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# Elementary Landscape Decomposition of the Hamiltonian Path Optimization Problem

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Slides of the presentation of the work in EvoCOP 2014 conference (Granda, Spain)

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### Elementary Landscape Decomposition of the Hamiltonian Path Optimization Problem

1. 1. 1 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Elementary Landscape Decomposition of the Hamiltonian Path Optimization Problem Darrell Whitley and Francisco Chicano Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work
2. 2. 2 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 • A landscape is a triple (X,N, f) where Ø X is the solution space Ø N is the neighbourhood operator Ø f is the objective function Landscape Definition Landscape Definition Elementary Landscapes Landscape decomposition Apps. The pair (X,N) is called configuration space s0 s4 s7 s6 s2 s1 s8 s9 s5 s32 0 3 5 1 2 4 0 7 6 • The neighborhood operator is a function N: X →P(X) • Solution y is neighbor of x if y ∈ N(x) • Regular and symmetric neighborhoods • d=|N(x)| ∀ x ∈ X • y ∈ N(x) ⇔ x ∈ N(y) • Objective function f: X →R (or N, Z, Q) Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work
3. 3. 3 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 • An elementary function is an eigenvector of the graph Laplacian (plus constant) • Graph Laplacian: • Elementary function: eigenvector of Δ (plus constant) Elementary Landscapes: Formal Definition s0 s4 s7 s6 s2 s1 s8 s9 s5 s3 Adjacency matrix Degree matrix Depends on the configuration space Eigenvalue Landscape Definition Elementary Landscapes Landscape decomposition Apps. Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work
4. 4. 4 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Elementary Landscapes: Characterizations Linear relationship • An elementary landscape is a landscape for which where • Grover’s wave equation Eigenvalue ba Depend on the problem/instance Landscape Definition Elementary Landscapes Landscape decomposition Apps. def a b Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work
5. 5. 5 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Elementary Landscapes: Examples Problem Neighbourhood d λ Symmetric TSP 2-opt n(n-3)/2 n-1 swap two cities n(n-1)/2 2(n-1) Antisymmetric TSP inversions n(n-1)/2 n(n+1)/2 swap two cities n(n-1)/2 2n Graph α-Coloring recolor 1 vertex (α-1)n 2α Graph Matching swap two elements n(n-1)/2 2(n-1) Graph Bipartitioning Johnson graph n2/4 2(n-1) NEAS bit-flip n 4 Max Cut bit-flip n 4 Weight Partition bit-flip n 4 Landscape Definition Elementary Landscapes Landscape decomposition Apps. Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work
6. 6. 6 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 • What if the landscape is not elementary? • Any landscape can be written as the sum of elementary landscapes • There exists a set of eigenfunctions of Δ that form a basis of the function space (Fourier basis) Landscape Decomposition Landscape Definition Elementary Landscapes Landscape decomposition Apps. X X X e1 e2 Elementary functions (from the Fourier basis) Non-elementary function f Elementary components of f f < e1,f > < e2,f > < e2,f > < e1,f > Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work
7. 7. 7 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Landscape Decomposition: Examples Problem Neighbourhood d Components General TSP inversions n(n-1)/2 2 swap two cities n(n-1)/2 2 Subset Sum Problem bit-flip n 2 MAX-kSAT bit-flip n k NK-landscapes bit-flip n K+1 Radio Network Design bit-flip n max. nb. of reachable antennae Frequency Assignment change 1 frequency (α-1)n 2 QAP swap two elements n(n-1)/2 3 Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Landscape Definition Elementary Landscapes Landscape decomposition Apps.
8. 8. 8 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 • Selection operators usually take into account the fitness value of the individuals • We could use the average fitness value in the neighborhood of a solution instead Selecting by Average Fitness Value X Neighborhoods avg avg Minimizing Fitness-based selection Average-based selection Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Landscape Definition Elementary Landscapes Landscape decomposition Apps.
9. 9. 9 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 • In elementary landscapes the traditional and the new operator are the same! Recall that... • However, they are not the same in non-elementary landscapes. If we have n elementary components, then: • This selection strategy could be useful for plateaus Elementary components X Minimizing avg avg Fitness-based selection Average-based selection ? Selecting by Average Fitness Value Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Landscape Definition Elementary Landscapes Landscape decomposition Apps.
