Chvátal Gomory Rounding and Integrality Gaps

1. ChvátalGomory Rounding and Integrality Gaps Mohit Singh Kunal Talwar MSR NE, McGill MSR SV TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAA

2. Approximation Algorithm Design Cleverly define Lower Bound on Optimum Think hard Show that Write natural linear program Think hard Show that Write natural linear program Add cleverly designed constraints to get Think hard Show that

3. When no good algorithms known • Would like to establish limits on what we can hope to do • APX-hardness: usually good evidence. Often unavailable • LP Gaps show limits of specific LP • Show gap between and Write natural linear program Think hard Show that Write natural linear program Add cleverly designed constraints to get Think hard Show that

4. Cut Generating Procedures • Ways to automate getting tighter relaxations • Lovász-Schrijver • Sherali-Adams • Lassere • Chvátal-Gomory • Often (at least retrospectively), improve LP/SDP gaps • Matching, MaxCut, Sparsest Cut, Unique Label Cover [Arora-Bollobás-Lovász 2002] Can we establish limits for these procedures? Write natural linear program Add cleverly designed constraints to get Think hard Show that

5. Gap for LS, SA etc. • [Arora-Bollobás-Lovász2002] • Vertex Cover: Large Class of LPs has integrality gap .Implies gaps for LS, SA. • GMTT07,DK07,S08,CMM09,MS09,T09,RS09,KS09,CL10 LS/SA/Lassere/LS+ gaps for several problems • MaxCut • Unique Label Cover • Sparsest Cut • CSP • LIN • Matching • …

6. This talk: What about Chvátal-Gomory • Hypergraphmatching in k-uniform hypergraphs • rounds of CG bring gap down to • [Chan Lau 10] SA gap is at least even after rounds.

7. This talk: What about Chvátal-Gomory • Gaps remain large for many rounds of CG • Vertex Cover: Gap ) after rounds • MaxCut: Gap after rounds • Unique Label Cover: Gap after rounds • -: Gap after rounds • Same as SA gaps.

8. Defining Chvátal-Gomory Cuts • [Gomory 1958] • For a polyhedron • Let where • Let • is polyhedron obtained after j rounds of CG

9. Hypergraph Matching • -uniform hypergraph: • Each edge with • Goal: find largest subset of disjoint edges s.t.

10. Hypergraph matching • Graph maximum matching • SA takes rounds to get within • CG gets to integer hull in 1 round • APX hard • -inapproximable • )approximation • [Chan Lau 10] Gap after rounds of SA • There is a poly size LP with gap

11. Intersecting family • is an intersecting family if for all • [Chan Lau 10] LP + intersecting has gap at most

12. Intersecting family via CG • is an intersecting family if for all • Fix • valid for • valid for • valid for • valid for • So valid for

13. Small families suffice • Extremal combinatorics result • For any intersecting family in a k-regular hypergraph, there is one of size • Implies that Integrality gap of is bounded by I.e. for hypergraph matching, round CG is nearly a factor of two better than round SA.

14. Max Cut LP

15. Max Cut SA gap [CharikarMakarychevMakarychev 09] for any subset s.t. a distribution overs solutions such that • is integral for any Survives rounds of SA

16. Max Cut CG • Observation: for any constraint in , are integers and (can add arbitrary positive multiple of to remove negative coefficients and get stronger constraint) Main idea: show that is feasible for

17. Proof by induction • Base case: k=0. Inspection • Induction Step. Need to show in holds for Case 1: Case 2:

18. : • Let for some Recall a distribution overs solutions s.t. • is integral for any • For each , • For , set • For , set , for arbitrary fixed • For , set • New integral. Agrees with on . • . Therefore done.

19. : • By definition, -1 valid for • Therefore done.

20. Conclusions • Similar proofs for unique games, CSPs, VC • CG hierarchy often not much better than SA • Noticeably better for Hypergraph matching • What other problems show large gap between clever LP and LS/SA? Does CG capture them?