**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Person# Carsten Moldenhauer

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related units

Loading

Courses taught by this person

Loading

Related research domains

Loading

Related publications

Loading

People doing similar research

Loading

Courses taught by this person

No results

Related publications (9)

Loading

Loading

Loading

People doing similar research (37)

Related units (3)

Related research domains (3)

Approximation algorithm

In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable

Planar graph

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it c

Dual graph

In the mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair

We consider the Node-weighted Steiner Forest problem on planar graphs. Demaine et al. showed that a generic primal-dual algorithm gives a 6-approximation. We present two different proofs of an approximation factor of~$3$. Then, we draw a connection to Goemans and Williamson who apply the primal-dual algorithm to feedback problems on planar graphs. We reduce these problems to Node-weighted Steiner Forest and show that for the graphs in the reductions their respective linear programming relaxations are equivalent to each other. This explains why both type of problems can be approximated with primal-dual methods. We show that the largest sub-determinant of a matrix $A\in Q^{d\times n}$ can be approximated with a factor of $O(\log d)^{d/2}$ in polynomial time. This problem also subsumes the problem of finding a simplex of largest volume in the convex hull of $n$ points in $Q^d$ for which we obtain the same approximation guarantee. The previously best known approximation guarantee for both problems was $d^{(d-1)/2}$ by Khachiyan. We further show that it is NP-hard to approximate both problems with a factor of~$c^d$ for some explicit constant~$c$. We highlight the importance of sub-determinants in combinatorial optimization by showing their significance in two problems. First, the Stable Set problem asks for a maximum cardinality set of pairwise non-adjacent vertices. The problem is NP-hard to approximate with factor $n^{1-\epsilon}$ for any $\epsilon>0$, where $n$ is the number of vertices. We restrict to instances where the sub-determinants of the constraint matrix are bounded by a function in~$n$. This is equivalent to restricting to graphs with bounded odd cycle packing number $ocp$, the maximum number of vertex-disjoint odd cycles in the graph. We obtain a polynomial-time approximation scheme for graphs with $ocp=o(n/\log n)$. Further, we obtain an $\alpha$-approximation algorithm for general graphs where~$\alpha$ smoothly increases from a constant to $n$ as $ocp$ grows from $O(n/\log n)$ to $n/3$. In contrast, we show that assuming the exponential-time hypothesis, Stable Set cannot be solved in polynomial time if $ocp=\Omega(\log^{1+\epsilon} n)$ for some $\epsilon>0$. Second, we consider coloring $n$ intervals with $k$ colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is at most one. This problem induces a totally unimodular constraint matrix and is therefore efficiently solvable. We construct a fast algorithm with running time $O(n \log n + kn\log k)$.

, , ,

2014