MAX-2-SAT: How Good is Tabu Search in the Worst-Case? Monaldo Mastrolilli and Luca Maria Gambardella IDSIA, Lugano (Switzerland)
Main result (Max-2-Sat) we give the first theoretical evidence of the advantage of a tabu search strategy over the basic local search alone that critically depends on the tabu list length T. AAAI-04 2
Max-2-Sat Input: n binary variables: A,B,C,… m clauses A + C A + B ~C + ~B ~B + ~A Goal: find assignment to variables that satisfies the max number of clauses : A=1 ; B=0 ; C=0 AAAI-04 3
Local Search: 1–flip Neighborhood A=1 ; B=0 ; C=1 A=1 ; B=1 ; C=1 ---------------- ---------------- A + C A + C A + B A + B ~C + ~B ~C + ~B ~B + ~A ~B + ~A A basic local search LS starts with any given assignment, and then repeatedly changes (“flips”) the assignment of a variable that leads to the largest decrease in the total number of unsatisfied clauses. AAAI-04 4
Local Search: worst-case A local optimal solution is defined as a state whose local neighborhood does not include a state that is strictly better. Let Opt loc denote the number of satisfied clauses at a local optimum of an instance of Max-2-Sat. 2 A=0 ; B=1 ; C=1 Opt loc ≥ Opt ---------------- 3 A + B A + C [Hansen, Jaumard 1990] ~B + ~C AAAI-04 5
Tabu Search = LS + memory [Glover 1986] IDEA: keep track of the T most recent flipped IDEA variables (tabu variables unless improvement) TS(T) • tabu search with tabu tenure at most T • 1-flip • choose the best non tabu + asp. criterion • random choice for equivalent candidates • if all moves tabu, choose the “less” tabu AAAI-04 6
Literature TSAT TSAT Mazure, Sais and Gregoire: Tabu Search for Sat, AAAI-97 • TSAT is extremely good and simple! • In their empirical study they found that the length of the tabu list is crucial!! • They show (experimentally) that the optimal length of the tabu list is linear with respect to the number n of variables. AAAI-04 7
Aim of this work The practical superiority of tabu search over local search alone has been already shown experimentally several times. A natural question addressed here is to understand if this superiority holds also from the worst-case point of view. Moreover, it is well known that a critical parameter of tabu techniques is the tabu list length. ISSUE: is TS(T) better than LS ISSUE even in the worst case? AAAI-04 8
Approximation Ratio Let c=n/T be the ratio between the number of variables n and the maximum tabu tenure T. We study the approximation ratio R(c) of TS(T) as a function of c. R ( c ) = BestF ound (TS(T)) . Opt AAAI-04 9
Approximation Ratio Upper Bound Theorem Starting from any arbitrary initial solution, the approximation ratio of TS ( T ) for MAX-2-SAT is bounded from above by 2 c 2 R R R ( c ) = 3 c 2 − 2 c + 2 . (1) 0.775 0.75 0.725 0.7 2 c=n/T 0.675 3 20 40 60 80 100 AAAI-04 10 c c
Approximation Ratio Upper Bound Bad News: Corollary: For MAX-2SAT, the asymptotic ratio of TS(T) is 2/3 for any T = o(n) (=sub linear). Recall the performanc e ratio of LS : 2 Opt Opt ≥ loc 3 AAAI-04 11
Approximation Ratio Upper Bound Is there a value of T = Θ ( n ) such that TS ( T ) exhibits a worst-case superiority over the basic local search alone? The Figure suggests that interesting values can be found when the ratio c = n/T is “small”. R R 0.775 0.75 0.725 0.7 0.675 20 40 60 80 100 c c AAAI-04 12
Tabu Search vs Local Search Good News: strong separation !! THEOREM: For MAX-2SAT, TS(n) achieves a performance ratio of 3/4 in O(nm) steps. Recall the performanc e ratio of LS : 2 Opt Opt ≥ loc 3 AAAI-04 13
