9,432 research outputs found
Partitioning into Isomorphic or Connected Subgraphs
This thesis deals mainly with the partitioning and connectedness of graphs. First, we show that the problem of partitioning the nodes of a graph into a specific number of subsets such that the induced subgraphs on these sets are isomorphic to one another is NP-complete. If the induced subgraphs have to be connected, the problem remains NP-complete. Then we inspect some special graph classes for which the problem is solvable in polynomial time.
Afterwards, we deal with the problem of defining a polytope by incidence vectors of nodes, which induce a connected graph. We inspect some facet-defining inequalities and their general structure. For some graph classes we state the full description.
We then proceed to the problem of partitioning the nodes of a graph into a given number of parts such that the induced graphs are connected. For the corresponding polytope we show the dimension and some facet defining inequalities. This theoretical inspection is advanced by the problem of partitioning a graph into different parts such that the parts induce a connected graph in order to maximize the induced cut. We introduce different ideas for solving this problem in SCIP and show the numerical results.
This leads to interesting problems on MIPs in general. As the problem in literature generally deals with the feasible region, we focus on the objective function. To do that, we inspect the problem of finding MIPs for problems with nonlinear objective functions. We discuss properties and requirements showing the existence or non-existence of particular formulations.
Lastly, we inspect the problem of partitioning the nodes of a graph such that all but one class are isomorphic. This problem becomes interesting if the part not inducing the isomorphism is minimized. For this purpose we also introduce a technique, which generates the parts by brute-force. Instead of partitioning the graph into isomorphic parts, we proceed to the problem of similar graphs. In this case we inspect different similarities and show algorithms which implement these
P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches
While the P vs NP problem is mainly approached form the point of view of
discrete mathematics, this paper proposes reformulations into the field of
abstract algebra, geometry, fourier analysis and of continuous global
optimization - which advanced tools might bring new perspectives and approaches
for this question. The first one is equivalence of satisfaction of 3-SAT
problem with the question of reaching zero of a nonnegative degree 4
multivariate polynomial (sum of squares), what could be tested from the
perspective of algebra by using discriminant. It could be also approached as a
continuous global optimization problem inside , for example in
physical realizations like adiabatic quantum computers. However, the number of
local minima usually grows exponentially. Reducing to degree 2 polynomial plus
constraints of being in , we get geometric formulations as the
question if plane or sphere intersects with . There will be also
presented some non-standard perspectives for the Subset-Sum, like through
convergence of a series, or zeroing of fourier-type integral for some natural . The last discussed
approach is using anti-commuting Grassmann numbers , making nonzero only if has a Hamilton cycle. Hence,
the PNP assumption implies exponential growth of matrix representation of
Grassmann numbers. There will be also discussed a looking promising
algebraic/geometric approach to the graph isomorphism problem -- tested to
successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure
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