A Completeness Theory for Polynomial (Turing) Kernelization

Abstract

The framework of Bodlaender et al. (J Comput Sys Sci 75(8):423–434, 2009) and Fortnow and Santhanam (J Comput Sys Sci 77(1):91–106, 2011) allows us to exclude the existence of polynomial kernels for a range of problems under reasonable complexity-theoretical assumptions. However, some issues are not addressed by this framework, including the existence of Turing kernels such as the “kernelization” of leaf out-branching $(k)$ that outputs $n$ instances each of size poly$(k)$. Observing that Turing kernels are preserved by polynomial parametric transformations (PPTs), we define two kernelization hardness hierarchies by the PPT-closure of problems that seem fundamentally unlikely to admit efficient Turing kernelizations. This gives rise to the MK- and WK-hierarchies which are akin to the M- and W-hierarchies of parameterized complexity. We find that several previously considered problems are complete for the fundamental hardness class WK[1], including Min Ones $d$ -SAT $(k)$, Binary NDTM Halting $(k)$, Connected Vertex Cover $(k)$, and Clique parameterized by $k \log n$. We conjecture that no WK[1]-hard problem admits a polynomial Turing kernel. Our hierarchy subsumes an earlier hierarchy of Harnik and Naor that, from a parameterized perspective, is restricted to classical problems parameterized by witness size. Our results provide the first natural complete problems for, e.g., their class $VC_1$; this had been left open.

Appendix: Problem Zoo

Below we provide problem statements to all problems discussed in the paper. We adopt the notation that appends brackets at the end of problem names to specify the parameterization used for the specific problem. For instance, Connected Vertex Cover $(k)$ denotes the Connected Vertex Cover problem parameterized by the number $k$ of vertices in the solution.

${\Phi }$-SAT:

Input: A formula $\phi \in \Phi$ with $n$ variables, and an integer $k$.

Task: Decide whether $\phi$ is satisfiable.

${\Phi }$-WSAT:

Input: A formula $\phi \in \Phi$ with $n$ variables, and an integer $k$.

Task: Decide whether $\phi$ is satisfiable by an assignment of Hamming weight $k$ (an assignment that assigns exactly $k$ variables the boolean value 1).

Binary NDTM Halting:

Input: A Turing machine $M$ of size $n$ with a binary alphabet, and an integer $k$.

Task: Decide whether $M$ halts on the empty string in $k$ steps.

Clique:

Input: A graph $G$ with $n$ vertices, and an integer $k$.

Task: Decide whether $G$ has a clique of size $k$ (a pairwise adjacent subset of $k$ vertices).

Capacitated Vertex Cover:

Input: A graph $G$ with $n$ vertices, a capacity function $\alpha : V(G) \rightarrow \mathbb {N}$, and an integer $k$.

Task: Decide whether $G$ has a capacitated vertex cover of size $k$ (a subset of $k$ vertices $S$ that are incident with each edge $G$, and such that each vertex $v \in S$ is incident with at most $\alpha (v)$ edges).

Connected Vertex Cover:

Input: A graph $G$ with $n$ vertices, and an integer $k$.

Task: Decide whether $G$ has a connected vertex cover of size $k$ (a connected subset of $k$ vertices $S$ that are incident with each edge of $G$).

Directed Multicolored Cycle:

Input: A directed graph $G$, a coloring function $c:V \rightarrow [k]$, and an integer $k$.

Task: Decide whether $G$ has a multicolored directed cycle of length $k$ (a directed cycle which includes exactly one vertex from each color).

Directed Multicolored Path:

Input: A directed graph $G$, a coloring function $c:V \rightarrow [k]$, and an integer $k$.

Task: Decide whether $G$ has a multicolored directed path of length $k$ (a directed path which includes exactly one vertex from each color).

Disjoint Cycles:

Input: A graph $G$ with $n$ vertices, and an integer $k$.

Task: Decide whether $G$ contains $k$ pairwise disjoint cycles.

Disjoint Factors:

Input: A $n$-character string $S$ over the alphabet $[k]$.

Task: Decide whether there exists a set of $k$ non-overlapping substrings $S_1,\ldots ,S_k$ of $S$ such that $S_i$ is of the form $i \cdots i$ for every alphabet symbol $i \in [k]$.

Disjoint Paths:

Input: A graph $G$ with $n$ vertices, and $k$ pairs of vertices $(s_1,t_1),\ldots ,(s_k,t_k)$.

Task: Decide whether $G$ contains $k$ pairwise disjoint paths connecting $s_i$ to $t_i$ for all $i \in [k]$.

Dominating Set:

Input: A graph $G$ with $n$ vertices, and an integer $k$.

Task: Decide whether $G$ has a dominating set of size $k$ (a set $D$ of $k$ vertices for which every vertex not in $D$ has a neighbor in $D$).

Exact Hitting Set:

Input: A hypergraph $(V,\mathcal {E})$ with $|V|=n$ and $|\mathcal {E}|=m$.