Weighted Max-2-Sat We note that the same analysis and approximation ratio can be obtained for the weighted case of Max-2-Sat, although the time complexity might be only pseudopolynomially bounded in the input size (simply take multiple copies of the clauses). AAAI-04 14
Future Work In this work we analyzed the worst-case behavior of tabu search as a function of the tabu list length for Max-2-Sat. It would be interesting to understand if a similar analysis can be provided for the general MAX-SAT problem. An interesting starting case is Max-3-Sat. AAAI-04 15
Theoretical Lens: proof THEOREM: For MAX-2SAT, TS(n) achieves a performance ratio of 3/4 in O(nm) steps. Proof sketch Proof sketch • S(j): solution at step j • C h (j): subset of clauses with exactly h literals satisfied by S(j) (e.g. C 0 (j) is the set of unsat. clauses) NOTE: the presence of the aspiration criterion ensures that the best found solution by TS(n) is also a local optimum for LS AAAI-04 16
Theoretical Lens: proof (ctd) Assume that at step k we reach a local optimum. Two possibilities: (a) S(k) is a 3/4 approximate solution ==> done!! (a) (b) otherwise. (b) If case (b) (b) we will prove that in at most n steps a better solution is found (= the number of satisfied clauses is increased by at least 1). NOTE: Opt ≤ m, hence in at most O(mn) steps we are in case (a) (a) !! AAAI-04 17
Theoretical Lens: proof (ctd) Assumption: At step k we reach a local optimum S(k) Assumption and we are in case (b). (For simplicity, assume tabu list empty at step k.) Goal: prove that in at most n steps a better solution Goal than S(k) is found. OBSERVE: there is h h ≤ n such that at step k flip one not tabu variable that appears in C 0 (k k) k k+1 flip one not tabu variable that appears in C 0 (k+1 k+1) k+1 … k+h- -1 1 flip one not tabu variable that appears in C 0 (k+h k+h- -1 1) k+h k+h all variables in C 0 (k+h) have been flipped during the k+h last h ≤ n (i.e. all tabu) AAAI-04 18
Theoretical Lens: proof (ctd) … at step k+h we reach a solution such that all the variables in the not satisfied clauses, i.e. C 0 (k+h), have been flipped during the last h steps. C ( ) k C ( k h) ∩ + = ∅ 0 0 C ( ) k C ( k h) ∩ + = ∅ 1 0 all unsatisfied clauses at step k are sat. at step k+h all clauses in C 1 (k) are still sat. at step k+h AAAI-04 19
Theoretical Lens: proof (ctd) C ( ) k C ( k h) ∩ + = ∅ 0 0 C ( ) k C ( k h) ∩ + = ∅ 1 0 | C ( k h)| m | C ( )| k | C ( )| k + ≤ − − 0 0 1 Two possibilities: (1) |C 0 (k)| > |C 0 (k+h)| ==> done!! (1) (2) |C 0 (k)| ≤ |C 0 (k+h)|. (2) AAAI-04 20
Theoretical Lens: proof (ctd) | C ( k h)| m | C ( )| k | C ( )| k + ≤ − − 0 0 1 (2) |C 0 (k)| ≤ |C 0 (k+h)| (2) Known: 2|C 0 (k)| ≤ |C 1 (k)| since S(k) loc. opt. m | C ( k h)| | C ( )| k | C ( )| k ≥ + + + 0 0 1 2 | C ( )| k | C ( )| k ≥ + 0 1 4 | C ( )| k ≥ 0 1 1 ..a contradiction !! | C ( )| k m O pt ⇒ ≤ ≤ 4 4 0 AAAI-04 21
Tabu Search Origin It was first suggested by Glover, F. (1986) “Future paths for integer programming and links to artificial intelligence”, Computers & Operations Research, Vol. 13, pp. 533-549. The basic ideas of TS have also been sketched by: Hansen, P. “The steepest ascent mildest descent heuristic for combinatorial programming”, Congress on Numerical Methods in Combinatorial Optimization , Capri, Italy, 1986. AAAI-04 22
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