Fundamental Problems on Bounded-Treewidth Graphs: The Real Source of Hardness.

Summary

It is known for many algorithmic problems that if a tree decomposition of width t is given in the input, then the problem can be solved with exponential dependence on t. A line of research initiated by Lokshtanov, Marx, and Saurabh [SODA 2011] produced lower bounds showing that in many cases known algorithms already achieve the best possible exponential dependence on t, assuming the Strong Exponential-Time Hypothesis (SETH). The main message of this paper is showing that the same lower bounds can already be obtained in a much more restricted setting: informally, a graph consisting of a block of t vertices connected to components of constant size already has the same hardness as a general tree decomposition of width t. Formally, a (σ,δ)-hub is a set Q of vertices such that every component of Q has size at most σ and is adjacent to at most δ vertices of Q. We explore if the known tight lower bounds parameterized by the width of the given tree decomposition remain valid if we parameterize by the size of the given hub. - For every ε > 0, there are σ,δ > 0 such that Independent Set (equivalently Vertex Cover) cannot be solved in time (2-ε)^p⋅ n, even if a (σ, δ)-hub of size p is given in the input, assuming the SETH. This matches the earlier tight lower bounds parameterized by width of the tree decomposition. Similar tight bounds are obtained for Odd Cycle Transversal, Max Cut, q-Coloring, and edge/vertex deletions versions of q-Coloring. - For every ε > 0, there are σ,δ > 0 such that △-Partition cannot be solved in time (2-ε)^p ⋅ n, even if a (σ, δ)-hub of size p is given in the input, assuming the Set Cover Conjecture (SCC). In fact, we prove that this statement is equivalent to the SCC, thus it is unlikely that this could be proved assuming the SETH. - For Dominating Set, we can prove a non-tight lower bound ruling out (2-ε)^p ⋅ n^𝒪(1) algorithms, assuming either the SETH or the SCC, but this does not match the 3^p⋅ n^{𝒪(1)} upper bound. Thus our results reveal that, for many problems, the research on lower bounds on the dependence on tree width was never really about tree decompositions, but the real source of hardness comes from a much simpler structure. Additionally, we study if the same lower bounds can be obtained if σ and δ are fixed universal constants (not depending on ε). We show that lower bounds of this form are possible for Max Cut and the edge-deletion version of q-Coloring, under the Max 3-Sat Hypothesis (M3SH). However, no such lower bounds are possible for Independent Set, Odd Cycle Transversal, and the vertex-deletion version of q-Coloring: better than brute force algorithms are possible for every fixed (σ,δ).