The Graph Minor Algorithm with Parity Conditions

@article{Kawarabayashi2011TheGM,
  title={The Graph Minor Algorithm with Parity Conditions},
  author={K. Kawarabayashi and B. Reed and P. Wollan},
  journal={2011 IEEE 52nd Annual Symposium on Foundations of Computer Science},
  year={2011},
  pages={27-36}
}
We generalize the seminal Graph Minor algorithm of Robertson and Seymour to the parity version. We give polynomial time algorithms for the following problems:\begin{enumerate}\itemthe parity $H$-minor (Odd $K_k$-minor) containment problem, and\itemthe disjoint paths problem with $k$ terminals and the parity condition for each path, \end{enumerate}as well as several other related problems. We present an $O(m \alpha(m,n) n)$ time algorithm for these problems for any fixed $k$, where $n,m$ are the… Expand
The odd Hadwiger's conjecture is "almost" decidable
TLDR
The last odd minor of G, H, is indeed a minimal counterexample to the odd Hadwiger's conjecture for the case of t, and the result says that a minimal countederexamples satisfies the lsat conclusion. Expand
Packing Edge-Disjoint Odd Eulerian Subgraphs Through Prescribed Vertices in 4-Edge-Connected Graphs
TLDR
The Erdos--Posa property for edge-disjoint packing of S-closed walks with parity constraints in 4-edge-connected graphs is shown and a fixed-parameter algorithm for finding edge- Disjoint walks satisfying the conditions in a 4- edge-connected graph for a parameter $k$. Expand
Erdös-Pósa property and its algorithmic applications: parity constraints, subset feedback set, and subset packing
TLDR
This paper extends both Thomassen's result and Reed's result on the Erdos-Posa theorem to cycles that are required to go through a subset of S, and gives the first fixed parameter algorithms for the two problems: the feedback set problem with respect to the S-cycles, and the cycle packing problem. Expand
Parameterized Tractability of Multiway Cut with Parity Constraints
TLDR
This paper shows that Parity Multiway Cut is fixed parameter tractable (FPT) by giving an algorithm that runs in time $f(k)n^{{\mathcal{O}}(1)$ and shows that instances of this problem with solutions of size ${\cal O}(\log \log n)$ can be solved in polynomial time. Expand
Hitting topological minors is FPT
TLDR
This work improves upon the algorithm of Golovach et al. Expand
A Tight Lower Bound for Edge-Disjoint Paths on Planar DAGs
TLDR
Under the Exponential Time Hypothesis (ETH), it is shown that there is no f (k) ·no(k) algorithm for EDGE-DISJOINT PATHS on planar DAGs, where k is the number of terminal pairs, n is thenumber of vertices and f is any computable function. Expand
Ear-decompositions and the complexity of the matching polytope
TLDR
It is shown that deciding whether $\beta(G) \leq 1$ can be executed efficiently by looking at any ear-decomposition starting with an odd circuit and performing basic modulo-2 computations, and that computing $\beta$ is a Fixed-Parameter-Tractable problem (FPT). Expand
Asymptotic Equivalence of Hadwiger's Conjecture and its Odd Minor-Variant
Hadwiger’s conjecture states that every Kt-minor free graph is (t − 1)-colorable. A qualitative strengthening of this conjecture raised by Gerards and Seymour, known as the Odd Hadwiger’s conjecture,Expand
Forbidden minors for tight cycle relaxations
We study the problem of optimizing an arbitrary weight function w z over the metric polytope of a graph G = (V,E), a well-known relaxation of the cut polytope. We define the signed graph (G,E−),Expand
Fixed-parameter tractability for subset feedback set problems with parity constraints
TLDR
This paper further generalizes the subset feedback set problem to one with the parity constraints, and shows the fixed parameter tractability. Expand
...
1
2
3
...

References

SHOWING 1-10 OF 98 REFERENCES
The disjoint paths problem in quadratic time
TLDR
The time complexity of all the algorithms with the most expensive part depending on Robertson and [email protected]?s algorithm can be improved to O(n^2), for example, the membership testing for minor-closed class of graphs. Expand
Odd cycle packing
TLDR
The integrality gap of the standard LP-relaxation of the odd cycle packing problem is Θ (√n), and it is proved that there is an O(m1/2)-approximation algorithm for the node- and arc- directed even cycle packing problems, which almost matches the hardness result. Expand
A simpler algorithm and shorter proof for the graph minor decomposition
TLDR
A simplified algorithm for finding the decomposition based on a new constructive proof of the decompose theorem for graphs excluding a fixed minor H, which runs in time O(n3), as does the original algorithm of Robertson and Seymour. Expand
Packing Digraphs with Directed Closed Trails
  • P. Balister
  • Computer Science, Mathematics
  • Combinatorics, Probability and Computing
  • 2003
It has been shown [2] that if n is odd and m1,…,mt are integers with mi[ges ]3 and [sum ]i=1tmi=|E(Kn)| then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m1,…,mt. ThisExpand
A nearly linear time algorithm for the half integral parity disjoint paths packing problem
TLDR
This is the first polynomial time algorithm for this problem, and generalizes polynometric time algorithms by Kleinberg and Kawarabayashi and Reed [20], respectively, for the half integral disjoint paths packing problem, i.e., without the parity requirement. Expand
The linkage problem for group-labelled graphs
textabstractThis thesis aims to extend some of the results of the Graph Minors Project of Robertson and Seymour to "group-labelled graphs". Let $\Gamma$ be a group. A $\Gamma$-labelled graph is anExpand
A shorter proof of the graph minor algorithm: the unique linkage theorem
TLDR
This paper provides a new and much simpler proof of the correctness of the Graph Minor Algorithm and proves the "Unique Linkage Theorem" without using Graph Minors structure theorem. Expand
Approximation algorithms and hardness results for cycle packing problems
TLDR
A lower bound of Ω(log n/loglog n) is proved for the integrality gap of edge-disjoint cycle packing and the approximability of νe(G) in directed graphs, improving upon the previously known APX-hardness result for this problem. Expand
On the odd-minor variant of Hadwiger's conjecture
TLDR
It is shown that, for every l, if a graph contains no odd K"l-expansion then its chromatic number is O(llogl), and that given a graph and a subset S of its vertex set, either there are k vertex-disjoint odd paths with endpoints in S, or there is a set X of at most 2k-2 vertices such that every odd path with both ends in S contains a vertex in X. Expand
Approximating the list-chromatic number and the chromatic number in minor-closed and odd-minor-closed classes of graphs
TLDR
There is a polynomial time approximation algorithm for the list-chromatic number of graphs without Kk-minors and there is an O(n2500k) algorithm for deciding whether or not any 2500k- chromatic graph contains an odd-Kk- Minors. Expand
...
1
2
3
4
5
...