dumb!

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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Example of a csp C E D B F A G H Va Vb Vc Vd Ve Vf Vg Vh 1 = red 2 = blue 3 = green Variables and Instantiation Order Checking back

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That was good old fashioned chronological backtracking instantiate a variable check current against past variables to see if okay current? past? Future? If not okay try another value If no values left go back to previous variable

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Example of a csp C E D B F A G H 1 = red 2 = blue 3 = green E1 E2 E3 E4 What would have happened if we had the E* intermediate variables? i.e. it falls back on E4, then E3, towards E?

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Its all just depth first search, right?

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BT Thrashes! pastpast futurefuture current variable v[i] conflict with v[h] past variable v[h] future variable v[j]

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BT Assume that when we instantiate the current variable v[i] we check against all past variables from v[1] to v[i-1] to check if consistent consistent := false for x in domain[i] while not(consistent) // find a consistent value begin consistent := true v[i] := x for h in (1.. i-1) while consistent // check backwards begin consistent := (check(v[i],v[h]) end if not(consistent) then delete(x,domain[i]) end // did we find a good value?

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John Gaschnigs BJ Make a small modification to BT. Remember the last variable we check against If the loop terminates with consistent false then jump back to v[h], where h := lastCheck[i] This is then the deepest variable we failed against consistent := false maxCheck[i] := 0 for x in domain[i] while not(consistent) // find a consistent value begin consistent := true v[i] := x for h in (1.. i-1) while consistent // check backwards begin consistent := (check(v[i],v[h]) maxCheck := max(h,maxCheck[i]) end if not(consistent) then delete(x,domain[i]) end // did we find a good value?

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John Gaschnigs BJ If the loop terminates with consistent true... lastCheck[i] = i-1 we then step back! consistent := false maxCheck[i] := 0 for x in domain[i] while not(consistent) // find a consistent value begin consistent := true v[i] := x for h in (1.. i-1) while consistent // check backwards begin consistent := (check(v[i],v[h]) maxCheck := max(h,maxCheck[i]) end if not(consistent) then delete(x,domain[i]) end // did we find a good value? BJ jumps then steps back. It can only jump after it has moved forwards!

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BJ reduces thrashing 1 2 3 4 5 6 maxCheck[i] 0123401234 4 Jump back to v[4] then step back

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Rina Dechters GBJ If there are no values remaining for v[i] Jump back to v[h], where v[h] is the deepest variable connected to v[i] in the constraint graph If there are no values remaining for v[h] Jump back to v[g], where v[g] is the deepest variable connected to v[h] or v[i] in the constraint graph If there are no values remaining for v[g] Jump back to v[f], where v[f] is the deepest variable connected to v[g] or v[h] or v[i] in the constraint graph Graph-based backjumping, exploits topology of constraint graph What happens if constraint graph is a clique?

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We want something that can jump and jump again, something that takes into consideration what caused a dead-end, not something that just looks at the topology of the constraint graph Combine BJ and GBJ

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CBJ Remember your conflicts, and when you have used them forget them. When we instantiate v[i] := x and check(v[i],v[h]) and it fails v[i] is in conflict with v[h] add h to the set confSet[i] confSet[i] is then the set of past variables that conflict with values in the domain of v[i]

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CBJ If there are no values remaining for v[i] Jump back to v[h], where v[h] is the deepest variable in conflict with v[i] The hope: re-instantiate v[h] will allow us to find a good value for v[i] Conflict-directed backjumping, exploits failures within the search process What happens if: constraint graph is dense, tight, or highly consistent? If there are no values remaining for v[h] Jump back to v[g], where v[g] is the deepest variable in conflict with v[i] or v[h] The hope: re-instantiate v[g] will allow us to find a good value for v[i] or a good value for v[h] that will be good for v[i] If there are no values remaining for v[g] Jump back to v[f], where v[f] is the deepest variable in conflict with v[i] or v[h] or v[g] The hope: re-instantiate v[f] will allow us to find a good value for v[i] or a good value for v[h] that will be good for v[i] or a good value for v[g] that will be good for v[h] and v[i]

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CBJ consistent := false confSet[i] := {0} for x in domain[i] while not(consistent) // find a consistent value begin consistent := true v[i] := x for h in (1.. i-1) while consistent // check backwards begin consistent := (check(v[i],v[h]) if not(consistent) then confSet[i] := confSet[i] {h} end if not(consistent) then delete(x,domain[i]) end

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CBJ When jumping back from v[i] to v[h] update conflict sets confSet[h] := confSet[h] confSet[i] \ {h} confSet[i] := {0} That is, when we jump back from v[h] jump back to a variable that is in conflict with v[h] or with v[i] Throw away everything you new on v[i] Reset all variables from v[h+1] to v[i] (i.e. domain and confSet)

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CBJ 1 2 3 4 5 6 {4,1,0} {2,0} conflict set

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CBJ 1 2 3 4 5 6 {2,1,0} conflict set

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CBJ (reduce thrashing) 1 2 3 4 5 6 {2,1,0} Jump back to deepest past variable in confSet (call it h) and then combine confSet[i] with confSet[h] History: Konkrat and V Beek, Gent and Underwood

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Forward Checking 1 2 3 4 5 6 7 9 8 NOTE: arrows go forward!

