Proportionally dense subgraph of maximum size: Complexity and approximation

Abstract

We define a proportionally dense subgraph (PDS) as an induced subgraph of a graph with the property that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the graph. We prove that the problem of finding a PDS of maximum size is APX-hard on split graphs, and NP-hard on bipartite graphs. We also show that deciding if a PDS is inclusion-wise maximal is co-NP-complete on bipartite graphs. Nevertheless, we present a simple polynomialtime (2− 2 ∆+1 )-approximation algorithm for the problem, where ∆ is the maximum degree of the graph. Finally, we show that all Hamiltonian cubic graphs with n vertices (except two) have a PDS of size ⌊ 2n+1 3 ⌋, which we prove to be an upper bound on the size of a PDS in cubic graphs.

Graphs without a partition into two proportionally dense subgraphs

“…Hence, only the vertices inside the PDS must be satisfied. In [4], the authors prove that Max PDS is NP-hard on bipartite and split graphs, and propose a polynomial-time (2− 2 ∆+1 )approximation algorithm, where ∆ is the maximum degree of the graph. They also show that deciding if a subset of vertices can be a (proper) subset of the vertices of a PDS is co-NP-complete on bipartite graphs.…”