# Symmetry Handling in Binary Programs via Polyhedral Methods

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1 Symmetry Handling in Binary Programs via Polyhedral Methods Christopher Hojny joint work with Marc Pfetsch Technische Universität Darmstadt Department of Mathematics 26 International Conference on Mathematical Software ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods

2 Motivation Consider a necklace with n black () and white () beads. Associate with a necklace a vector in {, } n. necklaces x, x {, } n are equal cyclic shift γ such that γ(x) = x.? = ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

3 Motivation Consider a necklace with n black () and white () beads. Associate with a necklace a vector in {, } n. necklaces x, x {, } n are equal cyclic shift γ such that γ(x) = x.? = shift ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

4 Motivation Consider a necklace with n black () and white () beads. Associate with a necklace a vector in {, } n. necklaces x, x {, } n are equal cyclic shift γ such that γ(x) = x.? = 2 shifts ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

5 Motivation Consider a necklace with n black () and white () beads. Associate with a necklace a vector in {, } n. necklaces x, x {, } n are equal cyclic shift γ such that γ(x) = x.? = 3 shifts ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

6 Motivation Consider a necklace with n black () and white () beads. Associate with a necklace a vector in {, } n. necklaces x, x {, } n are equal cyclic shift γ such that γ(x) = x.? = 3 shifts Question: How can we characterize (the convex hull of) a representative system of distinct necklaces? ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

7 Outline Symmetry Detection Representative Systems Symretopes and Symresacks Numerical Experiments ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

8 Symmetry Detection Consider binary program (BP) max{c x : Ax b, x {, } n }. We can distinguish two types of permutation symmetries of BP: Problem Symmetry Group Contains all permutations γ S n with Aγ(x) b Ax b and c γ(x) = c x. Formulation Symmetry Group Contains all permutations γ S n for which there is σ S m s.t. γ(c) = c, σ(b) = b, and A σ(i),γ(j) = A i,j. ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 4

9 Symmetry Detection Consider binary program (BP) max{c x : Ax b, x {, } n }. We can distinguish two types of permutation symmetries of BP: Problem Symmetry Group Contains all permutations γ S n with Aγ(x) b Ax b and c γ(x) = c x. NP-hard Formulation Symmetry Group Contains all permutations γ S n for which there is σ S m s.t. γ(c) = c, σ(b) = b, and A σ(i),γ(j) = A i,j. Graph-Isomorphism-hard ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 4

10 How to Determine Formulation Symmetries? Build an auxillary colored graph and determine its color preserving automorphisms. max x +x 2 x 2 x 2x +2x x x 2 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 5

11 How to Determine Formulation Symmetries? Build an auxillary colored graph and determine its color preserving automorphisms. max x +x 2 x 2 x 2x +2x x x 2 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 5

12 How to Determine Formulation Symmetries? Build an auxillary colored graph and determine its color preserving automorphisms. max x +x 2 x 2 x 2 2x +2x Automorphism group can be determined, e.g., with bliss, nauty, and saucy. x x 2 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 5

13 Outline Symmetry Detection Representative Systems Symretopes and Symresacks Numerical Experiments ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 6

14 Orbit Representative Systems Given a symmetry group Γ of a binary program (BP), how can we determine a representative system of the orbits where x is feasible for BP? Γ(x) = {γ(x) : γ Γ}, Idea: Add for each γ Γ to BP. a γ x := n (2 n γ(i) 2 n i ) x i i= ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 7

15 Pros and Cons Pros a γ x cuts off symmetric solutions {x {, } n : Ax b} γ Γ {x Rn : a γ x } is lexmax representative system of BP, see [Friedman, 27] Cons coefficients of a γ x grow exponentially large possibly many inequalities, one for each γ Γ ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 8

16 Outline Symmetry Detection Representative Systems Symretopes and Symresacks Numerical Experiments ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 9

17 Symretopes Definition Given Γ S n, the symretope w.r.t. Γ is the polytope S(Γ) := conv ( {x {, } n : a γ x γ Γ} ). ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods

