When polynomial approximation meets exact computation

Abstract

We outline a relatively new research agenda aiming at building a new approximation paradigm by matching two distinct domains, the polynomial approximation and the exact solution of NP-hard problems by algorithms with guaranteed and non-trivial upper complexity bounds. We show how one can design approximation algorithms achieving ratios that are “forbidden” in polynomial time (unless a very unlikely complexity conjecture is confirmed) with worst-case complexity much lower than that of an exact computation.

Appendix

1.1 A list of combinatorial problems

max independent set::

Given a graph G(V, E), max independent set consists of finding a set $S \subseteq V$ of maximum size such that for any $(u,v) \in S \times S$, $(u,v) \notin E$.

max clique::

Given a graph G(V, E), max clique consists of finding a set $K \subseteq V$ of maximum size such that for any $(u,v) \in K \times K$, $(u,v) \in E$.

min set cover::

Given a set C of cardinality n and a system $\mathcal {S} = \{S_1, \ldots , S_m\} \subset 2^C$, min set cover consists of determining a minimum size subsystem $\mathcal {S}'$ such that $\cup _{S \in \mathcal {S}'}S = C$.

min vertex cover::

Given a graph G(V, E), min vertex cover consists of finding a set $C \subseteq V$ of minimum size such that, for every $(u,v) \in E$, either u, or v belongs to C.

min coloring::

Given a graph G(V, E), min coloring consists of partitioning the vertex-set V of G into a minimum number of independent sets.

min independent dominating set::

Given a graph G(V, E), min independent dominating set consists of finding the smallest independent set of G that is maximal for the inclusion.

capacitated dominating set::

Given a graph G(V, E) with each of its vertex v equipped with a number c(v) that represents the number of the other vertices that v can dominate, a set $S \subset V$ is a capacitated dominating set if there exists a function $f_{S}: V \setminus S \rightarrow S$ such that $f_S(v)$ is a neighbor of v for each $v \in V \setminus S$ and $|f^{-1}_S(v)| \leqslant c(v)$. The goal here is to determine a capacitated dominating set of the smallest possible size.

min bandwidth::

Given a graph G(V, E), the min bandwidth problem consists of labeling the vertices of V with distinct integers $f(v_i)$, in such a way that the quantity $\max \{|f(v_i) - f(v_j)|: v_iv_j \in E\}$ is minimized.

max tsp::

Given a complete edge-weighted graph G, the objective is to determine a maximum-weight Hamiltonian cycle of G.

bin packing::

Given an infinite number of bins $B_1, \ldots ,$ with capacity 1, and a list of n items with sizes $a_1,\dots ,a_n \in ~[0, 1]$, the objective is to pack them into a minimum number of bins with the constraint $\sum _{i \in B_j}a_i \leqslant 1$.

max sat::

Given a set of m disjunctive clauses over a set of n variables, max sat consists of finding a truth assignment for the variables that maximizes the number of satisfied clauses.

k-sat::

Given a set of m disjunctive clauses, each of them containing at most k literals), over a set of n variables, does there exist a truth-assignment on the variables simultaneously satisfying all of them?

hamiltonian path::

Given a graph G(V, E), does there exist a Hamiltonian path (i.e., a simple path that visits all the vertices of V) in G?