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911 Backtracking
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Backtracking is based on the observation that it is often possible to reject a solution by looking at just a small portion of it For example, if an instance of SAT contains the clause (x 1 x2 ), then all assignments with x1 = x2 = 0 (ie, false) can be instantly eliminated To put it differently, by quickly checking and discrediting this partial assignment, we are able to prune a quarter of the entire search space A promising direction, but can it be systematically exploited Here s how it is done Consider the Boolean formula (w, x, y, z) speci ed by the set of clauses (w x y z), (w x), (x y), (y z), (z w), (w z) We will incrementally grow a tree of partial solutions We start by branching on any one variable, say w:
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Initial formula w=0 w=1
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Plugging w = 0 and w = 1 into , we nd that no clause is immediately violated and thus neither of these two partial assignments can be eliminated outright So we need to keep 266
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branching We can expand either of the two available nodes, and on any variable of our choice Let s try this one:
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Initial formula w=0 w=1
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This time, we are in luck The partial assignment w = 0, x = 1 violates the clause (w x) and can be terminated, thereby pruning a good chunk of the search space We backtrack out of this cul-de-sac and continue our explorations at one of the two remaining active nodes In this manner, backtracking explores the space of assignments, growing the tree only at nodes where there is uncertainty about the outcome, and stopping if at any stage a satisfying assignment is encountered In the case of Boolean satis ability, each node of the search tree can be described either by a partial assignment or by the clauses that remain when those values are plugged into the original formula For instance, if w = 0 and x = 0 then any clause with w or x is instantly satis ed and any literal w or x is not satis ed and can be removed What s left is (y z), (y), (y z) Likewise, w = 0 and x = 1 leaves with the empty clause ( ) ruling out satis ability Thus the nodes of the search tree, representing partial assignments, are themselves SAT subproblems This alternative representation is helpful for making the two decisions that repeatedly arise: which subproblem to expand next, and which branching variable to use Since the bene t of backtracking lies in its ability to eliminate portions of the search space, and since this happens only when an empty clause is encountered, it makes sense to choose the subproblem that contains the smallest clause and to then branch on a variable in that clause If this clause happens to be a singleton, then at least one of the resulting branches will be terminated (If there is a tie in choosing subproblems, one reasonable policy is to pick the one lowest in the tree, in the hope that it is close to a satisfying assignment) See Figure 91 for the conclusion of our earlier example More abstractly, a backtracking algorithm requires a test that looks at a subproblem and quickly declares one of three outcomes: 1 Failure: the subproblem has no solution 2 Success: a solution to the subproblem is found 3 Uncertainty 267 (), (y z),
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Figure 91 Backtracking reveals that is not satis able
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(w x y z), (w x), (x y), (y z), (z w), (w z) w=0 (x y z), (x), (x y), (y z) x=0 (y z), (y), (y z) y=0 (z), (z) z=0 () z=1 () y=1 () x=1 (), (y z) w=1 (x y), (y z), (z), (z) z=0 (x y), () z=1 (x y), (y), ()
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In the case of SAT, this test declares failure if there is an empty clause, success if there are no clauses, and uncertainty otherwise The backtracking procedure then has the following format Start with some problem P0 Let S = {P0 }, the set of active subproblems Repeat while S is nonempty: choose a subproblem P S and remove it from S expand it into smaller subproblems P 1 , P2 , , Pk For each Pi : If test(Pi ) succeeds: halt and announce this solution If test(Pi ) fails: discard Pi Otherwise: add Pi to S Announce that there is no solution For SAT, the choose procedure picks a clause, and expand picks a variable within that clause We have already discussed some reasonable ways of making such choices With the right test, expand, and choose routines, backtracking can be remarkably effective in practice The backtracking algorithm we showed for SAT is the basis of many successful satis ability programs Another sign of quality is this: if presented with a 2 SAT instance, it will always nd a satisfying assignment, if one exists, in polynomial time (Exercise 91)!
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