64 research outputs found
A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO)
AbstractThe “roof dual” of a QUBO (Quadratic Unconstrained Binary Optimization) problem has been introduced in [P.L. Hammer, P. Hansen, B. Simeone, Roof duality, complementation and persistency in quadratic 0–1 optimization, Mathematical Programming 28 (1984) 121–155]; it provides a bound to the optimum value, along with a polynomial test of the sharpness of this bound, and (due to a “persistency” result) it also determines the values of some of the variables at the optimum. In this paper we provide a graph-theoretic approach to provide bounds, which includes as a special case the roof dual bound, and show that these bounds can be computed in O(n3) time by using network flow techniques. We also obtain a decomposition theorem for quadratic pseudo-Boolean functions, improving the persistency result of [P.L. Hammer, P. Hansen, B. Simeone, Roof duality, complementation and persistency in quadratic 0–1 optimization, Mathematical Programming 28 (1984) 121–155]. Finally, we show that the proposed bounds (including roof duality) can be applied in an iterated way to obtain significantly better bounds. Computational experiments on problems up to thousands of variables are presented
Generalized roof duality and bisubmodular functions
Consider a convex relaxation of a pseudo-boolean function . We
say that the relaxation is {\em totally half-integral} if is a
polyhedral function with half-integral extreme points , and this property is
preserved after adding an arbitrary combination of constraints of the form
, , and where \gamma\in\{0, 1, 1/2} is a
constant. A well-known example is the {\em roof duality} relaxation for
quadratic pseudo-boolean functions . We argue that total half-integrality is
a natural requirement for generalizations of roof duality to arbitrary
pseudo-boolean functions. Our contributions are as follows. First, we provide a
complete characterization of totally half-integral relaxations by
establishing a one-to-one correspondence with {\em bisubmodular functions}.
Second, we give a new characterization of bisubmodular functions. Finally, we
show some relationships between general totally half-integral relaxations and
relaxations based on the roof duality.Comment: 14 pages. Shorter version to appear in NIPS 201
Generalized roof duality
AbstractThe roof dual bound for quadratic unconstrained binary optimization is the basis for several methods for efficiently computing the solution to many hard combinatorial problems. It works by constructing the tightest possible lower-bounding submodular function, and instead of minimizing the original objective function, the relaxation is minimized. However, for higher-order problems the technique has been less successful. A standard technique is to first reduce the problem into a quadratic one by introducing auxiliary variables and then apply the quadratic roof dual bound, but this may lead to loose bounds.We generalize the roof duality technique to higher-order optimization problems. Similarly to the quadratic case, optimal relaxations are defined to be the ones that give the maximum lower bound. We show how submodular relaxations can efficiently be constructed in order to compute the generalized roof dual bound for general cubic and quartic pseudo-boolean functions. Further, we prove that important properties such as persistency still hold, which allows us to determine optimal values for some of the variables. From a practical point of view, we experimentally demonstrate that the technique outperforms the state of the art for a wide range of applications, both in terms of lower bounds and in the number of assigned variables
Potts model, parametric maxflow and k-submodular functions
The problem of minimizing the Potts energy function frequently occurs in
computer vision applications. One way to tackle this NP-hard problem was
proposed by Kovtun [19,20]. It identifies a part of an optimal solution by
running maxflow computations, where is the number of labels. The number
of "labeled" pixels can be significant in some applications, e.g. 50-93% in our
tests for stereo. We show how to reduce the runtime to maxflow
computations (or one {\em parametric maxflow} computation). Furthermore, the
output of our algorithm allows to speed-up the subsequent alpha expansion for
the unlabeled part, or can be used as it is for time-critical applications.
To derive our technique, we generalize the algorithm of Felzenszwalb et al.
[7] for {\em Tree Metrics}. We also show a connection to {\em -submodular
functions} from combinatorial optimization, and discuss {\em -submodular
relaxations} for general energy functions.Comment: Accepted to ICCV 201
Decomposition of Trees and Paths via Correlation
We study the problem of decomposing (clustering) a tree with respect to costs
attributed to pairs of nodes, so as to minimize the sum of costs for those
pairs of nodes that are in the same component (cluster). For the general case
and for the special case of the tree being a star, we show that the problem is
NP-hard. For the special case of the tree being a path, this problem is known
to be polynomial time solvable. We characterize several classes of facets of
the combinatorial polytope associated with a formulation of this clustering
problem in terms of lifted multicuts. In particular, our results yield a
complete totally dual integral (TDI) description of the lifted multicut
polytope for paths, which establishes a connection to the combinatorial
properties of alternative formulations such as set partitioning.Comment: v2 is a complete revisio
Reducing Binary Quadratic Forms for More Scalable Quantum Annealing
Recent advances in the development of commercial quantum annealers such as the D-Wave 2X allow solving NP-hard optimization problems that can be expressed as quadratic unconstrained binary programs. However, the relatively small number of available qubits (around 1000 for the D-Wave 2X quantum annealer) poses a severe limitation to the range of problems that can be solved. This paper explores the suitability of preprocessing methods for reducing the sizes of the input programs and thereby the number of qubits required for their solution on quantum computers. Such methods allow us to determine the value of certain variables that hold in either any optimal solution (called strong persistencies) or in at least one optimal solution (weak persistencies). We investigate preprocessing methods for two important NP-hard graph problems, the computation of a maximum clique and a maximum cut in a graph. We show that the identification of strong and weak persistencies for those two optimization problems is very instance-specific, but can lead to substantial reductions in the number of variables
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