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In this work a convex relaxation of a subgraph isomorphism problem is proposed, which leads to a new lower bound that can provide a proof that a subgraph isomorphism between two graphs can not be found. The bound is based on a semidefinite programming relaxation of a combinatorial optimisation formulation for subgraph isomorphism and is explained in detail. We consider subgraph isomorphism problem instances of simple graphs which means that only the structural information of the two graphs is exploited and other information that might be available (e.g., node positions) is ignored. The bound is based on the fact that a subgraph isomorphism always leads to zero as lowest possible optimal objective value in the combinatorial problem formulation. Therefore, for problem instances with a lower bound that is larger than zero this represents a proof that a subgraph isomorphism can not exist. But note that conversely, a negative lower bound does not imply that a subgraph isomorphism must be present and only indicates that a subgraph isomorphism can not be excluded. In addition, the relation of our approach and the reformulation of the largest common subgraph problem into a maximum clique problem is discussed.

The subgraph isomorphism problem is a well-known combinatorial optimization problem and often involves the problem of finding the appropriate matching too. It is also of particular interest in computer vision where it can be exploited to recognise objects. For example, if an object in an image is represented by a graph, the object could be identified as subgraph within a possibly larger scene graph. Several approaches have been proposed to tackle the subgraph isomorphism problem and we refer to a few [

The main contribution of this paper lies in the convex relaxation of a subgraph isomorphism problem and the identification of a lower bound for this optimization problem. The computation of that bound is based on the SDP approximation of a combinatorial optimization formulation for subgraph isomorphism. The combinatorial optimization formulation and its convex relaxation is explained in detail. The approach is designed to find a subgraph isomorphism which maps the entire node-set of the possibly smaller graph to a subset of nodes in the second graph. We also discuss an interesting relation to an approach that is based on a reformulation of the largest common subgraph problem into a largest clique problem [

After providing the notation we use, we introduce a combinatorial quadratic optimization formulation for the subgraph isomorphism problem that can be interpreted as an error-correcting graph matching approach.

The integer optimization problem we end up with is generally an indefinite quadratic integer optimization problem which is known to be NP-hard [

In this paper, we consider simple graphs

A subgraph isomorphism is a mapping

In this section we propose and show the correctness of a combinatorial problem formulation for finding a subgraph isomorphism. The general idea is to find a bipartite matching between the set of nodes from the smaller graph to a subset of nodes of the larger graph. The matching is evaluated by an objective function that can be interpreted as a comparison of the structure between all possible node pairs in the first graph and the structure of the node pairs (to which the nodes are matched) in the second graph. A matching that leads to no structural differences has no costs and represents a subgraph isomorphism. Mathematically, the evaluation can be formulated as a quadratic objective function

The constraints make use of the matrices

Using double indices a nonzero vector element

An illustration of the 0,1-indicator vector with

The

Relational Structure Matrix:

Complementary Adjacency Matrices

We show this as similar representations will be used to illustrate some particular matrices that appear later in this paper.

An example graph and its adjacency matrix

In the following, we show that a 0,1-solution vector

The minimal value of the combinatorial optimization problem (

The elements of

A solution with the minimal value of zero of the quadratic optimization problem (

In order to prove this we look closer at the term

Due to the constraints,

The node

The nodes

The subcases we have to consider in B include all four possible structural configurations between the two pairs of nodes (

If the two nodes

Otherwise, if

The mappings which correspond to configuration case I and II are visualised in Figure

Analogously, the structure comparison term penalises assignments where pairs of nodes (

Finally, if

Figure

The itemisation of these four possible cases shows that only mappings that lead to a change in the structure are penalised with a cost. Structure preserving mappings which are compatible with a subgraph isomorphism are without costs and if all mappings are structure preserving it represents a subgraph isomorphism.