18 Symretopes Definition Given Γ S n, the symretope w.r.t. Γ is the polytope S(Γ) := conv ( {x {, } n : a γ x γ Γ} ). If Γ is given by generators, the optimization problem over symresacks is NP-hard. There is Γ S n such that the coefficients of facet inequalities grow exponentially. To avoid exponential coefficients, find tractable IP-formulation for symretopes with small coefficients. Idea: Consider /-knapsack polytope induced by a γ x. ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods

19 Symresacks Definition Given γ S n, the symresack w.r.t. γ is the polytope P γ := conv ( {x {, } n : a γ x } ). Properties non-standard knapsack polytopes, feasible points can be completely characterized by minimal cover inequalities and box constraints, in general, there are exponential many minimal covers ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods

20 Separation Complexity Theorem The separation problem of minimal cover inequalities for P γ and x R n can be solved in time O(n 2 ). Consequences: Symmetry handling is possible with {, ±}-inequalities: separate minimal cover inequalities of P γ instead of adding a γ x, separation is possible in time O( Γ n 2 ), avoid exponential coefficients. ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

21 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

22 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

23 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Infeasibility Detection x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

24 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Infeasibility Detection x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

25 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Feasibility Detection x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

26 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Feasibility Detection x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

27 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

28 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

29 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

30 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

31 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

32 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

33 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Theorem The propagation problem for P γ can be solved in time O(n). ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

34 Outline Symmetry Detection Representative Systems Symretopes and Symresacks Numerical Experiments ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 4

35 Implementation implemented constraint handler for symresacks in SCIP 3.2. containing separation of minimal cover inequalities, propagation rules, resolution methods if propagation detected infeasibility, implemented constraint handler for orbisacks [Kaibel and Loos, 2] which are symresacks for γ = (, 2)(3, 4)... (2n, 2n) use automated symmetry detection methods of [Pfetsch and Rehn, 25], use CPLEX 2.6. as LP-solver ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 5

36 Numerical Experiments tests were run on Linux cluster with Intel i3 3.2GHz dual core processors, 8GB memory, single thread time limit: 36 seconds test instances MIPLIB 2 benchmark SONET network design [Sherali and Smith, 2] Wagon Load Balancing [Ghoniem and Sherali, 2] ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 6

37 Numerical Experiments: MIPLIB 2 benchmark number of solved instances, default 6/87 prop sepa sepa+prob orbisacks symresacks speed up of solution time orbisacks symresacks reduction of nodes orbisacks symresacks ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 7

38 Numerical Experiments: SONET number of solved instances, default 5/5 prop sepa sepa+prob orbisacks symresacks speed up of solution time orbisacks symresacks reduction of nodes orbisacks symresacks ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 8

39 Numerical Experiments: Wagon Load Balancing number of solved instances, default 2/2 prop sepa sepa+prob orbisacks symresacks 2 2 speed up of solution time orbisacks..3. symresacks..2.2 reduction of nodes orbisacks..4. symresacks.95.. ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 9

40 Thank you for your attention! ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

41 Literature Friedman, E. J. (27). Fundamental domains for integer programs with symmetries. In Dress, A., Xu, Y., and Zhu, B., editors, Combinatorial Optimization and Applications, volume 466 of Lecture Notes in Computer Science, pages Springer Berlin Heidelberg. Ghoniem, A. and Sherali, H. D. (2). Defeating symmetry in combinatorial optimization via objective perturbations and hierarchical constraints. IIE Transactions, 43(8): Kaibel, V. and Loos, A. (2). Finding descriptions of polytopes via extended formulations and liftings. In Mahjoub, A. R., editor, Progress in Combinatorial Optimization. Wiley. Pfetsch, M. E. and Rehn, T. (25). A computational comparison of symmetry handling methods for mixed integer programs. Sherali, H. D. and Smith, J. C. (2). Improving discrete model representations via symmetry considerations. Management Science, 47(): ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

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