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1 2 3 4 5 6 7 9 8

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Check Forwards, Jump Back! 1 2 3 4 5 6 7 9 8 There are no values in cd[6] compatible with v[9] get more values into cd[9] (undo v[1]?) OR get more values into cd[6] (undo v[4]) … and if that doesnt work? undo v[3] so cd[4] gets value compatible with cd[6] that is then compatible with cd[9]

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CBJ Variants BM-CBJ, FC-CBJ, MAC-CBJ

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CBJ DkC If we jump from v[i] to v[h] and confSet[i] = {0,h} then remove value(v[h]) from domain(h) value(v[h]) is 2-inconsistent wrt v[i] If we jump from v[h] to v[g] and confSet[h] = {0,g} then remove value(v[g]) from domain(g) value(v[g]) is 3-inconsistent wrt v[i] and v[h] If we jump from v[g] to v[f] and confSet[g] = {0,f} then remove value(v[f]) from domain(f) value(v[f]) is 4-inconsistent wrt v[i] and v[h] and v[g] What happens if the problem is highly consistent? See JAIR 14 2001, Xinguang Chen & Peter van Beek

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CBJ ATMS If we jump from v[i] to v[g] and confSet[h] {0.. g-1} then do NOT reset domain(h) and do NOT reset confSet(h) Consider the past variables as assumptions and confSet[i] as an explanation Down side, we have more work to do. This is a kind of learning (what kind?)

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CBJ ~ DB confSet[x,i] gives the past variable in conflict with v[i] := x Finer grained: on jumping back we can deduce better what values to return to domains Down side, we have more work to do. This is an algorithm between CBJ and DB

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The bridge and the long jumpFunny things about cbj

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Value ordering on insoluble problems can have an effect Problem: V1 to V7, each with domain {A,B} nogoods {(1A,7A),(3A,7B),(5A,7B),(6A,7A),(6A,7B),(6B,7A),(6B,7B)} Var Val confSet V1 A V2 A V3 A V4 A V5 A V6 A V7 A/B {1,3} Var Val confSet V1 A V2 A V3 B V4 A V5 A V6 A V7 A/B {1,5} Var Val confSet V1 A V2 A V3 B V4 A V5 B V6 A V7 A/B {6} Var Val confSet V1 A V2 A V3 B V4 A V5 B V6 B V7 A/B {6} Finally V6 has no values and cbj jumps to V0

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Funny things about cbjValue ordering on insoluble problems can have an effect Problem: V1 to V7, each with domain {A,B} nogoods {(1A,7A),(3A,7B),(5A,7B),(6A,7A),(6A,7B),(6B,7A),(6B,7B)} Var Val confSet V1 B V2 B V3 B V4 B V5 B V6 B V7 A/B {6} Var Val confSet V1 B V2 B V3 B V4 B V5 B V6 A V7 A/B {6} Finally V6 has no values and cbj jumps to V0 We now order domains and choose B then A! Value ordering made a difference to an insoluble problem!

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Conflicting claims Bessier & Regin CP96: CBJ is nothing but an overhead random problems as evidence Smith & Grant IJCAI95: CBJ helps minimise occurrence of EHPs random problems as evidence Chen & van Beek JAIR 2001: CBJ is a tiny overhead When it makes a difference it is a HUGE difference random & real problems as evidence

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New CBJ I believe all state of the art sat solvers are using cbj (or have rediscovered cbj but dont know it) CBJ for QSAT: see recent AIJ conflict and solution directed!

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Who is not using cbj? Constraint programming! We dont jump and we dont learn

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Is speed everything? No How about explanations and retraction?

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Why is cbj not in CP? Need to propagate laterally (see MAC-CBJ tech report) but this is no big deal Need to get explanations out of constraints! Not just writing a good constraint propagator but a good constraint explainer! Maybe there is not yet the demand for retraction and explanation (but I dont believe that)

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So? Paper rejected from IJCAI91 (written in 1990) I was a Lisp programmer at the time (it shows) I think the experiments were very good (so there!) Nice study of influence of topological parameter on search cost. In conclusion I forgot to say CBJ was new … why? I like BMJ, it is cool (I was smart for 1 day) I think CBJ is pretty (natural, discovered, not invented) I like FC-CBJ (I can understand it) I identify work to be done (researchers love that (why?)) … and I make errors re-dvos (researchers love that (why?)) I put my results in perspective (trash them :) I got encouragement (Nadel) and help (Ole and Peter) I got a whole load of background (Rina) But it hurt … why did it take 3 years to get somebody to read it? It is still alive! We are not done yet.

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