The Primal Pathwidth SETH

Abstract

Motivated by the importance of dynamic programming (DP) in parameterized complexity, we consider several fine-grained questions, such as the following examples: (i) can Dominating Set be solved in time $(3-ε)^{pw}n^{O(1)}$? (where $pw$ is the pathwidth) (ii) can Coloring be solved in time $pw^{(1-ε)pw}n^{O(1)}$? (iii) can a short reconfiguration between two size-$k$ independent sets be found in time $n^{(1-ε)k}$? Such questions are well-studied: in some cases the answer is No under the SETH, while in others coarse-grained lower bounds are known under the ETH. Even though questions such as the above seem "morally equivalent" as they all ask if a simple DP can be improved, the problems concerned have wildly varying time complexities, ranging from single-exponential FPT to XNLP-complete. This paper's main contribution is to show that, despite their varying complexities, these questions are not just morally equivalent, but in fact they are the same question in disguise. We achieve this by putting forth a natural complexity assumption which we call the Primal Pathwidth-Strong Exponential Time Hypothesis (pw-SETH) and which states that 3-SAT cannot be solved in time $(2-ε)^{pw}n^{O(1)}$, for any $ε>0$, where $pw$ is the pathwidth of the primal graph of the input. We then show that numerous fine-grained questions in parameterized complexity, including the ones above, are equivalent to the pw-SETH, and hence to each other. This allows us to obtain sharp fine-grained lower bounds for problems for which previous lower bounds left a constant in the exponent undetermined, but also to increase our confidence in bounds which were previously known under the SETH, because we show that breaking any one such bound requires breaking all (old and new) bounds; and because we show that the pw-SETH is more plausible than the SETH.

Figures (3)

• Figure 1: Summary of connections between other conjectures and the $\textrm{pw}$-SETH
• Figure 2: The consistency gadget used in the reduction of \ref{['lem:c4-2']}. The four boxes contain the backbone vertices $v_{j,i_1}$ and we concentrate on a specific variable $x$, which is represented by the vertices $x_j, x_{j+1}, x_{j+2}, x_{j+3}$. Dashed lines represent strong edges. The intended assignment is that either we place in the hitting set $x_j, x_j^b, x_{j+1}, x_{j+1}^b, x_{j+2}, \ldots$, or the vertices $x_j^a, x_j^c, x_{j+1}^a, x_{j+1}^c, \ldots$. This is achieved because we give the construction sufficient budget to delete half of the vertices of the top row, and for any two consecutive vertices we have to delete at least one (as they have either a strong edge or two common neighbors). To see that the construction has the claimed pathwidth consider the sequence of bags that resemble the shaded area, that is, take $v_{j,i_1}, v_{j,i_1+1},\ldots$ and also $v_{j+1,1},\ldots,v_{j+1,i_1}$. For each strong edge in the top row, there exists such a bag such that the endpoints of the strong edge have all their neighbors in the bag.
• Figure 3: Consistency gadget from \ref{['lem:sattolc']}. On the left we have vertex $x_{i,j}$ and on the right vertex $x_{i,j+1}$.

Theorems & Definitions (50)

• Conjecture 1: $\textrm{pw}$-SETH
• Lemma 2
• Lemma 3
• Definition 4
• Theorem 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10