The Triangle-free 2-matching
Polytope Of Subcubic Graphs
Kristóf Bérczi
Egerváry Research Group (EGRES)
Eötvös Loránd University
Budapest
ISMP 2012
Motivation
Hamiltonian cycle problem
Relaxation:
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.
n-4
n-4
G
G
n-4
n-4
Degrees at most 2 in G.
No cycle of length 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
Previous work
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
Polyhedral descriptions
The b-factor polytope
Def.: 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.
K
F
matching
The b-factor polytope
matching
Def.: The b-factor polytope is the convex hull of
incedence vectors of b-factors.
matchings
Thm.: matching
The b-factor polytope is determined by
The C(≤)k-free case
The 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!
matchings
Triangle-free 2-factors
Thm.:
(Hartvigsen and Li ’07) matching
Conjecture:
For subcubic G, the ∆-free 2-factor polytope is
determined by
NOT TRUE !!!
Subcubic graphs
Problem with degrees
„Usual” way of proof:
G
3
∆ -free 2-factors
G’
3
3
∆ -free 2-matchings
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-matchings
Thm.: (Hartvigsen and Li ’12)
For subcubic G, the ∆-free 2-matching polytope is
determined by
New proof
Perfect matchings
Thm.: (Edmonds ‘65)
The p.m. polytope is determined by
Proof:
(Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver)
Another proof
Thm.: (Edmonds ‘65)
The p.m. polytope is determined by
Proof:
(Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver)
Another proof
Thm.: (Edmonds ‘65)
The p.m. polytope is determined by
Proof:
(Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver)
Plan
Tricky !
Technical
…
Are
inequalities
true for x’?
Define
shrinking
Shrink the
complement,
Extend
put
convex
OR combinations
combination
together
to the
original
problem
Define
tightness
Hartvigsen
and Li
Yipp !
Shrinking
Shrinking a tight ∆
Shrinking a tight tri-comb
Conclusions
Now:
• New proof for the description of the ∆-free 2matching 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!