M.Sc Student | Kulik Ariel |
---|---|

Subject | Submodular and Linear Maximization with Knapsack Constraints |

Department | Department of Computer Science |

Supervisor | Professor Hadas Shachnai |

Full Thesis text |

Submodular maximization generalizes many fundamental problems in discrete optimization, including Max-Cut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks.

In this work
we consider the problem of maximizing any submodular function subject to *d *knapsack constraints, where *d *is a fixed constant. For
short, we call this problem SUB. We establish a strong relation between the
discrete problem and its continuous relaxation, obtained through *extension by expectation *of
the submodular function. Formally, we show that, for any non-negative
submodular function, an *α*-approximation
algorithm for the continuous relaxation implies a randomized (*α−ε*)-approximation algorithm for
SUB. We use this relation to improve the best known approximation ratio for the
problem to 1*/*4 − *ε*, for any *ε > *0, and to obtain a
nearly optimal (1−*e*^{−1}−*ε*)−approximation ratio for the
monotone case, for any *ε
> *0. We further show that the probabilistic domain defined by a
continuous solution can be reduced to yield a polynomial size domain, given an
oracle for the extension by expectation. This leads to a deterministic version
of our technique.

Our approach has a potential of wider applicability, which we demonstrate on the examples of the Generalized Assignment Problem and Maximum Coverage with additional knapsack constraints.

We also consider the special case of
SUB in which the objective function is *linear*.
In this case, our problem reduces to the classic *d-dimensional knapsack *problem. It is known that,
unless *P *= *NP*, there is no *fully polynomial time approximation
scheme *for *d*-dimensional
knapsack, already for *d *=
2. The best known result is a *polynomial
time approximation scheme (PTAS) *due to Frieze and Clarke (*European J. of Operational Research, 100-
109, 1984*) for the case where *d
*≥ 2 is some
fixed constant. A fundamental open question is whether the problem admits an *efficient PTAS (EPTAS)*.

We resolve this question by showing that there is no EPTAS
for *d* dimensional
knapsack, already for *d *=
2, unless *W*[1] = *FPT*. Furthermore, we show that
unless all problems in SNP are solvable in sub-exponential time, there is no
approximation scheme for two-dimensional knapsack whose√

running time is *f*(1*/ε*)|I|^{o}^{(√1/ε)}, for any function *f*. Together, the two results suggest that a
significant improvement over the running time of the scheme of Frieze and
Clarke is unlikely to exist.