The Triangle-free 2-matching Polytope Of Subcubic Graphs Kristóf Bérczi Egerváry Research Group (EGRES) Eötvös Loránd University Budapest ISMP 2012
Feb 22, 2016
The Triangle-free 2-matching Polytope Of Subcubic Graphs
Kristóf BércziEgerváry Research Group (EGRES)
Eötvös Loránd UniversityBudapest
ISMP 2012
Motivation
Hamiltonian cycle problemRelaxation:Find a subgraph • with degrees = 2• containing no „short” cycles (length at most k)
Fisher, Nemhauser, Wolsey ‘79:how solutions for the weighted version approximate the optimal TSP
Remark:for k > n/2 the relax. and the HCP are equivalent
Connectivity augmentation
Problem: Make G k-node-connected by adding a minimum number of new edges.
k = n-1: trivial (complete graph)
k = n-2: maximal matching in G
k=n-3:Deleting n-4 nodes G remains connected.
G G
Degrees at most 2 in G.No cycle of length 4.
n-4 n-4
n-4 n-4
Definitions
G=(V,E) undirected, simple, b:V→Z+
Def.: A b-matching is a subset F⊆E s.t. dF(v) ≤ b(v) for each node v. If = holds everywhere, then F is a b-
factor.
If b=t for each node: t-matching.
Examples: b=1
b=2
Let K be a list of forbidden subgraphs.Def.: A K-free b-matching contains no
member of K.Def.: A C(≤)k-free 2-matching contains no
cycle of length (at most) k.• Hamiltonian relax.: C≤k-free 2-factor • Node-conn. aug.: C4-free 2-matching
Notation: C3=∆, C4=◊
Example: k=3
Papadimitriu ‘80:• NP-hard for k ≥ 5
Vornberger ‘80:• NP-hard in cubic graphs for k ≥ 5• NP-hard in cubic graphs for k = 4 with weights
Hartvigsen ’84:• Polynomial algorithm for k=3
Hartvigsen and Li ‘07, Kobayashi ‘09:• Polynomial algorithm for k=3 in subcubic graphs with general weigths
Nam ‘94:• Polynomial algorithm for k=4 if ◊’s are node-disjoint
Hartvigsen ‘99, Király ’01, Pap ’05, Takazawa ‘09:• Results for bipartite graphs and k=4
Frank ‘03, Makai ‘07:• Kt,t-free t-matchings in bipartite graphs
B. and Kobayashi ’09, Hartvigsen and Li ‘11:• Polynomial algorithm for k=4 in subcubic graphs
B. and Végh ’09, Kobayashi and Yin ‘11:• Kt,t- and Kt+1-free t-matchings in degree-bounded graphs
Previous work
Polyhedral descriptions
The b-factor polytopeDef.: The b-factor polytope is the convex hull of
incedence vectors of b-factors.
Def.: (K,F) is a blossom if K⊆V, F⊆δ(K) and b(K)+|F| is odd.
F
K
The b-factor polytopeDef.: The b-factor polytope is the convex hull of
incedence vectors of b-factors.
Thm.:The b-factor polytope is determined by
matching
matching
matchings
matching
The C(≤)k-free caseThe weighted C(≤)k-free 2-matching (factor) problem
is NP-hard for k ≥ 4
What about k = 3 ???
Problem: Give a description of the ∆-free 2-matching (factor) polytope.
UNSOLVED!
Triangle-free 2-factorsThm.: (Hartvigsen and Li ’07) For subcubic G, the ∆-free 2-factor polytope is
determined by
NOT TRUE !!!
Conjecture:
matchings
matching
Subcubic graphsProblem with degrees„Usual” way of proof:
G G’
∆ -free 2-factors ∆ -free 2-matchings
3 3 3
Tri-combs Def.: (K,F,T) is a tri-comb if K⊆V, T is a
set of ∆’s „fitting” K, F⊆δ(K) and |T|+|F| is odd.
Triangle-free 2-matchingsThm.: (Hartvigsen and Li ’12) For subcubic G, the ∆-free 2-matching polytope is
determined by
New proof
Perfect matchingsThm.: (Edmonds ‘65)The p.m. polytope is determined by
Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver)
Another proofThm.: (Edmonds ‘65)The p.m. polytope is determined by
Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver)
Another proofThm.: (Edmonds ‘65)The p.m. polytope is determined by
Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver)
Hartvigsen and Li
Define tightness
Plan
Define
shrinking
Are inequalities true for
x’?
Tricky !
Technical…
Yipp !
Extend convex
combination to the original problem
OR
Shrink the complement, put combinations together
Shrinking
Shrinking a tight ∆
Shrinking a tight tri-comb
Conclusions
Now:• New proof for the description of the ∆-free 2-
matching polytope of subcubic graphs• Slight generalization
– list of triangles– b-matching; on nodes of triangles b = 2– not subcubic; degrees of triangle nodes ≤ 3
Open problems:• Algorithm for maximum ◊-free 2-matching• Description of the ∆-free 2-matching polytope
in general graphs
Thank you for your attention!