10. 10. 10 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 • Let {x0, x1, ...} a simple random walk on the configuration space where xi+1∈N(xi) • The random walk induces a time series {f(x0), f(x1), ...} on a landscape. • The autocorrelation function is defined as: • The autocorrelation length and coefficient: • Autocorrelation length “conjecture”: Autocorrelation s0 s4 s7 s6 s2 s1 s8 s9 s5 s3 2 0 3 5 1 2 4 0 7 6 The number of local optima in a search space is roughly Solutions reached from x0 after l moves Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Landscape Definition Elementary Landscapes Landscape decomposition Apps.
11. 11. 11 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Hamiltonian Path Optimization (HPO) Definition Neighborhoods • In TSP we search for a Hamiltonian circuit with minimal cost • In HPO we search for a Hamiltonian path with minimal cost • Applications: •  DNA fragment assembly (usually not solved as such) •  DNA linkage marker sequencing (solved as HPO) 6 1 2 3 5 4 (1,2,3,4,5) 6 1 2 3 5 4 (1,2,3,4,5) d ¯f = ↵ + X c2C w(c) f(⇡) = n 1X i=1 w⇡(i),⇡(i+1) + w⇡(n),⇡(1) Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work
12. 12. 12 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Neighborhoods • We investigate two different neighborhoods: (reduced) reversals and swaps 2 4 3 5 1 6 2 1 5 3 4 6 (a) Reversal 2 4 3 5 1 6 2 1 3 5 4 6 (b) Swap Fig. 1. Examples of reversal and swap for a permutation of size 6. Component Model ey and Sutton developed a “component” based model that makes it easy to Reversal f a more general problem: the Quadratic Assignment Problem ven two matrices r and w the ﬁtness function of QAP is: fQAP (⇡) = nX i,j=1 ri,jw⇡(i),⇡(j). (3) serve that (3) generalizes (2) if w is the weight matrix for HPO ri,j = j i+1, where is the Kronecker delta. HPO has applica- ormatics, in particular in DNA fragment assembling [10] and the f radiation hybrid maps [1]. ape for Reversals utation ⇡ and two positions i and j with 1  i < j  n, we can mutation ⇡0 by reversing the elements between i and j (inclusive). new permutation is deﬁned as: ⇡0 (k) = ⇢ ⇡(k) if k < i or k > j, ⇡(j + i k) if i  k  j. (4) ) illustrates the concept of reversal. The reversal neighborhood rmutation ⇡ contains all the permutations that can be formed by sals to ⇡. Each reversal can be identiﬁed by a pair [i, j], which are d ending positions of the reversal. We use square brackets in the tinguish them from swaps. Then, we have |NR(⇡)| = n(n 1)/2. Swap that is incident on vertex i is sampled once. Since this is true fo follows that every edge not in the current solution is sampled twi for each of the vertices in which it is incident. Therefore, summ neighbors: d · Avg(f(y))y2N(x) = d · f(x) (n 2)f(x) + 2 P c2 Computing the average over the neighborhood and taking i result of Lemma 2: Avg(f(y)) y2N(x) = f(x) n 2 n(n 1)/2 1 f(x) + 2 n(n 1)/2 1 X c2 = f(x) + n n(n 1)/2 1 ( ¯f f(x)). 5 Landscape Structure for Swaps Given a permutation ⇡, we can build a new permutation ⇡0 b positions i and j in the permutation. The new permutation is d ⇡0 (k) = 8 < : ⇡(k) if k 6= i and k 6= j, ⇡(i) if k = j, ⇡(j) if k = i. Figure 1(b) illustrates the concept of swap. The swap neighbo a permutation ⇡ contains all the permutations that can be form swaps to ⇡. Each swap can be identiﬁed by the pair (i, j) of positio cardinality of the swap neighborhood is |NS(⇡)| = n(n 1)/2. U stated, we will refer always to the swap neighborhood in this se 2 4 3 5 1 6 2 1 5 3 4 6 (a) Reversal 2 4 3 5 1 6 2 1 3 5 4 6 (b) Swap Fig. 1. Examples of reversal and swap for a permutation of size 6. 4.1 Component Model Whitley and Sutton developed a “component” based model that makes • We call it “reduced” reversals because we omit the reversal of the whole permutation Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Definition Neighborhoods
