Part II Strong lift-and-project cutting planes. Vienna, January 2012

Transcription

1 Part II Strong lift-and-project cutting planes Vienna, January 2012

2 The Lovász and Schrijver M(K, K) Operator Let K be a given linear system in 0 1 variables. for any pair of inequalities αx β 0 and α x β 0, compute the product inequality : ( β α T ) ( ) β Y 0. products x i x j, for all 1 i < j n are replaced with variables x ij terms x 2 i, for 1 i n, are replaced with x i (valid when x i is binary.) α This yields an extended formulation which is provably stronger than the original (Lovász and Schrijver 91).

3 The M(QSTAB(G), QSTAB(G)) relaxation Recall that Ω is the collection of all (maximal) cliques { 1 QST AB(G) := i C x i 0 (C Ω) x i 0 (i V ) M(QSTAB(G), QSTAB(G)): CV Is : x i + j C:{i,j} Ē x ij 0 (C Ω, i V \ C) CP Is : i C x i 1 (C Ω) i C C x i {i,j} Ē(C:C ) x ij 1 (C, C Ω) x ij = 0 ({i, j} E) x ij 0 ({i, j} Ē) x i 0 (i V )

4 On the strength of N(QSTAB(G), QSTAB(G)) non-compact: n Ω CVIs and Ω ( Ω 1)/2 CPIs separation of CVIs and CPIs NP-hard in the strong sense (Giandomenico 06) projection N(QSTAB(G), QSTAB(G)) onto the x-space: - stronger than the Sherali-Adams relaxation: N(QSTAB(G), QSTAB(G)) N(QSTAB(G)) N(FRAC(G)) - neither contains nor is contained in T H(G) or N + (FRAC(G)) Theorem (Giandomenico, Letchford, Rossi, S 09) N(QSTAB(G), QSTAB(G)) implies all web and antiweb inequalities, together with various lifted versions

5 On the strength of N(QSTAB(G), QSTAB(G)) Relaxation T H(G) implied inequalities clique [Grotschel, Lovász and Schrijver 88] N +(F RAC(G)) clique, odd-cycle, odd-antihole and odd-wheel [Lovász and Schrijver 91] web [Giandomenico and Letchford 06] N(QST AB(G), QST AB(G)) web and antiweb and their lifted versions edge, clique, odd-hole, odd-antihole [Giandomenico, Letchford, Rossi, S. 09]

6 Benders reformulation N(QSTAB, QSTAB) max 1 T x + 0 T y max 1 T x + η s. t. s. t. cliques Ax 1 Ax 1 CVIs + CPIs Bx + Dy d v T (d Bx) η, v EXT(Q) v T (d Bx) 0, v RAY(Q) x R n +, y R p + x R n +, η R where EXT(Q) and RAY(Q) contain resp. the extreme points and extreme rays of the slave polyhedron Q := {v R p : v T D η, v 0} in our case, η = 0 and Q = {v R p : v T D 0, v 0} is a polyhedral cone Ext(Q) contains only the zero vector the optimality cuts disappear

7 Benders reformulation N(QSTAB, QSTAB) max 1 T x max 1 T x s. t. s. t. cliques Ax 1 Ax 1 CVIs + CPIs Bx + Dy d v T (d Bx) 0, v RAY RAY := x R n +, y R p + {extreme rays of v T D 0, v 0} x R n + Clever selection of Benders cuts accomplished by a Cut Generating Linear Program (CGLP): min v T (d Bx ) v T D 0 i=1,...,p v i = 1 v 0

8 Benders reformulation N(QSTAB, QSTAB) max 1 T x max 1 T x s. t. s. t. cliques Ax 1 Ax 1 CVIs + CPIs Bx + Dy d v T (d Bx) 0, v RAY RAY := x R n +, y R p + {extreme rays of v T D 0, v 0} x R n + Clever selection of Benders cuts accomplished by a Cut Generating Linear Program (CGLP): min v T (d Bx ) v T D 0 i=1,...,p v i = 1 v 0

9 Cut Generating LP i vertex, H, K cliques. u Ki v HK x i (1 x (H K)) 0... (i, j) Ē a (i,j),(k,i) b (i,j)(h,k) = 1 a (i,j),(k,i) = 1 if j K and 0 otherwise; b (i,j)(h,k) = 1 if i H and j K and 0 otherwise

