On Subexponential and FPT-Time Inapproximability

Abstract

Fixed-parameter algorithms, approximation algorithms and moderately exponential algorithms are three major approaches to algorithm design. While each of them being very active in its own, there is an increasing attention to the connection between these different frameworks. In particular, whether Independent Set would be better approximable once endowed with subexponential-time or FPT-time is a central question. In this article, we provide new insights to this question using two complementary approaches; the former makes a strong link between the linear PCP conjecture and inapproximability; the latter builds a class of equivalent problems under approximation in subexponential time.

Appendix

1.1 Definitions of the Problems Considered

Vertex Cover

Input: :

A graph $G=(V,E)$.

Goal: :

Find a smallest vertex cover of $G$, i.e., a set $C$ of vertices such that for every $e=\{u,v\} \in E$, $C\cap \{u,v\}\ne \emptyset$.

Independent Set

Input: :

A graph $G=(V,E)$.

Goal: :

Find a largest independent set in $G$, i.e., a set of vertices which are pairwise nonadjacent.

Dominating Set

Input: :

A graph $G=(V,E)$.

Goal: :

Find a smallest dominating set in $G$, i.e., a set of vertices $S$ such that every vertex in $V \setminus S$ has a neighbor in $S$.

Independent Dominating Set

Input: :

A graph $G=(V,E)$.

Goal: :

Find a smallest set of vertices which is simultaneously an independent set and a dominating set in $G$.

Generalized Dominating Set

Input: :

A graph $G=(V,E)$ with a partition $V=(V_1, V_2, V_3)$ (some of the sets being possibly empty).

Goal: :

Find a smallest set of vertices $V'\subseteq V_1\cup V_2$ which dominate all vertices in $V_2\cup V_3$.

Bipartite Subgraph

Input: :

A graph $G=(V,E)$.

Goal: :

Find an induced bipartite subgraph of $G$ containing a maximum number of vertices.

Coloring

Input: :

A graph $G=(V,E)$.

Goal: :

Find a proper (vertex-)coloring of $G$, i.e., a coloring where no adjacent vertices get the same color, using a smallest number of colors.

Max Cut

Input: :

A graph $G=(V,E)$.

Goal: :

Find a set $S \subseteq V$ such that the number of edges having exactly one endpoint in $S$ is maximized.

3Sat

Input: :

A 3CNF $\phi$ on the variable set $V$.

Question: :

Does there exist a truth assignment of $V$ satisfying all clauses of $\phi$?

Max- $k$ Sat

Input: :

A CNF $\phi$ on the variable set $V$ containing at most $k$ literals per clause.

Goal: :

Find a truth assignement of $V$ that satisfies a maximum number of clauses.

Min-Sat

Input: :

A CNF $\phi$ on the variable set $V$.

Goal: :

Find a truth assignement of $V$ that satisfies a minimum number of clauses.

Max-3Lin

Input: :

A system $Az=b$ of linear equations in the variable set $V$ over $\mathbb {F}_2$, each equation involving exactly 3 variables.

Goal: :

Find an assignment of values to $V$ satisfying a maximum number of equations.

Max- $c$ -Csp

Input: :

A collection of $m$ boolean functions on the variable set $V$, where each function depends on at most $c$ variables.

Goal: :

Find a boolean assignment of $V$ that satisfies a maximum number of equations.

Max- $c$ -Csp Above Average

Input: :

A collection of $m$ boolean functions on the variable set $V$, where each function depends on at most $c$ variables, and a nonnegative integer $k$.

Parameter: :

$k$.

Question: :

Does there exist a boolean assignment of $V$ that satisfies at least $\rho \cdot m$ functions, where $\rho$ is the average fraction of functions satisfied by a uniform random assignment?

Set Packing

Input: :

A universe $\mathcal {U}$ and a collecton $\mathcal {F}$ of subsets of $\mathcal {U}$.

Goal: :

Find a maximum number of sets from $\mathcal {U}$ which are pairwise disjoint.

Set Cover

Input: :

A universe $\mathcal {U}$ and a collecton $\mathcal {F}$ of subsets of $\mathcal {U}$.

Goal: :

Find a minimum number of sets from $\mathcal {F}$ whose union is $\mathcal {U}$.

1.2 Some Words About MaxSNP and $\mathbf{L}$-Reductions

By Fagin’s Theorem, NP is characterized as the class of graph problems expressible in existential second-order logic. In this logic, one can quantify existentially and universally over the vertices but one is restricted to existential quantification over sets of vertices. In SNP (for Strict NP), the quantification over vertices can only be universal.

We now introduce $L$-reductions which are linear reductions mostly preserving approximation schemata.

Let $\varPi _A$ and $\varPi _B$ be two optimization problems, $v_A$ and $v_B$ being the two corresponding functions mapping a solution to its value. An $L$-reduction is defined by two functions $f$ and $g$ computable in polynomial-time and two constants $\alpha$ and $\beta$ such that:

• $f$ maps instances of $\varPi _A$ to instances of $\varPi _B$.

• $g$ maps solutions of $f(I)$ to solutions of $I$.

• OPT$_B(f(I)) \leqslant \alpha$OPT$_A(I)$.

• for every solution $S$ of $f(I)$, $|$OPT$_A(I)-v_A(g(S))| \leqslant \beta |$OPT$_B(f(I))-v_B(S)|$.

MaxSNP is the class of problems that $L$-reduce to a maximization version of a SNP problems.

A problem $\varPi$ is MaxSNP-hard if all the MaxSNP problems $L$-reduce to $\varPi$.

A problem is MaxSNP-complete if it is both MaxSNP-hard and in MaxSNP.