The quest for finding Hamiltonian cycles

Transcription

1 The quest for finding Hamiltonian cycles Giang Nguyen School of Mathematical Sciences University of Adelaide

2 Travelling Salesman Problem Given a list of cities and distances between cities, what is the shortest route that visits every city exactly once and returns to the origin city?

3 Travelling Salesman Problem Given a list of cities and distances between cities, what is the shortest route that visits every city exactly once and returns to the origin city? Hamiltonian Cycle Problem Given a list of cities and connections between cities, is there a route that visits every city exactly once and returns to the origin city?

4 Travelling Salesman Problem Given a list of cities and distances between cities, what is the shortest route that visits every city exactly once and returns to the origin city? Hamiltonian Cycle Problem Given a list of cities and connections between cities, is there a route that visits every city exactly once and returns to the origin city?

5 Travelling Salesman Problem Given a list of cities and distances between cities, what is the shortest route that visits every city exactly once and returns to the origin city? Hamiltonian Cycle Problem Given a list of cities and connections between cities, is there a route that visits every city exactly once and returns to the origin city? Both problems are NP-hard.

6 TSP and HCP are NP-hard

7 TSP and HCP are NP-hard P Problems for which a solution can be found in polynomial time

8 TSP and HCP are NP-hard P NP Problems for which a solution can be found in polynomial time Problems for which a solution can be verified in polynomial time

9 TSP and HCP are NP-hard P NP P = NP? Problems for which a solution can be found in polynomial time Problems for which a solution can be verified in polynomial time One of the seven Millennium Prize Problems

10 TSP and HCP are NP-hard P NP P = NP? Problems for which a solution can be found in polynomial time Problems for which a solution can be verified in polynomial time One of the seven Millennium Prize Problems NP-hard Finding a polynomial-time algorithm to solve any NP-hard problem would lead to polynomial-time algorithms for all NP problems.

11 Solving the Hamiltonian Cycle Problem for a given graph (Graph Theory)

12 Solving the Hamiltonian Cycle Problem for a given graph (Graph Theory) Optimising an expected function over a set of actions permitted on the graph (Markov Decision Processes)

13 Solving the Hamiltonian Cycle Problem for a given graph (Graph Theory) Optimising an expected function over a set of actions permitted on the graph (Markov Decision Processes) Maximising a determinant function over a set of doubly stochastic matrices defined on the graph (Linear Algebra)

14 Hamiltonian Cycle Problem and Markov Decision Processes

15 Embedding the HCP in a Markov decision process

16 Embedding the HCP in a Markov decision process State space S = {1, 2,..., N}

17 Embedding the HCP in a Markov decision process State space S = {1, 2,..., N} Action space A(i)

18 Embedding the HCP in a Markov decision process State space S = {1, 2,..., N} Action space A(i)

19 Embedding the HCP in a Markov decision process State space S = {1, 2,..., N} Action space A(i) Policy f chooses a A(i) at state i with probability f ia

20 Embedding the HCP in a Markov decision process State space S = {1, 2,..., N} Action space A(i) Policy f chooses a A(i) at state i with probability f ia

21 Embedding the HCP in a Markov decision process State space S = {1, 2,..., N} Action space A(i) Policy f chooses a A(i) at state i with probability f ia

22 Embedding the HCP in a Markov decision process State space S = {1, 2,..., N} Action space A(i) Policy f chooses a A(i) at state i with probability f ia

23 Embedding the HCP in a Markov decision process State space S = {1, 2,..., N} Action space A(i) Policy f chooses a A(i) at state i with probability f ia

24 Embedding the HCP in a Markov decision process State space S = {1, 2,..., N} Action space A(i) Policy f chooses a A(i) at state i with probability f ia

25 Embedding the HCP in a Markov decision process State space S = {1, 2,..., N} Action space A(i) Policy f chooses a A(i) at state i with probability f ia

26 Embedding the HCP in a Markov decision process State space S = {1, 2,..., N} Action space A(i) Policy f chooses a A(i) at state i with probability f ia Transition probability matrix P(f)

27 A deterministic policy f and its transition probability matrix P(f) =

28 A randomised policy f 1 2 α 1 α 5 6 and its transition probability matrix P(f) = α 0 1 α α 0 α α 0 1 α α 0 α 0

29 Maps the policy space into a discounted occupation measure space X β, where x ia (f) := {v[(i βp(f)) 1 ]} i f ia Discount factor β (0, 1) Initial state distribution v = (1, 0,..., 0)

30 Maps the policy space into a discounted occupation measure space X β, where x ia (f) := {v[(i βp(f)) 1 ]} i f ia Discount factor β (0, 1) Initial state distribution v = (1, 0,..., 0) HCP Constrained discounted MDP (Feinberg, 2000)

31 Quadratic program (QP) a A(i) x ia β b A(i) a A(1) x bi = δ i1, i = 1,..., N (1) x ia 0, i = 1,..., N, a A(i) (2) 1 x 1a = 1 β N (3) x ia x ib = 0, i = 1,..., N, a b A(i) (4)

32 Quadratic program (QP) a A(i) x ia β b A(i) a A(1) x bi = δ i1, i = 1,..., N (1) x ia 0, i = 1,..., N, a A(i) (2) 1 x 1a = 1 β N (3) x ia x ib = 0, i = 1,..., N, a b A(i) (4) (1) & (2): Solution x is feasible on the given graph

33 Quadratic program (QP) a A(i) x ia β b A(i) a A(1) x bi = δ i1, i = 1,..., N (1) x ia 0, i = 1,..., N, a A(i) (2) 1 x 1a = 1 β N (3) x ia x ib = 0, i = 1,..., N, a b A(i) (4) (1) & (2): Solution x is feasible on the given graph (3): Property of a Hamiltonian cycle

34 Quadratic program (QP) a A(i) x ia β b A(i) a A(1) x bi = δ i1, i = 1,..., N (1) x ia 0, i = 1,..., N, a A(i) (2) 1 x 1a = 1 β N (3) x ia x ib = 0, i = 1,..., N, a b A(i) (4) (1) & (2): Solution x is feasible on the given graph (3): Property of a Hamiltonian cycle (4): Deterministic

35 Linear programming relaxation (LP) a A(i) x ia β b A(i) a A(1) x bi = δ i1, i = 1,..., N (1) x ia 0, i = 1,..., N, a A(i) (2) x 1a = 1 1 β N (3) x ia x ib = 0, i = 1,..., N, a b A(i)

36 Basic feasible solutions

37 Basic feasible solutions

38 A 1-randomized policy is deterministic at every vertex except j.

39 A 1-randomized policy is deterministic at every vertex except j. splitting vertex j: two outgoing edges (j, a) and (j, b)

40 A 1-randomized policy is deterministic at every vertex except j. splitting vertex j: two outgoing edges (j, a) and (j, b) splitting probabilities: probability of choosing an edge at j

41 A 1-randomized policy is deterministic at every vertex except j. splitting vertex j: two outgoing edges (j, a) and (j, b) splitting probabilities: probability of choosing an edge at j Can be decomposed into two deterministic policies.

42 A deterministic policy can be one of the three types:

43 A deterministic policy can be one of the three types: a Hamiltonian cycle

44 A deterministic policy can be one of the three types: a Hamiltonian cycle a short-cycle policy: has a cycle going through vertex

45 A deterministic policy can be one of the three types: a Hamiltonian cycle a short-cycle policy: has a cycle going through vertex a noose-cycle policy: has no cycle going through vertex

46 Theorem A basic feasible solution to (LP) is either: a Hamiltonian cycle, or 1-randomized policy that is a convex combination of a short-cycle policy and a noose-cycle policy.

47 Theorem A basic feasible solution to (LP) is either: a Hamiltonian cycle, or 1-randomized policy that is a convex combination of a short-cycle policy and a noose-cycle policy

48 Theorem A basic feasible solution to (LP) is either: a Hamiltonian cycle, or 1-randomized policy that is a convex combination of a short-cycle policy and a noose-cycle policy. Theorem (i) For such a pair of short-cycle and noose-cycle policies, there exists only one convex combination that is a basic feasible solution. (ii) Formulae for splitting probabilities are derived.

49 Proposed algorithm: LP relaxation + branch and fix

50 Proposed algorithm: LP relaxation + branch and fix

51 Proposed algorithm: LP relaxation + branch and fix

52 Proposed algorithm: LP relaxation + branch and fix

53 Proposed algorithm: LP relaxation + branch and fix

54 Hamiltonian Non-Hamiltonian Graph Executed Possible Run Size Branches Branches Time Dodecahedron : Knight s Tour :38 Petersen :01 Coxeter :23

55 Hamiltonian Non-Hamiltonian Average Min Max Graph Executed Executed Executed Possible Size Branches Branches Branches Branches

56 Room for improvement Introduce good objective functions for the (LP). Impose constraints on splitting probabilities to eliminate more 1-randomized policies. Consider contracted graphs to reduce graph size at each iteration.

57 Hamiltonian Cycle Problem and Linear Algebra

58 A deterministic policy f and its transition probability matrix P(f) =

59 A randomised policy f 1 2 α 1 α 5 6 and its transition probability matrix P(f) = α 0 1 α α 0 α α 0 1 α α 0 α 0

60 Theorem (Borkar, Ejov & Filar, 2006) DS: all doubly stochastic policies HCP is equivalent to minimize [(I P ε + 1 N J) 1 ] 11 over all P DS.

61 Computational difficulties with optimisation algorithms Evaluating inverses [(I P ε + 1 N J) 1 ] 11 := [A 1 (P ε )] 11 Dense Hessian matrices, due to perturbation

62 Computational difficulties with optimisation algorithms Evaluating inverses [(I P ε + 1 N J) 1 ] 11 := [A 1 (P ε )] 11 Dense Hessian matrices, due to perturbation Applying the adjoint form of the inverse: [A 1 (P ε )] 11 = det A 11(P ε ) det A(P ε )

63 Computational difficulties with optimisation algorithms Evaluating inverses [(I P ε + 1 N J) 1 ] 11 := [A 1 (P ε )] 11 Dense Hessian matrices, due to perturbation Applying the adjoint form of the inverse: [A 1 (P ε )] 11 = det A 11(P ε ) det A(P ε ) Replace minimizing [A 1 (P ε )] 11 with maximizing det A(P ε ) and drop perturbation

64 Theorem For any graph G and any policy P F, 0 det A(P) k, where k is the length of the longest cycle in G.

65 Theorem For any graph G and any policy P F, 0 det A(P) k, where k is the length of the longest cycle in G. Non-Hamiltonian Graph (of size N = 10, and k = 9) det A(P) = 9 det A(P) = 8 det A(P) = 0

66 Theorem For any graph G and any policy P F, 0 det A(P) k, where k is the length of the longest cycle in G. Hamiltonian Graph (of size N = 10, and k = 10) det A(P) = 10 det A(P) = 10 det A(P) = 8

67 Theorem For any graph G and any policy P F, 0 det A(P) k, where k is the length of the longest cycle in G. Positive gap between non-hamiltonian and Hamiltonian graphs [ ] [ ] max det A(P) + 1 max det A(P) P non-ham P Ham

68 HCP is equivalent to maximize det A(P) over all P DS.

69 HCP is equivalent to maximize det A(P) over all P DS. Graph of det A(P λ ) where P λ = λ 1P 1 + λ 2P 2 + λ 3P 3.

70 HCP is equivalent to maximize det A(P) over all P DS. Non-linear & non-concave Extreme points correspond to deterministic policies Graph of det A(P λ ) where P λ = λ 1P 1 + λ 2P 2 + λ 3P 3.

71 HCP is equivalent to maximize det A(P) over all P DS. Non-linear & non-concave Extreme points correspond to deterministic policies Empirically, no interior max or min Graph of det A(P λ ) where P λ = λ 1P 1 + λ 2P 2 + λ 3P 3.

72 Analogous interpretation for eigenvalues Every stochastic matrix P has at least one eigenvalue λ N = 1 (Perron Frobenius Theorem) Corollary For any stochastic N N matrix P, (1 λ i ) N, N 1 i=1 where λ i are the eigenvalues of P, excluding λ N = 1.

73 Theorem Consider the symmetric linear perturbation: P ε := (1 ε)p + ε N J. For any graph G, any stochastic P F, and ε [0, 1) 0 det A(P ε ) 1 (1 ε)k ε = k + ε( ) + ε 2 ( ) + O(ε 3 ), where k is the length of the longest cycle in G. This demonstrates the robustness of the determinant function.

74 S: stochastic policies, P : stationary distribution matrix, ε [0, 1) Theorem HCP is equivalent to maximize det(i P ε + 1 N J) over all P S. Theorem HCP is equivalent to minimize Trace[(I P ε + P ) 1 ] over all P S.