Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs
Jana Novotná, Karolina Okrasa, Michał Pilipczuk, Paweł Rzążewski, Erik Jan van Leeuwen, Bartosz Walczak · Algorithmica · 2020
Abstract Let $${\mathcal {C}}$$ C and $${\mathcal {D}}$$ D be hereditary graph classes. Consider the following problem: given a graph $$G\in {\mathcal {D}}$$ G ∈ D , find a largest, in terms of the number of vertices, induced subgraph of G that belongs to $${\mathcal {C}}$$ C . We prove that it can be solved in $$2^{o(n)}$$ 2 o ( n ) time, where n is the number of vertices of G , if the following conditions are satisfied: the graphs in $${\mathcal {C}}$$ C are sparse, i.e., they have linearly many edges in terms of the number of vertices; the graphs in $${\mathcal {D}}$$ D admit balanced separators of size governed by their density, e.g., $${\mathcal {O}}(\varDelta )$$ O ( Δ ) or $${\mathcal {O}}(\sqrt{m})$$ O ( m ) , where $$\varDelta$$ Δ and m denote the maximum degree and the number of edges, respectively; and the considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph. This leads, for example, to the following corollaries for specific classes $${\mathcal {C}}$$ C and $${\mathcal {D}}$$ D : a largest induced forest in a $$P_t$$ P t -free graph can be found in $$2^{\tilde{{\mathcal {O}}}(n^{2/3})}$$ 2 O ~ ( n