19 research outputs found
New bounds for the max--cut and chromatic number of a graph
We consider several semidefinite programming relaxations for the max--cut
problem, with increasing complexity. The optimal solution of the weakest
presented semidefinite programming relaxation has a closed form expression that
includes the largest Laplacian eigenvalue of the graph under consideration.
This is the first known eigenvalue bound for the max--cut when that is
applicable to any graph. This bound is exploited to derive a new eigenvalue
bound on the chromatic number of a graph. For regular graphs, the new bound on
the chromatic number is the same as the well-known Hoffman bound; however, the
two bounds are incomparable in general. We prove that the eigenvalue bound for
the max--cut is tight for several classes of graphs. We investigate the
presented bounds for specific classes of graphs, such as walk-regular graphs,
strongly regular graphs, and graphs from the Hamming association scheme
Efficient encoding of the weighted MAX k-CUT on a quantum computer using QAOA
The weighted MAX k-CUT problem consists of finding a k-partition of a given
weighted undirected graph G(V,E) such that the sum of the weights of the
crossing edges is maximized. The problem is of particular interest as it has a
multitude of practical applications. We present a formulation of the weighted
MAX k-CUT suitable for running the quantum approximate optimization algorithm
(QAOA) on noisy intermediate scale quantum (NISQ)-devices to get approximate
solutions. The new formulation uses a binary encoding that requires only
|V|log_2(k) qubits. The contributions of this paper are as follows: i) A novel
decomposition of the phase separation operator based on the binary encoding
into basis gates is provided for the MAX k-CUT problem for k >2. ii) Numerical
simulations on a suite of test cases comparing different encodings are
performed. iii) An analysis of the resources (number of qubits, CX gates) of
the different encodings is presented. iv) Formulations and simulations are
extended to the case of weighted graphs. For small k and with further
improvements when k is not a power of two, our algorithm is a possible
candidate to show quantum advantage on NISQ devices.Comment: 14 page
Stochastic graph partitioning: quadratic versus SOCP formulations
International audienceWe consider a variant of the graph partitioning problem involving knapsack constraints with Gaussian random coefficients. In this new variant, under this assumption of probability distribution, the problem can be traditionally formulated as a binary SOCP for which the continuous relaxation is convex. In this paper, we reformulate the problem as a binary quadratic constrained program for which the continuous relaxation is not necessarily convex. We propose several linearization techniques for latter: the classical linearization proposed by Fortet (Trabajos de Estadistica 11(2):111–118, 1960) and the linearization proposed by Sherali and Smith (Optim Lett 1(1):33–47, 2007). In addition to the basic implementation of the latter, we propose an improvement which includes, in the computation, constraints coming from the SOCP formulation. Numerical results show that an improvement of Sherali–Smith’s linearization outperforms largely the binary SOCP program and the classical linearization when investigated in a branch-and-bound approach
On relaxations of the max -cut problem formulations
A tight continuous relaxation is a crucial factor in solving mixed integer
formulations of many NP-hard combinatorial optimization problems. The
(weighted) max -cut problem is a fundamental combinatorial optimization
problem with multiple notorious mixed integer optimization formulations. In
this paper, we explore four existing mixed integer optimization formulations of
the max -cut problem. Specifically, we show that the continuous relaxation
of a binary quadratic optimization formulation of the problem is: (i) stronger
than the continuous relaxation of two mixed integer linear optimization
formulations and (ii) at least as strong as the continuous relaxation of a
mixed integer semidefinite optimization formulation. We also conduct a set of
experiments on multiple sets of instances of the max -cut problem using
state-of-the-art solvers that empirically confirm the theoretical results in
item (i). Furthermore, these numerical results illustrate the advances in the
efficiency of global non-convex quadratic optimization solvers and more general
mixed integer nonlinear optimization solvers. As a result, these solvers
provide a promising option to solve combinatorial optimization problems. Our
codes and data are available on GitHub
Orbitopal Fixing
The topic of this paper are integer programming models in which a subset of
0/1-variables encode a partitioning of a set of objects into disjoint subsets.
Such models can be surprisingly hard to solve by branch-and-cut algorithms if
the order of the subsets of the partition is irrelevant, since this kind of
symmetry unnecessarily blows up the search tree. We present a general tool,
called orbitopal fixing, for enhancing the capabilities of branch-and-cut
algorithms in solving such symmetric integer programming models. We devise a
linear time algorithm that, applied at each node of the search tree, removes
redundant parts of the tree produced by the above mentioned symmetry. The
method relies on certain polyhedra, called orbitopes, which have been
introduced bei Kaibel and Pfetsch (Math. Programm. A, 114 (2008), 1-36). It
does, however, not explicitly add inequalities to the model. Instead, it uses
certain fixing rules for variables. We demonstrate the computational power of
orbitopal fixing at the example of a graph partitioning problem.Comment: 22 pages, revised and extended version of a previous version that has
appeared under the same title in Proc. IPCO 200
A class of spectral bounds for Max k-Cut
International audienceIn this paper we introduce a new class of bounds for the maximum -cut problem on undirected edge-weighted simple graphs. The bounds involve eigenvalues of the weighted adjacency matrix together with geometrical parameters. They generalize previous results on the maximum (2-)cut problem and we demonstrate that they can strictly improve over other eigenvalue bounds from the literature. We also report computational results illustrating the potential impact of the new bounds