3 research outputs found
On solving the MAX-SAT using sum of squares
We consider semidefinite programming (SDP) approaches for solving the maximum
satisfiability problem (MAX-SAT) and the weighted partial MAX-SAT. It is widely
known that SDP is well-suited to approximate the (MAX-)2-SAT. Our work shows
the potential of SDP also for other satisfiability problems, by being
competitive with some of the best solvers in the yearly MAX-SAT competition.
Our solver combines sum of squares (SOS) based SDP bounds and an efficient
parser within a branch & bound scheme.
On the theoretical side, we propose a family of semidefinite feasibility
problems, and show that a member of this family provides the rank two
guarantee. We also provide a parametric family of semidefinite relaxations for
the MAX-SAT, and derive several properties of monomial bases used in the SOS
approach. We connect two well-known SDP approaches for the (MAX)-SAT, in an
elegant way. Moreover, we relate our SOS-SDP relaxations for the partial
MAX-SAT to the known SAT relaxations.Comment: 26 pages, 5 figures, 8 tables, 2 appendix page
On solving MAX-SAT using sum of squares
We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability (MAX-SAT) problem and weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiability problems by being competitive with some of the best solvers in the yearly MAX-SAT competition. Our solver combines sum of squares (SOS)–based SDP bounds and an efficient parser within a branch-and-bound scheme. On the theoretical side, we propose a family of semidefinite feasibility problems and show that a member of this family provides the rank-two guarantee. We also provide a parametric family of semidefinite relaxations for MAX-SAT and derive several properties of monomial bases used in the SOS approach. We connect two well-known SDP approaches for (MAX)-SAT in an elegant way. Moreover, we relate our SOS-SDP relaxations for partial MAX-SAT to the known SAT relaxations
On the generalized Ï‘-number and related problems for highly symmetric graphs
This paper is an in-depth analysis of the generalized Ï‘-number of a graph. The generalized Ï‘-number, Ï‘k(G), serves as a bound for both the k-multichromatic number of a graph and the maximum k-colorable subgraph problem. We present various properties of Ï‘k(G), such as that the sequence (Ï‘k(G))k is increasing and bounded from above by the order of the graph G. We study Ï‘k(G) when G is the strong, disjunction, or Cartesian product of two graphs. We provide closed form expressions for the generalized Ï‘-number on several classes of graphs including the Kneser graphs, cycle graphs, strongly regular graphs, and orthogonality graphs. Our paper provides bounds on the product and sum of the k-multichromatic number of a graph and its complement graph, as well as lower bounds for the k-multichromatic number on several graph classes including the Hamming and Johnson graphs