Note that due to the symmetry of the adjacency matrices the quadratic cost term

Finally, the sum (

List of all outcomes of the

Node configurations | Cost | ||||
---|---|---|---|---|---|

I: | 1 | 0 | 0 | 1 | 0 |

II: | 1 | 1 | 0 | 0 | 1 |

III: | 0 | 0 | 1 | 1 | 1 |

IV: | 0 | 1 | 1 | 0 | 0 |

(a) (Case I) adjacent nodes

I: Good assignment (no costs)

II: Bad assignment (costly)

(a) (Case III) a pair of nodes

III: Bad assignment (costly)

IV: Good assignment (no costs)

We wish to emphasise that the minimisation of (

In the next section we discuss in detail how we obtain the semidefinite relaxation of (

A randomly created subgraph problem instance with

In the following we explain the convex relaxation of the combinatorial isomorphism approach (

The constraint

In order to obtain an appropriate SDP relaxation for the combinatorial subgraph matching problem, we start with the reformulation of the objective function of (

Here we exploited the invariance of the trace operator under cyclic exchange. Besides being symmetric, the matrix

The relational structure matrix

Here the relational structure matrix

In the convex relaxation (

(a) In order to get a tight relaxation, we exploit the fact that

Here we make use of the Kronecker delta

(b) We restrict the first element in the matrix

The matrix

(c) The equality constraints

This defines

(d) All integer solutions

The indices

Even if the number of indices is high the structure of a single matrix is fairly simple as every matrix has only two nonzero elements. For all these constraints the corresponding constants

The shown matrix

Altogether this sums up to

The most computational effort within the SDP approach is needed for the computation of the solution of the SDP relaxation (

Mean computation time for three SDP solvers needed to solve the SDP relaxations of our experiments. The CSDP-solver fits best our needs.

Problem size ( | CSDP 6.0.1 | DSDP 5.8 | PENSDP 2.2 |
---|---|---|---|

7/15 | ( | ( | ( |

The CSDP-solver is a variant of the predictor corrector algorithm suggested by Helmberg et al. [

In [

Using that, we compute that a subgraph matching problem instance with

According to [

In this section we discuss the connections that can be drawn between our subgraph isomorphism approach and the maximum clique formulation for finding a maximum common subgraph isomorphism. Details about the maximum clique search in arbitrary graphs can be found for example in [

Equation (

Bounds for the example problem. The largest clique that can be obtained in the related maximum clique problem is

7 | 105 | 0.346 | 6 | 0.855 | 61.602 | 38.344 | 35 | 52.50 |

For the illustrative example (see Figure

The distribution of the objective values for the subgraph isomorphism problem which is shown in Figure

For a further investigation of the bound (

The experiments reveal that for various problem instances, the relaxation is tight enough to conclude that no subgraph isomorphism can exist. We obtained 123 problem instances with a lower bound

Summarisation of the results. We performed 1000 (

Preproc. | |||||||
---|---|---|---|---|---|---|---|

Non | 7/15 | 1000 | 661 | 123 ( | 877 | 538 | |

Pruning | 7/15 | 1000 | 661 | 327 ( | 673 | 334 |

None of the upper bounds for the maximum clique size comes close to the trivial bound

Bound | Mean bound | Min. | Max | |
---|---|---|---|---|

7/15 | 53.31 | 74.55 | ||

7/15 | 30.69 | 54.34 | ||

7/15 | 32 | 54 | ||

7/15 | 52.50 | 54.50 |

In the next section we investigate the improvements to the bound computation when mappings are removed from the problem formulation that cannot lead to a subgraph isomorphism.

We implemented a simple pruning technique to reduce the dimension of the SDP problem size. The basic idea of this is to eliminate all mappings

Applying the above-described technique to the problem instances in the previous section, the size of the SDP problem matrices reduces from

We expect that the bound could be slightly tightened by including also the

In this paper we proposed a convex relaxation bound to the subgraph isomorphism problem and showed that the bound is not only of theoretical interest but also applies to several instances of subgraph matching problems. It would be interesting to investigate which criteria a subgraph matching problem has to fulfill to result in a tight relaxation. Such insights could be useful in the process of creating or obtaining object graphs from images for object recognition tasks. At the current stage, reasonable tight bounds result from semidefinite problems with a problem matrix size of up to

This research was partly supported by Marie Curie Intra-European Fellowships within the 6th European Community Framework Programme and an