] In the derivation of the spectral graph partitioning algorithm, we relax a combinatorial optimization problem to a continuous optimization problem. This relaxation has the following effects. The combinatorial problem requires an exact bisection of the graph, but the continuous algorithm can produce (after rounding) partitions that aren’t perfectly balanced The combinatorial problem cannot be modified to accommodate vertices that have different masses, whereas the continuous problem can The combinatorial problem requires finding eigenvectors, whereas the continuous problem requires only matrix multiplication The combinatorial problem is NP-hard, but the continuous problem can be solved in polynomial time

] In the derivation of the spectral graph partitioning algorithm, we relax a combinatorial optimization problem to a continuous optimization problem. This relaxation has the following effects. The combinatorial problem requires an exact bisection of the graph, but the continuous algorithm can produce (after rounding) partitions that aren’t perfectly balanced The combinatorial problem cannot be modified to accommodate vertices that have different masses, whereas the continuous problem can The combinatorial problem requires finding eigenvectors, whereas the continuous problem requires only matrix multiplication The combinatorial problem is NP-hard, but the continuous problem can be solved in polynomial time

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Polynomial Time

The problems for which the deterministic polynomial-time algorithm exists belong to the complexity class (p) and it is central in the field of computational complexity theory. According to Cobham’s thesis, polynomial time is a synonym for “feasible”, “fast”, and “efficient”.

Question

] In the derivation of the spectral graph partitioning algorithm, we relax a combinatorial optimization
problem to a continuous optimization problem. This relaxation has the following effects.
The combinatorial problem requires an exact bisection of the graph, but the continuous algorithm can produce (after rounding) partitions
that aren’t perfectly balanced
The combinatorial problem cannot be modified to accommodate vertices that have different
masses, whereas the continuous problem can
The combinatorial problem requires finding
eigenvectors, whereas the continuous problem requires only matrix multiplication
The combinatorial problem is NP-hard, but
the continuous problem can be solved in polynomial time

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