Abstract
Given a vertex-weighted graph \(G = \langle V, E \rangle \), the minimum weighted vertex cover (MWVC) problem is to choose a subset of vertices with minimum total weight such that every edge in the graph has at least one of its endpoints chosen. While there are good solvers for the unweighted version of this NP-hard problem, the weighted version—i.e., the MWVC problem—remains understudied despite its common occurrence in many areas of AI—like combinatorial auctions, weighted constraint satisfaction, and probabilistic reasoning. In this paper, we present a new solver for the MWVC problem based on a novel reformulation to a series of SAT instances using a primal-dual approximation algorithm as a starting point. We show that our SAT-based MWVC solver (SBMS) significantly outperforms other methods.
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Notes
- 1.
While the MVC is approximable within a constant factor, this has no implications on the MIS problem. In fact, the MIS problem is one of the hardest combinatorial problems and has no polynomial-time constant-factor approximation algorithm unless P = NP [29].
- 2.
This inapproximability result is tighter under the unique games conjecture [16].
- 3.
It suffices for this reward to be greater than the sum of the weights of all vertices in the graph.
- 4.
We can compute much more informed lower and upper bounds as explained later.
- 5.
We skip a detailed discussion of this transformation since it is similar to the works of various authors mentioned later.
- 6.
In fact, this approach is employed by CircuitTSAT, a state-of-the-art solver for disjunctive temporal reasoning problems [24].
- 7.
UG-hard means “Unique Games-hard”, i.e., hard under the unique games conjecture.
- 8.
frb53-24-1 and frb59-26-4 are the only two exceptions with a gap of 2.
- 9.
See the second paragraph on page 18 of [5] that states “... MaxCLQdyn+EFL+SCR is not evaluated on BHOSLIB benchmark which is much harder and requires more effective technologies for exact algorithms ...”.
- 10.
Gurobi was competitive with SBMS on these five instances.
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Acknowledgments
The research at USC was supported by NSF under grant numbers 1409987 and 1319966 and a MURI under grant number N00014-09-1-1031. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the sponsoring organizations, agencies or the U.S. government.
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Xu, H., Kumar, T.K.S., Koenig, S. (2016). A New Solver for the Minimum Weighted Vertex Cover Problem. In: Quimper, CG. (eds) Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2016. Lecture Notes in Computer Science(), vol 9676. Springer, Cham. https://doi.org/10.1007/978-3-319-33954-2_28
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