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!