Task: Decide whether $(V,\mathcal {E})$ has an exact hitting set (a subset $S \subseteq V$ such that $|S \cap E| = 1$ for all $E \in \mathcal {E}$).

Exact Set Cover:

Input: A hypergraph $(V,\mathcal {E})$ with $|V|=n$ and $|\mathcal {E}|=m$.

Task: Decide whether $(V,\mathcal {E})$ has an exact set cover (a subset $\mathcal {S}\subseteq \mathcal {E}$ of pairwise disjoint edges with $\bigcup \mathcal {S}= V$).

Independent Set:

Input: A graph $G$ with $n$ vertices, and an integer $k$.

Task: Decide whether $G$ has an independent set of size $k$ (a pairwise non-adjacent subset of $k$ vertices).

Hitting Set:

Input: A hypergraph $(V,\mathcal {E})$ with $|V|=n$ and $|\mathcal {E}|=m$, and an integer $k$.

Task: Decide whether $G$ has a hitting set of size $k$ (a subset $S \subseteq V$ of size $k$ with $S \cap E \ne \emptyset$ for all $E \in \mathcal {E}$).

Local Circuit SAT:

Input: A circuit $C$ over $k+k\log m$ variables and of size $k+k\log m$, and a string $S$.

Task: Decide whether there is a list of $k$ positions $i_1,\ldots ,i_k$ in $S$ such that feeding the contents of the positions to the first $k$ inputs, and the binary expansions of $i_1,\ldots ,i_k$ to the remaining inputs, causes $C$ to accept.

Min Ones $d$-SAT:

Input: A formula $\phi \in \Gamma _{1,d}$ with $n$ variables, and an integer $k$.

Task: Decide whether $\phi$ is satisfiable by an assignment of Hamming weight at most $k$.

Multicolored $\Phi$-WSAT:

Input: A formula $\phi \in \Phi$ over a variable set $X$ of size $n$, a coloring function $c:X \rightarrow [k]$, and an integer $k$.

Task: Decide whether $\phi$ is satisfiable by an multicolored assignment of Hamming weight $k$ (an assignment where no two variables of same color are assigned a 1).

Multicolored Clique:

Input: A graph $G=(V,E)$ with $|V|=n$, a coloring function $c: V \rightarrow [k]$, and an integer $k$.

Task: Decide whether $G$ has a multicolored clique of size $k$ (a clique containing exactly one vertex of each color).

Multicolored Cycle:

Input: A graph $G$, a coloring function $c:V \rightarrow [k]$, and an integer $k$.

Task: Decide whether $G$ has a multicolored cycle of length $k$ (a cycle which includes exactly one vertex from each color).

Multicolored Hitting Set:

Input: A hypergraph $(V,\mathcal {E})$ with $|V|=n$ and $|\mathcal {E}|=m$, a coloring function $c:V \rightarrow [k]$, and an integer $k$.

Task: Decide whether $G$ has a multicolored hitting set of size $k$ (a hitting set which includes exactly one vertex from each color).

Multicolored Path:

Input: A graph $G$, a coloring function $c:V \rightarrow [k]$, and an integer $k$.

Task: Decide whether $G$ has a multicolored path of length $k$ (a path which includes exactly one vertex from each color).

NDTM Halting:

Input: A Turing machine $M$ of size $n$, and an integer $k$.

Task: Decide whether $M$ halts on the empty string in $k$ steps.

Perfect Deletion:

Input: A graph $G$ on $n$ vertices, and an integer $k$.

Task: Decide whether $G$ has at most $k$ vertices $S$ such that $G-S$ is perfect.

Set Cover:

Input: A hypergraph $(V,\mathcal {E})$ with $|V|=n, \,|\mathcal {E}|=m$, and $\max _{E\in \mathcal {E}}|E|=d$. Also, an integer $k$.

Task: Decide whether $(V,\mathcal {E})$ has a set cover of size $k$ (a subset $\mathcal {S}\subseteq \mathcal {E}$ of $k$ edges with $\bigcup \mathcal {S}= V$).

Small Subset Sum:

Input: An integer $k$, a set $S$ of integers of size at most $2^k$, and an integer $t$.

Task: Decide whether there are at most $k$ distinct integers in $S$ that sum up to $t$.

Steiner Tree:

Input: A graph $G=(V,E)$ with $|V|=n$, a set of $t$ terminals $T \subseteq V$, a set of $\ell$ non-terminals $N \subseteq V$, and an integer $k$.

Task: Decide whether there is a subset of at most $k$ non-terminals $N' \subseteq N$ such that $G[T \cup N']$ is connected.

Unique Coverage:

Input: A hypergraph $(V,\mathcal {E})$ with $|V|=n$ and $|\mathcal {E}|=m$, and an integer $k$.

Task: Decide whether there exists a subset $\mathcal {E}' \subseteq E$ such that at least $k$ vertices are contained in exactly one edge in $\mathcal {E}'$.

Weakly Chordal Deletion:

Input: A graph $G$ on $n$ vertices, and an integer $k$.

Task: Decide whether $G$ has at most $k$ vertices $S$ such that $G-S$ is weakly chordal.