Convex and linear models of NP-problems

Reducing the NP-problems to the convex/linear analysis on the Birkhoff polytope. Introduction Since the classical works of J. Edmonds [2, and others], linear modeling became a common technique in combinatorial optimization [8, 9, 13, 14, 15, 16, and others]. Often, the linear models are expressed with some constrains on the incidence vector. The major benefit of this approach is the symmetry of the resulting model: the resulting equations are an invariant under relabeling. The major disadvantage of the approach is difficulty to express the constrains explicitly due to their size and structural complexity [8, 9, 13, and others]. In this work, the Subgraph Isomorphism Problem [3, 4, 7, 11] is taken as a basic NP-problem. The adjacency and incidence matrices are used to express the linear and convex models explicitly. In such asymmetric models, the unknown is a relabeling. The relabeling is presented with an unknown permutation matrix. That reduces the NP-problems to the linear/convex analysis on the Birkhoff polytope [1]. 1. Adjacency matrix models Let’s take the Subgraph Isomorphism Problem [3, 4, 7]: whether a given multi digraph g contains a subgraph which is isomorphic with another given multi digraph s. That is a NP-complete problem. Let n and m be powers of vertex-sets of g and s, appropriately. Based on a node labeling/enumeration, let’s construct adjacency matrices of these digraphs matrices G and S, appropriately. In terms of these matrices, the problem is a compatibility problem for the following quadratic system where the unknown is permutation matrix X = (xij)n×n: (1.1) PmnX GXP mn ≥ S Here, matrix Pmn is a truncation: Pmn = (Um 0)m×n, where matrix Um is the union matrix of size m×m. Permutation matrix X presents all n! possible ways to label vertices of (multi) digraph g. Compatibility of system 1.1 means that there is such a vertex labeling of digraph g that the appropriate adjacency matrix of that digraph will “cover” the adjacency matrix of (multi) digraph s. Such a labeling of g would make the positive solution for instance (g, s) the self evident. Obviously, if the system is 2000 Mathematics Subject Classification. Primary 68Q15, 68R10, 90C57.