1

The GSAT (Greedy Satisfiability) algorithm can be used to find a solution to a search problem encoded in CNF. I'm aware that since GSAT is greedy, it is incomplete (which means there would be cases where a solution might exist, but GSAT cannot find it). From the following link, I learned that this can happen when flipping variables greedily traps us in a cycle such as I → I' → I'' → I.

http://www.dis.uniroma1.it/~liberato/ar/incomplete/incomplete.html

I've been trying quite hard to come up with an actual instance that can show this, but have not been able to (and could not find examples elsewhere). Any help would be much appreciated. Thanks :)

P.S. I'm not talking about "hard" k-SAT problems in which the ratio of variables to clauses approaches 4.3. I'm just looking for a simple example, possibly involving the least number of variables and/or clauses required.

sarora
  • 499
  • 5
  • 20
  • I think the algorithm at that link has a bug: after selecting the best variable to change, if there are still unsatisfied clauses remaining it will go to step *1*, which causes everything to be randomly reinitialised, instead of trying a second greedy step. – j_random_hacker Feb 27 '14 at 06:52
  • And even if that problem is fixed, it's not possible to state a problem instance that cannot be found by this algorithm, since it could get lucky with its initial, random assignment of values to variables. – j_random_hacker Feb 27 '14 at 06:53

2 Answers2

0

Take a small unsatisfiable formula with n variables and run GSAT for > 2^n steps. Since there are only 2^n different combinations to try, GSAT must repeat itself - it will not stop because the formula is not satisifed.

One small unsatisfiable formula is (A V B V C) ^ (~A V B V C) ^ (A V ~B V C) ^ (~A V ~B V C) ^ (A V B V ~C) ^ (~A V B V ~C) ^ (A V ~B V ~C) ^ (~A V ~B V ~C) - all 8 combinations of 3-variable terms.

In Knuth vol 4A section 7.1.1 equation 32 P 56 Knuth gives what he calls an interesting 8-clause formula with eight different variables.

mcdowella
  • 19,301
  • 2
  • 19
  • 25
  • S/he's looking for a *satisfiable* formula that cannot be solved via GSAT. (Although with the GSAT algorithm described at the link the OP provided, no satisfiable problem instance is guaranteed to be missed, since it could "get lucky" with its first, random variable assignment.) – j_random_hacker Feb 27 '14 at 06:56
  • 1
    Thank you for your answer/comments guys. As j_random_hacker mentioned, I am looking for a _satisfiable_ formula that will give GSAT a hard time. I understand that a solution may be found given enough time (since GSAT restarts with a random initial assignment), but I'm looking for a set of clauses and **some initial assignment** that will throw GSAT off (potentially throw it into a loop so that it has to restart). – sarora Feb 27 '14 at 07:05
0

What about the formula:

{x_1, x_2, -x_3}, {-x_1, x_2, -x_3}, {-x_2, -x_3}, {-x_2, -x_3}, {x_2, x_3}, {x_2, x_3}

This formula is satisfied via the assignment (0,1,0). However if one starts with the initial assignment (0,0,1) then one gets the scores (1,2,2) and therefore will flip x_1. Then one gets to the assignment (1,0,1) which again leads to the scores (1,2,2) and you are stuck. Then only a restart with another initial assignment will help you get out.

Of course this a little bit constructed due to the two doubled clauses but I am sure one can extend this easily to achieve a formula without repeated clauses.

Arne
  • 272
  • 2
  • 12