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A.: Computing globally optimal solutions for singlerow layout problems using semidefinite programming and cutting planes
 INFORMS J. Comput
, 2008
"... This paper is concerned with the singlerow facility layout problem (SRFLP). A globally optimal solution to the SRFLP is a linear placement of rectangular facilities with varying lengths that achieves the minimum total cost associated with the (known or projected) interactions between them. We demon ..."
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This paper is concerned with the singlerow facility layout problem (SRFLP). A globally optimal solution to the SRFLP is a linear placement of rectangular facilities with varying lengths that achieves the minimum total cost associated with the (known or projected) interactions between them. We demonstrate that the combination of a semidefinite programming relaxation with cutting planes is able to compute globally optimal layouts for large SRFLPs with up to thirty departments. In particular, we report the globally optimal solutions for two sets of SRFLPs previously studied in the literature, some of which have remained unsolved since 1988. Key words: singlerow facility layout, space allocation, semidefinite programming, cutting planes, combinatorial optimization
The Power of Semidefinite Programming Relaxations for MAXSAT
"... Abstract. Recently, Linear Programming (LP)based relaxations have been shown promising in boosting the performance of exact MAXSAT solvers. We compare Semidefinite Programming (SDP) based relaxations with LP relaxations for MAX2SAT. We will show how SDP relaxations are surprisingly powerful, pro ..."
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Abstract. Recently, Linear Programming (LP)based relaxations have been shown promising in boosting the performance of exact MAXSAT solvers. We compare Semidefinite Programming (SDP) based relaxations with LP relaxations for MAX2SAT. We will show how SDP relaxations are surprisingly powerful, providing much tighter bounds than LP relaxations, across different constrainedness regions. SDP relaxations can also be computed very efficiently, thus quickly providing tight lower and upper bounds on the optimal solution. We also show the effectiveness of SDP relaxations in providing heuristic guidance for iterative variable setting, significantly more accurate than the guidance based on LP relaxations. SDP allows us to set up to around 80 % of the variables without degrading the optimal solution, while setting a single variable based on the LP relaxation generally degrades the global optimal solution in the overconstrained area. Our results therefore show that SDP relaxations may further boost exact MAXSAT solvers. 1
Improved SDO Relaxations for SAT The Tseitin Instances on Toroidal Grid Graphs
"... The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. We consider instances of SAT specified by a set of boolean variables x1,..., xn and a propositional formula Φ = m∧ Cj, with each clause Cj having the form j=1 Cj = ∨ xk ∨ ∨ ¯xk ..."
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The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. We consider instances of SAT specified by a set of boolean variables x1,..., xn and a propositional formula Φ = m∧ Cj, with each clause Cj having the form j=1 Cj = ∨ xk ∨ ∨ ¯xk where Ij, Īj ⊆ {1,..., n}, Ij ∩ Īj = ∅, k∈Ij k ∈ Īj and ¯xi denotes the negation of xi. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. This research is concerned with the application of Semidefinite Optimization (SDO) to the SAT problem [4], particularly for proving unsatisfiability. The ultimate goal is a practical SDObased algorithm for solving SAT. Let P denote the set of all nonempty sets I ⊆ {1,..., n} such that the term ∏ xi appears in the instance’s satisfiability i∈I conditions. Introduce new variables xI: = ∏ xi for each I ∈ P, define the vector v: = (1, xI1,..., xI) P  T, and define the rankone matrix Y: = vvT whose rows and columns are indexed by ∅ ∪ P. By construction, Y∅,I = xI for all I ∈ P. Furthermore, YI1,I2 = YI3,I4 whenever I1∆I2 = I3∆I4, where Ii∆Ij denotes the symmetric difference of Ii and Ij. The tradeoff involved in adding such constraints to the SDO is that as the number of constraints increases, the SDO problems become increasingly more demanding computationally. We add the smaller set of constraints: Y ∅,I1 = YI2,I3, Y ∅,I2 = YI1,I3, and Y ∅,I3 = YI1,I2 (1) i∈I Consider a toroidal grid graph of the form