Orbital Conflict. Jeff Linderoth. Jim Ostrowski. Fabrizio Rossi Stefano Smriglio. When Worlds Collide. Univ. of Wisconsin-Madison

Transcription

1 Orbital Conflict When Worlds Collide Jeff Linderoth Univ. of Wisconsin-Madison Jim Ostrowski University of Tennessee Fabrizio Rossi Stefano Smriglio Univ. of L Aquila MIP 2014 Columbus, OH July 23, 2014 The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

2 In The Beginning Motivation 1 François Symmetry is was HOT Symmetry-exploiting branching methods have within the last 5 years or so made some significant impact in our ability to solve some instances of MIP The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

3 In The Beginning Motivation 1 François Symmetry is was HOT Symmetry-exploiting branching methods have within the last 5 years or so made some significant impact in our ability to solve some instances of MIP 2 Cutting planes are STRONG Cutting Planes for an appropriate (strong) relaxation of an instance are perhaps the most important ingredient in our ability to solve MIP The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

4 In The Beginning Motivation 1 François Symmetry is was HOT Symmetry-exploiting branching methods have within the last 5 years or so made some significant impact in our ability to solve some instances of MIP 2 Cutting planes are STRONG Cutting Planes for an appropriate (strong) relaxation of an instance are perhaps the most important ingredient in our ability to solve MIP 3 Can We Merge The Two? Can we exploit symmetry to build relaxations from which we can generate cutting planes? The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

5 In The Beginning Background Prior to Blastoff Primary Problem(s) of Focus SCP min x {0,1} n{1 n x Ax 1 m} or SPP max x {0,1} n{1 n x Ax 1 m} A {0, 1} m n F : The set of feasible solutions Π n : The collection of permutations of [n] = {1, 2,..., n} The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

6 In The Beginning Background Prior to Blastoff Primary Problem(s) of Focus SCP min x {0,1} n{1 n x Ax 1 m} or SPP max x {0,1} n{1 n x Ax 1 m} A {0, 1} m n F : The set of feasible solutions Π n : The collection of permutations of [n] = {1, 2,..., n} Note: 1 n x = 1 n π(x) π Π n is a symmetry of IP if it maps a feasible solution to another feasible solution (of the same value) x F π(x) F Symmetric Potato The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

7 In The Beginning Background The Symmetry Group The set of symmetries of the instance (with composition of permutations) forms the symmetry group Γ = { π Π n π(x) F x F} The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

8 In The Beginning Background The Symmetry Group The set of symmetries of the instance (with composition of permutations) forms the symmetry group Γ = { π Π n π(x) F x F} Γ is a property of the feasible region: F = Γ = Π n So finding Γ must be at least as hard as solving the instance itself The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

9 In The Beginning Background The Symmetry Group The set of symmetries of the instance (with composition of permutations) forms the symmetry group Γ = { π Π n π(x) F x F} Γ is a property of the feasible region: F = Γ = Π n So finding Γ must be at least as hard as solving the instance itself For our methods, we can work with any subgroup G Γ For SCP and SPP, we can use the symmetry group of the matrix A: G(A) def = {π Π n σ Π m such that P σap π = A} Given A of reasonable size, there exist software packages (nauty, saucy) that can compute (generators of) G(A) effectively Actual algorithm is exponential, but in general works quickly enough for our problems of interest Computing (all) the generators is impractical for large problems AFAIK, this is not what commercial solvers do The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

10 In The Beginning Background Definitions Orbit For a point z R n, the orbit of z under G is the set of all points to which z can be sent by permutations in G: orb(g, z) def = {π(z) π G}. The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

11 In The Beginning Background Definitions Orbit For a point z R n, the orbit of z under G is the set of all points to which z can be sent by permutations in G: orb(g, z) def = {π(z) π G}. Stabilizer The stabilizer of a set S in G is the set of permutations in G that send S to itself: Used to stabilize sauces stab(s, G) = {π G π(s) = S}. stab(s, G) is a subgroup of G The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

12 In The Beginning Background Last Notation And Some Notation Abuse Branch-and-bound node a = (F a 1, F a 0 ), F a 1 : Set of variables fixed to one F a 0 : Set of variables fixed to zero The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

13 In The Beginning Background Last Notation And Some Notation Abuse Branch-and-bound node a = (F a 1, F a 0 ), F a 1 : Set of variables fixed to one F a 0 : Set of variables fixed to zero We sometimes extend permutations to acts on sets (specifically on sets of variables) in the natural way: S Z, S Z, π(s) = S if and only if π( i S e i ) = i S e i. So I may talk about the orbit of a variable or a pair of variables : orb(g, {j}) def = orb(g, e j ) orb(g, {i, j}) def = orb(g, e i + e j ) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

14 In The Beginning Background Last Notation And Some Notation Abuse Branch-and-bound node a = (F a 1, F a 0 ), F a 1 : Set of variables fixed to one F a 0 : Set of variables fixed to zero We sometimes extend permutations to acts on sets (specifically on sets of variables) in the natural way: S Z, S Z, π(s) = S if and only if π( i S e i ) = i S e i. So I may talk about the orbit of a variable or a pair of variables : orb(g, {j}) def = orb(g, e j ) orb(g, {i, j}) def = orb(g, e i + e j ) I will also refer to the group stab(f a 1, G) The subgroup of G in which permutations do not shift indices of variables fixed to 1 outside of the set F a 1 As we fix variables (to 1) at node a, the symmetry remaining in the problem becomes stab(f a 1, G) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

15 In The Beginning Orbital Branching Review Orbital Branching Groups contain inverse elements, so {k} orb({j}, G) {j} orb({k}, G), and the union of the orbits of each variable {i} forms a partition of [n]. The orbits encode which variables are equivalent (symmetric) with respect to the symmetry G. The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

16 In The Beginning Orbital Branching Review Orbital Branching Groups contain inverse elements, so {k} orb({j}, G) {j} orb({k}, G), and the union of the orbits of each variable {i} forms a partition of [n]. The orbits encode which variables are equivalent (symmetric) with respect to the symmetry G. Orbital Branching Let O O be an orbit of the symmetry group of the IP. Surely we can branch as x i 1 or x i 0. i O i O The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

17 In The Beginning Orbital Branching Review Orbital Branching Groups contain inverse elements, so {k} orb({j}, G) {j} orb({k}, G), and the union of the orbits of each variable {i} forms a partition of [n]. The orbits encode which variables are equivalent (symmetric) with respect to the symmetry G. Orbital Branching Let O O be an orbit of the symmetry group of the IP. Surely we can branch as x i 1 or x i 0. i O i O If at least one variable i O is going to be one, and they are all equivalent, then you may as well pick (i ) one arbitrarily. x i = 1 or x i = 0 i O The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

18 In The Beginning Orbital Branching An Alternative View of Orbital Branching Suppose that you have found that the variables x e, x f, x g and x h share an orbit at node a, O = {e, f, g, h}. Then you can surely branch as: a e f g h x e = 1 x f = 1 x g = 1 x h = 1 j O x j = 0 The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

19 In The Beginning Orbital Branching An Alternative View of Orbital Branching Suppose that you have found that the variables x e, x f, x g and x h share an orbit at node a, O = {e, f, g, h}. Then you can surely branch as: a e f g h x e = 1 x f = 1 x g = 1 x h = 1 j O x j = 0 But the best solution you can find from nodes f, g, and h will be the same as the best solution you can find from node e In fact, solutions will be isomorphic Prune nodes f, g, and h The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

20 In The Beginning Orbital Branching Review Constraint Orbital Branching If π G, then imposing the constraint λ x λ 0 is equivalent to imposing the constraint π(λ) x λ 0 So if orb(g, λ) = {λ, µ 1, µ 2,..., µ K }, then the inequalities µ j x λ 0 are all equivalent to (symmetric with) λ x λ 0 The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

21 In The Beginning Orbital Branching Review Constraint Orbital Branching If π G, then imposing the constraint λ x λ 0 is equivalent to imposing the constraint π(λ) x λ 0 So if orb(g, λ) = {λ, µ 1, µ 2,..., µ K }, then the inequalities µ j x λ 0 are all equivalent to (symmetric with) λ x λ 0 Constraint Orbital Branching: branches on symmetric inequalities µ j x λ 0 µ j x λ µ j orb(g,λ) µ j orb(g,λ) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

22 In The Beginning Orbital Branching Review Constraint Orbital Branching If π G, then imposing the constraint λ x λ 0 is equivalent to imposing the constraint π(λ) x λ 0 So if orb(g, λ) = {λ, µ 1, µ 2,..., µ K }, then the inequalities µ j x λ 0 are all equivalent to (symmetric with) λ x λ 0 Constraint Orbital Branching: branches on symmetric inequalities µ j x λ 0 µ j x λ µ j orb(g,λ) µ j orb(g,λ) But all of the µ j x λ 0 are equivalent by symmetry The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

23 In The Beginning Orbital Branching Review Constraint Orbital Branching If π G, then imposing the constraint λ x λ 0 is equivalent to imposing the constraint π(λ) x λ 0 So if orb(g, λ) = {λ, µ 1, µ 2,..., µ K }, then the inequalities µ j x λ 0 are all equivalent to (symmetric with) λ x λ 0 Constraint Orbital Branching: branches on symmetric inequalities µ j x λ 0 µ j x λ µ j orb(g,λ) µ j orb(g,λ) But all of the µ j x λ 0 are equivalent by symmetry So branch on two-node disjunction (λ x λ 0 ) µ j x λ µ j orb(g,λ) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

24 In The Beginning Orbital Branching Review Constraint Orbital Branching If π G, then imposing the constraint λ x λ 0 is equivalent to imposing the constraint π(λ) x λ 0 So if orb(g, λ) = {λ, µ 1, µ 2,..., µ K }, then the inequalities µ j x λ 0 are all equivalent to (symmetric with) λ x λ 0 Constraint Orbital Branching: branches on symmetric inequalities µ j x λ 0 µ j x λ µ j orb(g,λ) µ j orb(g,λ) But all of the µ j x λ 0 are equivalent by symmetry So branch on two-node disjunction (λ x λ 0 ) µ j x λ µ j orb(g,λ) The orbital branching method is a special case of constraint orbital branching for (λ, λ 0 ) = ( e k, 1). Either x i 1 or x i 0 i orb(i) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

25 YASI (Yet Another Simple Idea) Orbital Conflict Relation to Constraint OB Two-variable Orbital Branching An Application of Constraint Orbital Branching 1 Pick a pair of variables (i, j). 2 Either x i + x j 2 or x i + x j 1 The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

26 YASI (Yet Another Simple Idea) Orbital Conflict Relation to Constraint OB Two-variable Orbital Branching An Application of Constraint Orbital Branching 1 Pick a pair of variables (i, j). 2 Either x i + x j 2 or x i + x j 1 3 But if (i, j ) orb(stab(f a 1, G), {i, j}), then x i + x j 2 is equivalent to x i + x j 2 The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

27 YASI (Yet Another Simple Idea) Orbital Conflict Relation to Constraint OB Two-variable Orbital Branching An Application of Constraint Orbital Branching 1 Pick a pair of variables (i, j). 2 Either x i + x j 2 or x i + x j 1 3 But if (i, j ) orb(stab(f a 1, G), {i, j}), then x i + x j 2 is equivalent to x i + x j 2 a x i + x j 2 x i + x j 1 (i, j ) orb(stab(f a 1, G), {i, j}) Again, this is a special case of orbital branching, where (λ, λ 0 ) = ( e i e j, 2). The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

28 YASI (Yet Another Simple Idea) Orbital Conflict Basic Idea Orbital Conflict Basic Idea At some nodes of the branch and bound tree, we get to add inequalities x i + x j 1 (i, j ) orb(stab(f a 1, G), {(i, j}) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

29 YASI (Yet Another Simple Idea) Orbital Conflict Basic Idea Orbital Conflict Basic Idea At some nodes of the branch and bound tree, we get to add inequalities x i + x j 1 (i, j ) orb(stab(f a 1, G), {(i, j}) Suppose orb(stab(f a 1, G), {i, j}) = {(i, j), (i, k), (j, k)} So on right branch, we add x i + x j 1, x i + x k 1, x j + x k 1 The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

30 YASI (Yet Another Simple Idea) Orbital Conflict Basic Idea Orbital Conflict Basic Idea At some nodes of the branch and bound tree, we get to add inequalities x i + x j 1 (i, j ) orb(stab(f a 1, G), {(i, j}) Suppose orb(stab(f a 1, G), {i, j}) = {(i, j), (i, k), (j, k)} So on right branch, we add x i + x j 1, x i + x k 1, x j + x k 1 But this (of course) implies that x i + x j + x k 1 x j x i + x j 1 x i xi + x k 1 x j + x k 1 x k The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

31 YASI (Yet Another Simple Idea) Orbital Conflict Basic Idea Orbital Conflict Basic Idea At some nodes of the branch and bound tree, we get to add inequalities x i + x j 1 (i, j ) orb(stab(f a 1, G), {(i, j}) Suppose orb(stab(f a 1, G), {i, j}) = {(i, j), (i, k), (j, k)} So on right branch, we add x i + x j 1, x i + x k 1, x j + x k 1 But this (of course) implies that x i + x j + x k 1 x j x i + x j 1 x i xi + x k 1 x j + x k 1 x k Conflict! These are just edges in a conflict graph! We can use integrality considerations to generate strong valid inequalities! Specifically, we can generate all inequalities that we would for the stable set (or set packing) problem: Clique inequalities, Odd Hole, Web, Anti-web, Rank inequalities, etc... The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

32 YASI (Yet Another Simple Idea) Orbital Conflict Basic Idea Blastoff! Potato Spaceship Hooray! We ve Achieved Our Goal! Use symmetry to induce structure from which we can use integrality to generate cutting planes The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

33 YASI (Yet Another Simple Idea) Orbital Conflict Basic Idea Blastoff! Potato Spaceship Hooray! We ve Achieved Our Goal! Use symmetry to induce structure from which we can use integrality to generate cutting planes Downsides I don t like branching on inequalities Of course the left branch (x i + x j 2) can be implemented by fixing variable bounds, but what about the right branch? The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

34 YASI (Yet Another Simple Idea) Orbital Conflict Improvements We Can Make It Better! We don t like branching on inequalities Thankfully, we don t have to. In fact, (even though we didn t think of it originally this way), you can just do regular orbital branching, and use symmetry considerations to enrich the conflict graph on the right (fix to zero) branch. Orbital Conflict The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

35 YASI (Yet Another Simple Idea) Orbital Conflict Improvements We Can Make It Better! We don t like branching on inequalities Thankfully, we don t have to. In fact, (even though we didn t think of it originally this way), you can just do regular orbital branching, and use symmetry considerations to enrich the conflict graph on the right (fix to zero) branch. Orbital Conflict Just do orbital branching (fixing x i = 1 on the left branch) a x i = 1 x i = 0 i orb(stab(f a 1, G), {i}) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

36 YASI (Yet Another Simple Idea) Orbital Conflict Improvements We Can Make It Better! We don t like branching on inequalities Thankfully, we don t have to. In fact, (even though we didn t think of it originally this way), you can just do regular orbital branching, and use symmetry considerations to enrich the conflict graph on the right (fix to zero) branch. Orbital Conflict Just do orbital branching (fixing x i = 1 on the left branch) On the right branch, for each variable j F a 1 compute the orbit of stab(f a 1 \ {j}, G), {i, j} And add all edges to the (local) conflict graph a x i = 1 x i = 0 i orb(stab(f a 1, G), {i}) x i + x j 1 j F a 1 (i, j ) orb(stab(f a 1 \ {j}, G), {i, j}) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

37 YASI (Yet Another Simple Idea) Orbital Conflict Improvements We Can Make It Better! We don t like branching on inequalities Thankfully, we don t have to. In fact, (even though we didn t think of it originally this way), you can just do regular orbital branching, and use symmetry considerations to enrich the conflict graph on the right (fix to zero) branch. Orbital Conflict Just do orbital branching (fixing x i = 1 on the left branch) On the right branch, for each variable j F a 1 compute the orbit of stab(f a 1 \ {j}, G), {i, j} And add all edges to the (local) conflict graph It should work if the conflict graphs are dense a x i = 1 x i = 0 i orb(stab(f a 1, G), {i}) x i + x j 1 j F a 1 (i, j ) orb(stab(f a 1 \ {j}, G), {i, j}) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

38 YASI (Yet Another Simple Idea) Orbital Conflict Improvements STS81 Branching Tree x 0 = 1 All vars = 0 x 1 = 1 Big orbit = 0 x 3 = 1 Big orbit = 0 x 4 = 1 x 4 = x 5 = x 7 = 0 x 9 = 1 Big orbit = 0 x 17 = 1 4-orbit = 0 The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

39 YASI (Yet Another Simple Idea) Orbital Conflict Improvements STS81 Branching Tree x 0 = 1 All vars = 0 x 1 = 1 Big orbit = 0 x 3 = 1 Big orbit = 0 x 4 = 1 x 4 = x 5 = x 7 = 0 x 9 = 1 Big orbit = 0 x 17 = 1 4-orbit = 0 So at node 13, inequalities x i + x j 1 will be added for (i, j) orb(stab({0, 1}, G), {4, 3}) (i, j) orb(stab({0, 3}, G), {4, 1}) (i, j) orb(stab({1, 3}, G), {4, 0}) (i, j) orb(stab({0, 1, 3}, G), {17, 9}) (i, j) orb(stab({0, 1, 9}, G), {17, 3}) (i, j) orb(stab({0, 3, 9}, G), {17, 1}) (i, j) orb(stab({1, 3, 9}, G), {17, 0}) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

40 YASI (Yet Another Simple Idea) Orbital Conflict Improvements Conflict Graph at node 13 for a SCP (STS81) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

41 YASI (Yet Another Simple Idea) Orbital Conflict Improvements Conflict Graph at node 25 for a SCP (STS81) The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

42 Computational Results Implementation Orbital Conflict as Graph Enhancement Advantages Just fixes variables Can still do orbital branching, isomorphism pruning, etc. The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

43 Computational Results Implementation Orbital Conflict as Graph Enhancement Advantages Just fixes variables Can still do orbital branching, isomorphism pruning, etc. Branching? Given ^x and orbits O 1, O 2,... O p at node a, which orbit should be choose? For general integer programs, the variable on which you branch can have a huge impact on the solution time The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

44 Computational Results Implementation Orbital Conflict as Graph Enhancement Advantages Just fixes variables Can still do orbital branching, isomorphism pruning, etc. Branching? Given ^x and orbits O 1, O 2,... O p at node a, which orbit should be choose? For general integer programs, the variable on which you branch can have a huge impact on the solution time In our experience, best orbit selection rules combine trying to improve the bound, but also try to keep the symmetry around at the subproblems 1 Strong Branching: Solve the child node(s) that would result from branching on orbit j and pick the best: j arg max ( e ^x z + j ) j {1,...p} 2 Keep Symmetry Left: Compute the size of the symmetry group that would result if you would branch on orbit j and choose the one that keeps the group the largest j ( arg max stab(f a 1 {i j }, G) ) j {1,...p} The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

45 Computational Results Implementation Implementation Details I just started coding 2 weeks ago Implemented in MINTO Yes, MINTO still exists! LPs solved by Clp Generators for original symmetry group of A generated by nauty All group operations (computing stabilizers, orbits, etc.) done by GAP. GAP has no callable library :-( Pass the command strings and output through a local socket connection A heuristic for separating for clique inequalities from the (local) conflict graph Code Is Primitive! The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

46 Computational Results Instances Computational Results Some Symmetric IPs (Binary) Error Correcting Codes (cod(n,d)): Find maximum number of (0, 1) n vectors such that Hamming distance between each pair is d Covering Design (cov(v,k,t)): v > k > t: Find minimum number of k-sets of {1,..., v} to cover all t-sets of {1,..., v}. Covering Code (codbt(b,t)): Find minimum number of codewords such that every word in the alphabet is at most a (Hamming) distance 1 from a codeword. Edge Coloring (flosn,jgt(n)): Show that you can t color edges of these graphs on n nodes with colors. Steiner Triple System: (sts(n)): Find the incidence width of a Steiner Triple System of order n Thanks Francois! The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

47 Computational Results Instances Computational Results Number of Nodes Name CPLEX OB OC # Cliques codbt codbt cov flosn52u flosn60u flosn84u jgt18u jgt30u sts sts sts The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

48 Computational Results Instances Computational Results Number of Nodes Name CPLEX OB OC # Cliques codbt codbt cov flosn52u flosn60u flosn84u jgt18u jgt30u sts sts sts The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

49 Computational Results Instances Computational Results Number of Nodes Name CPLEX OB OC # Cliques codbt codbt cov flosn52u flosn60u flosn84u jgt18u jgt30u sts sts sts Preliminary Conclusions In general, we see a nice reduction in the number of nodes The overhead isn t too large to do this Have to manage (and separate from) a local conflict graph Have to compute orbits of pairs instead of single variables The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

50 Computational Results STS405? Even Better!? For larger instances (that didn t finish in an hour), the idea looks even more promising We were generating many clique inequalities The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

51 Computational Results STS405? Even Better!? For larger instances (that didn t finish in an hour), the idea looks even more promising We were generating many clique inequalities World s Largest Potato Fame, Glory, and World Records sts405 is the smallest (in terms of number of variables) unsolved instance in MIPLIB2010. At it seems a long ways from being solved. Can we use our new orbital conflict idea to solve it? The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

52 Computational Results STS405? Even Better!? For larger instances (that didn t finish in an hour), the idea looks even more promising We were generating many clique inequalities World s Largest Potato Fame, Glory, and World Records sts405 is the smallest (in terms of number of variables) unsolved instance in MIPLIB2010. At it seems a long ways from being solved. Can we use our new orbital conflict idea to solve it? To my knowledge, the best known bounds on the solution are 211 z sts I ran CPLEX for a long time on a complemented version of the problem x i = 1 x i are the variables. Constraints are Ax 2 form, The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

53 Computational Results STS405? STS405 CPLEX Log Parallel mode: deterministic, using up to 24 threads. Nodes Cuts/ Node Left Objective IInf Best Integer Best Bound ItCnt Gap * * % Cuts: % Cuts: %... Found 46 symmetric permutations % Elapsed real time = sec. (tree size = 0.01 MB, solutions = 2) %... Elapsed real time = sec. (tree size = MB, solutions = 5) Nodefile size = MB ( MB after compression) % % The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

54 Computational Results STS405? Attempting to Solve Computing (full) set of generators for the symmetry group of sts405 takes nauty around 8 hours. Run orbital conflict (on one machine) using a combination of keep symmetry left and strong branching rules The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

55 Computational Results STS405? Attempting to Solve Computing (full) set of generators for the symmetry group of sts405 takes nauty around 8 hours. Run orbital conflict (on one machine) using a combination of keep symmetry left and strong branching rules Sad Potato I Couldn t Solve It I ran orbital conflict for 4 days. On one machine. And it didn t solve. It s a framework. The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

56 Computational Results STS405? Attempting to Solve Computing (full) set of generators for the symmetry group of sts405 takes nauty around 8 hours. Run orbital conflict (on one machine) using a combination of keep symmetry left and strong branching rules Happy Potato I Couldn t Solve It I ran orbital conflict for 4 days. On one machine. And it didn t solve. It s a framework. We improved lower bound to -135 (complemented), or 270 (real instance). So our team can happily report that 270 z STS If you believe our code is correct The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

57 The End Conclusions We give the first(?) method that exploits symmetry to induce structures from which one can generate strong valid inequalities based on integrality considerations Additional computational overhead is relatively modest Good reductions in number of nodes for smallish symmetric instances Many cliques generated and bounds improved for larger instances The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

58 The End Conclusions We give the first(?) method that exploits symmetry to induce structures from which one can generate strong valid inequalities based on integrality considerations Additional computational overhead is relatively modest Good reductions in number of nodes for smallish symmetric instances Many cliques generated and bounds improved for larger instances Ongoing Work Perhaps add additional cutting planes for set packing problem Fame and Glory! Solve some unsolved instances using orbital conflict in combination with some enumeration and wasting decades of CPU time The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25

59 The End Gu and a Muscular Potato Playing a Guitar. Any Questions? The Orbiteers (UW, UT, L Aquila) Orbital Conflict MIP / 25