763 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
A hybrid constraint programming and semidefinite programming approach for the stable set problem
This work presents a hybrid approach to solve the maximum stable set problem,
using constraint and semidefinite programming. The approach consists of two
steps: subproblem generation and subproblem solution. First we rank the
variable domain values, based on the solution of a semidefinite relaxation.
Using this ranking, we generate the most promising subproblems first, by
exploring a search tree using a limited discrepancy strategy. Then the
subproblems are being solved using a constraint programming solver. To
strengthen the semidefinite relaxation, we propose to infer additional
constraints from the discrepancy structure. Computational results show that the
semidefinite relaxation is very informative, since solutions of good quality
are found in the first subproblems, or optimality is proven immediately.Comment: 14 page
Approximate Graph Coloring by Semidefinite Programming
We consider the problem of coloring k-colorable graphs with the fewest
possible colors. We present a randomized polynomial time algorithm that colors
a 3-colorable graph on vertices with min O(Delta^{1/3} log^{1/2} Delta log
n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any
vertex. Besides giving the best known approximation ratio in terms of n, this
marks the first non-trivial approximation result as a function of the maximum
degree Delta. This result can be generalized to k-colorable graphs to obtain a
coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)}
log^{1/2} n) colors. Our results are inspired by the recent work of Goemans and
Williamson who used an algorithm for semidefinite optimization problems, which
generalize linear programs, to obtain improved approximations for the MAX CUT
and MAX 2-SAT problems. An intriguing outcome of our work is a duality
relationship established between the value of the optimum solution to our
semidefinite program and the Lovasz theta-function. We show lower bounds on the
gap between the optimum solution of our semidefinite program and the actual
chromatic number; by duality this also demonstrates interesting new facts about
the theta-function
Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications
In computer vision, many problems such as image segmentation, pixel
labelling, and scene parsing can be formulated as binary quadratic programs
(BQPs). For submodular problems, cuts based methods can be employed to
efficiently solve large-scale problems. However, general nonsubmodular problems
are significantly more challenging to solve. Finding a solution when the
problem is of large size to be of practical interest, however, typically
requires relaxation. Two standard relaxation methods are widely used for
solving general BQPs--spectral methods and semidefinite programming (SDP), each
with their own advantages and disadvantages. Spectral relaxation is simple and
easy to implement, but its bound is loose. Semidefinite relaxation has a
tighter bound, but its computational complexity is high, especially for large
scale problems. In this work, we present a new SDP formulation for BQPs, with
two desirable properties. First, it has a similar relaxation bound to
conventional SDP formulations. Second, compared with conventional SDP methods,
the new SDP formulation leads to a significantly more efficient and scalable
dual optimization approach, which has the same degree of complexity as spectral
methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton
methods, for the dual problem. Both of them are significantly more efficiently
than standard interior-point methods. In practice, the smoothing Newton solver
is faster than the quasi-Newton solver for dense or medium-sized problems,
while the quasi-Newton solver is preferable for large sparse/structured
problems. Our experiments on a few computer vision applications including
clustering, image segmentation, co-segmentation and registration show the
potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern
Analysis and Machine Intelligenc
Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference
We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF
inference problems. The core of our method is a very efficient bounding
procedure, which combines scalable semidefinite programming (SDP) and a
cutting-plane method for seeking violated constraints. In order to further
speed up the computation, several strategies have been exploited, including
model reduction, warm start and removal of inactive constraints.
We analyze the performance of the proposed method under different settings,
and demonstrate that our method either outperforms or performs on par with
state-of-the-art approaches. Especially when the connectivities are dense or
when the relative magnitudes of the unary costs are low, we achieve the best
reported results. Experiments show that the proposed algorithm achieves better
approximation than the state-of-the-art methods within a variety of time
budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page
Complex Semidefinite Programming and Max-k-Cut
In a second seminal paper on the application of semidefinite
programming to graph partitioning problems, Goemans and Williamson
showed in 2004 how to formulate and round a complex semidefinite program to give what is to date still the best-known approximation guarantee of .836008 for Max-3-Cut. (This approximation ratio was also achieved independently around the same time by De Klerk et
al..) Goemans and Williamson left open the problem of how to apply their techniques to Max-k-Cut for general k. They point out that it does not seem straightforward or even possible to formulate a good quality complex semidefinite program for the general Max-k-Cut problem, which presents a barrier for the further application of their techniques.
We present a simple rounding algorithm for the standard semidefinite
programmming relaxation of Max-k-Cut and show that it is equivalent to the rounding of Goemans and Williamson in the case of Max-3-Cut. This allows us to transfer the elegant analysis of Goemans and Williamson for Max-3-Cut to Max-k-Cut. For k > 3, the resulting approximation ratios are about .01 worse than the best known guarantees. Finally, we present a generalization of our rounding algorithm and conjecture (based on computational observations) that it matches the best-known guarantees of De Klerk et al
- …