MaxRes) to get a model that covers all of the edges while using the least number of vertices (if any). A Minimum Vertex Cover (MVC) is a vertex cover. The following are equivalent: (i) V0 is a clique of size k for the complement, G (ii) V0 is an independent set of size k for G (iii) V nV0 is a vertex cover of size n k for G. Lemma: Given an undirected graph G (V E) with n vertices and a subset V0 V of size k. (1) The computed set V C is a vertex cover for G. If Gk has girth 2k + 1, and there are no local inputs, then 0 C0 and 1 C1 are k-hop indistinguishable. 1: Clique, Independent set, and Vertex Cover. If that isn't the case, get a penalty of value 1 on the value of cover: (assert-soft (not vertex_i) :weight 1 :id cover)Īt last, solve the problem: (set-option :config opt.maxsmt_engine=maxres) only for optimathsatĪt this point, one can use any efficient MaxSAT/MaxSMT engine (e.g. explore the vertices which are at most k+1 hops away from each vertex v I. Third, assert that every edge edge_i_j must be covered: (assert edge_i_j)įourth, if an edge edge_i_j is covered, then either vertex i or vertex j must be true: (assert (=> edge_i_j (or vertex_i vertex_j)))įifth, for each vertex_i assert with a soft clause that vertex_i should be false. Second, declare a Boolean variable for each edge in the graph connecting a vertex i with a vertex j (assume undirected) (declare-fun edge_i_j () Bool) (Any vertex without edges should be omitted because it would just waste time.) I may be wrong, but I speculate that MaxSAT solvers may outperform MaxSMT ones on this particular problem.įirst, declare a Boolean variable for each vertex in the graph: (declare-fun vertex_1 () Bool) This would give you more choice when picking a solver to handle the problem. ![]() However, since it does not use any SMT feature, you can actually write the same thing with the standard WCNF format used at MaxSAT competitions. Note: my example is based on the SMT-LIB syntax and can be solved with a MaxSMT solver like, e.g., z3 and optimathsat. This is how I would approach the problem. Could anyone please help me to figure out what should I do next? Do I need a brand new encoding?įirst, let me premise that I have some trouble understanding your encoding, so i'll start from scratch. But looks like that I cannot simply change all clause from nCNF to 3CNF for getting the same result. 2.1 Approximation Algorithms for Vertex Cover The most widely-known 2-approximation algorithm for unweighted Vertex Cover works by greed-ily building a maximal matching in Gand adding all nodes adjacent to an edge in the matching to a cover. I am now thinking to reduce it from SAT further more, maybe to 3SAT. ![]() I can only get a result for any graph with < 20 vertices, otherwise it will just take minutes and hours for getting me a result. vertex cover: if some edge i-j is uncovered, then neither i nor j is tight. Now I am able to get the minimum vertex cover by using this encoding but the efficiency is pretty bad. So right now I am working on using SAT to resolve the minimum vertex cover problem, and here is my encoding for the graph G = įinally, at least one vertex in vertex cover should come from edges: For i and j in each edge e from EĪdd clause (x ∨ x ∨.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |