General Information
| Student: |
Pavel Dvořák |
| Office: |
444 |
| School: |
Charles University in Prague |
| E-mail: |
jajsem guess_what_is_here koblich.cz |
| Project: |
$L$-bounded cut |
Project Description
Let $(V, E, s, t, c)$ be a network. $L$-bounded cut $F$ is subset of $E$ such that network $(V, E \setminus F, s, t, c)$ does not contain any $s$-$t$ path of length $L$ or smaller. The problem is NP-hard for $4 \leq L \leq n^{1 - \epsilon}$. We know FPT algoritm for parameters $L$ and $tw$. We also know the problem parametrized only by $L$ is Para-NP-complete. Our goal is to prove hardness of the problem parametrized only by tree-width.
Current activities
- Week 1
- Choice of the problem
- Presentation about problem.
- Reading about FPT class [1].
- Reading about $W[P]$ class [2].
- Week 2
- Reading about $W$-hierarchy [3].
- Try to find hard problem for reduction
- Bridge workshop about geometry
- Reading the list of hard problems in $W$-hierarchy [4].
- Capacitated vertex cover is $W[1]$-hard [5]. in tree-width
- Week 3
- Trying to make reduction to capacitated vertex cover.
- Excellent talk from Patrick Devlin about generating functions.
- Bridge workshop about graph theory
- Discover that capacitated vertex cover is actually $W[1]$-hard in path-width.
- Week 4
- Capacitated vertex cover might not be a good candidate for reduction. I could not realised some gadget, where the size of cut can be controlled.
- Trying to make reduction from multicolor clique.
- Bridge workshop about ramsey theory.
- Making presentation for cultural day.
- Week 5
- Finish proof about $W[1]$-hardness.
- Reduction from multicolor clique.
- Talk about fourier tranform.
- Combinatorics bridge workshop.
- Week 6
- Writing a paper with the result.
- Fixing a minor error in the proof.
- Drawing fiqure of gadget.
- Talk about maritime security and robots, bridge workshop about number theory.
References
| [1] |
Jörg Flum, Martin Gröhe, Parametrized Complexity Theory, ISBN 3-540-29952-1, Chapter 1.
|
| [2] |
Jörg Flum, Martin Gröhe, Parametrized Complexity Theory, ISBN 3-540-29952-1, Chapter 3.
|
| [3] |
Jörg Flum, Martin Gröhe, Parametrized Complexity Theory, ISBN 3-540-29952-1, Chapter 7.
|
| [4] |
Marco Cesati, Compendium of Parameterized Problems.
|
| [5] |
Michael Dom, Daniel Lokshtanov, Saket Saurabh, and Yngve Villanger, Capacitated
domination and covering: A parameterized perspective, in IWPEC'08, vol. 5018 of LNCS,
Springer, 2008, pp. 78-90.
|
