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Schulz, Christian ; Ucar, Bora (Ed.)We experimentally evaluate the performance of several Max Cut approximation algorithms. In particular, we compare the results of the Goemans and Williamson algorithm using semidefinite programming with Trevisan’s algorithm using spectral partitioning. The former algorithm has a known .878 approximation guarantee whereas the latter has a .614 approximation guarantee. We investigate whether this gap in approximation guarantees is evident in practice or whether the spectral algorithm performs as well as the SDP. We also compare the performances to the standard greedy Max Cut algorithm which has a .5 approximation guarantee and two additional spectral algorithms. The algorithms are tested on ErdősRenyi random graphs, complete graphs from TSPLIB, and realworld graphs from the Network Repository. We find, unsurprisingly, that the spectral algorithms provide a significant speed advantage over the SDP. In our experiments, the spectral algorithms return cuts with values which are competitive with those of the SDP.Free, publiclyaccessible full text available July 1, 2023

Aardal, Karen ; Sanità, Laura (Ed.)This paper considers the interplay between semidefinite programming, matrix rank, and graph coloring. Karger, Motwani, and Sudan [10] give a vector program for which a coloring of the graph can be encoded as a semidefinite matrix of low rank. By complementary slackness conditions of semidefinite programming, if an optimal dual solution has sufficiently high rank, any optimal primal solution must have low rank. We attempt to characterize graphs for which we can show that the corresponding dual optimal solution must have sufficiently high rank. In the case of the original Karger, Motwani, and Sudan vector program, we show that any graph which is a ktree has sufficiently high dual rank, and we can extract the coloring from the corresponding lowrank primal solution. We can also show that if the graph is not uniquely colorable, then no sufficiently high rank dual optimal solution can exist. This allows us to completely characterize the planar graphs for which dual optimal solutions have sufficiently high dual rank, since it is known that the uniquely colorable planar graphs are precisely the planar 3trees. We then modify the semidefinite program to have an objective function with costs, and explore when we can create a cost functionmore »Free, publiclyaccessible full text available May 27, 2023

The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, for example, algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [de Klerk E, Sotirov R (2012) Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry. Math. Programming 133(1):75–91.] present a semidefinite program (SDP) based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program but generally dominates it on small instances. We provide a family of simplicial TSP instances that shows that the integrality gap of this SDP is unbounded. Second, we show that these simplicial TSP instances imply the unbounded integrality gap of every SDP relaxation of the TSP mentioned in the survey on SDP relaxations of the TSP in section 2 of Sotirov [Sotirov R (2012) SDP relaxations for some combinatorial optimization problems. Anjos MF, Lasserre JB, eds., Handbook on Semidefinite, Conic and Polynomial Optimization (Springer, New York), 795–819.]. In contrast, the subtour linear program performs perfectly on simplicial instances. The simplicial instances thus form a natural litmusmore »