10 Cut Generating LP: example 5-hole, x = (0.5,..., 0.5) obj {1, 2}{1, 2}{1, 2}{2, 3}{2, 3}{2, 3}{3, 4}{3, 4}{3, 4}{4, 5}{4, 5}{4, 5}{1, 5}{1, 5}{1, 5} η obj {1, 2}{1, 2}{1, 2}{1, 2}{2, 3}{2, 3}{2, 3}{3, 4}{3, 4}{4, 5} {2, 3}{3, 4}{4, 5}{1, 5}{3, 4}{4, 5}{1, 5}{4, 5}{1, 5}{1, 5} η

11 Cutting plane performance sparse graphs are tractable up to 150 vertices (cut generation is too slow for V = 200) some cuts are sparse and clean, but some other are quite dense sometimes cuts are helpful in branch-and-cut, but often confound it different normalization constraints may change things, but do not seen resolutive

12 Relaxing M(K, K) pick a vertex j and only the CPIs containing variables x jk, for {j, k} Ē (11 out of 21 CPIs are discarded) include the CVIs containing variables x hk covered by the chosen CPIs (12 out of 35 discarded) included C Ω(j) j 7 1 discarded 7 j C Ω(j) 2 C Ω(j) 5 3 C Ω(j) 4 4 included 7 j 1 discarded 7 j V(j) 2 C Ω(j) V(j) 3 C Ω(j) 4 4

13 The M j (K, K) relaxation V (j) denotes the neighborhood of j V, V (j) = V \ V (j) Ω(j) set of all maximal cliques containing j; Ω(j) = Ω \ Ω(j) max x i i V s.t. x i x ik 1 i C C x i + x i + {i,k} Ē(C:C ) k C:{i,k} Ē k C:{i,k} Ē x ik 0 x ik 0 (C Ω(j), C Ω(j)) (C Ω(j), i {j} V (j)) (C Ω(j), i V (j) V ( V (j))) x ik = 0 ({i, k} E) x ik 0 ({i, k} Ē) x i 0 (i V )

14 On the strength of the closure j V N j (K, K) Relaxation T H(G) N + (F RAC(G)) N(QSTAB(G), QSTAB(G)) j V N j (QSTAB(G), QSTAB(G)) implied inequalities clique [Grotschel, Lovász and Schrijver 88] clique, odd-cycle, odd-antihole and odd-wheel [Lovász and Schrijver 91] web [Giandomenico and Letchford 06] web and antiweb and their lifted versions edge, clique, odd-hole, odd-antihole [Giandomenico, Letchford, Rossi, S. 07] odd-hole and antiweb and their lifted versions edge, clique, odd-antihole a large class of web [Giandomenico, Rossi, S. 10]

15 Implementation (aggressive) clique separation heuristic Ω Ω(j) filtered by slack: prefer cliques tight to the current fractional point Cut generation (one round) for each j V : Step 1. build Ω(j), Ω(j) Ω, build CGLP Step 2. Solve CGLP; add a (eventually) violated Benders cut Step 3. Run the clique separation heuristic; add violated clique cuts (and cliques to Ω) implemented in the IBM Cplex 11.2 framework; all default CGLPs solved by Cplex hybaropt option 2 Intel Xeon 5150 processors clock 2.66 GHz, 4GB of RAM

16 Upper bounds comparison Graph α(g) UB clq θ(g) UB C(N) BV Time Time UB C(N) BV brock ,590 brock ,302 brock ,665 brock ,433 C * 391 * C * 8,908 * c-fat ,483 DSJC * 297 * DSJC * 27 * mann a mann a * 120 * mann a * 1,062 * hamming ,416 keller ,586 19,319 p hat ,287 p hat , ,428 p hat , ,995 san ,102 sanr ,576 sanr ,428 j N j outperforms θ(g)

17 Upper bounds comparison Sparse random graphs ( 5%), generation parameters as in Gruber & Rendl 03 Graph V E α(g) UB clq θ(g) UB C(N) ,

18 Branch-and-cut-results CPLEX default Clique-B&C C(N)-B&C Graph Time Subprob. Time Subprob. Time Subprob. brock , ,352 1, ,613 1, ,078 brock , , ,340 brock , , ,983 brock , , ,368 C , , ,783 C c-fat DSJC , , ,456 DSJC , mann a mann a , , ,185 mann a , , ,214 hamming keller , , ,602 p hat p hat , , ,258 p hat , ,090 3, ,319 san sanr , , , ,885 sanr , ,581 1, ,538 1, ,983