13. 13. 13 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Component Model (Whitley, Sutton, Howe) Component Model Elementariness • Let C be a set of components, each component having a weight w(c) • A solution x is a subset of C: • The fitness value of x is the sum of the weights of the components in x: • Let us consider a neighbor y of x, we can classify the components in x and y as: •  Components that are in both solutions •  Components that are in x and not in y (removed from x when we move to y) •  Components that are in y and not in x (included from C-x when we move to y) Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work 4.1 Component Model Whitley and Sutton developed a “component” based identify elementary landscapes [17]. Let C be a set of Each component c 2 C has a weight (or cost) denote is a subset of components and the evaluation functio to the sum of the weights of the components in x: f( C x denote the subset of components that do not of solution x. Note that the sum of the weights of computed by P c2C w(c) f(x). In the context of the wave equation can be expressed as: Avg(f(y)) y2N(x) = f(x) p1f(x) + p2 X c2C where p1 = ↵/d is the (sampling) rate at which com f(x) are removed from solution x to create a neig and p2 = /d is the rate at which components in th create a neighboring solution y 2 N(x). By simple a X wap for a permutation of size 6. nent” based model that makes it easy to be a set of “components” of a problem. ost) denoted by w(c). A solution x ✓ C ion function f(x) maps each solution x nts in x: f(x) = P c2x w(c). Finally, let hat do not contribute to the evaluation weights of the components in C x is ntext of the component model, Grover’s + p2 X c2C w(c) f(x) ! , t which components that contribute to eate a neighboring solution y 2 N(x), onents in the set C x are sampled to By simple algebra, ed a “component” based model that makes it easy to es [17]. Let C be a set of “components” of a problem. weight (or cost) denoted by w(c). A solution x ✓ C d the evaluation function f(x) maps each solution x the components in x: f(x) = P c2x w(c). Finally, let omponents that do not contribute to the evaluation sum of the weights of the components in C x is x). In the context of the component model, Grover’s ssed as: (x) p1f(x) + p2 X c2C w(c) f(x) ! , pling) rate at which components that contribute to 2 4 3 5 1 6 2 1 5 3 4 6 (a) Reversal 2 4 3 5 1 6 2 1 3 5 4 6 (b) Swap 6 1 2 3 5 4 (1,2,3,4,5) 1 2 3 4 5 6 6 1 2 3 5 4 (1,2,3,4,5) 1 2 3 4 5 6 x y
14. 14. 14 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Component Theorem (Whitley, Sutton, Howe) • Let us assume that generating a symmetric neighborhood it happens that: •  All the components IN x are removed the same number of times, α •  All the components in C-x (OUT of x) are included the same number of times, β • Then, the landscape is elementary • And the wave equation is: • where 6 1 2 3 5 4 (1,2,3,4,5) 1 2 3 4 5 6 Component Model y and Sutton developed a “component” based model that makes it easy to y elementary landscapes [17]. Let C be a set of “components” of a problem. omponent c 2 C has a weight (or cost) denoted by w(c). A solution x ✓ C bset of components and the evaluation function f(x) maps each solution x sum of the weights of the components in x: f(x) = P c2x w(c). Finally, let denote the subset of components that do not contribute to the evaluation tion x. Note that the sum of the weights of the components in C x is ted by P c2C w(c) f(x). In the context of the component model, Grover’s quation can be expressed as: Avg(f(y)) y2N(x) = f(x) p1f(x) + p2 X c2C w(c) f(x) ! , p1 = ↵/d is the (sampling) rate at which components that contribute to re removed from solution x to create a neighboring solution y 2 N(x), = /d is the rate at which components in the set C x are sampled to a neighboring solution y 2 N(x). By simple algebra, (f(y)) = f(x) p1f(x) + p2 X w(c) f(x) ! = f(x) + d ( ¯f f(x)), nents in x: f(x) = P c2x w(c). Finally, let s that do not contribute to the evaluation he weights of the components in C x is context of the component model, Grover’s x) + p2 X c2C w(c) f(x) ! , at which components that contribute to create a neighboring solution y 2 N(x), mponents in the set C x are sampled to ). By simple algebra, X 2C w(c) f(x) ! = f(x) + d ( ¯f f(x)), P c2C w(c) [16]. omputed by c2C w(c) f(x). In the context of the compon wave equation can be expressed as: Avg(f(y)) y2N(x) = f(x) p1f(x) + p2 X c2C w(c) f where p1 = ↵/d is the (sampling) rate at which components (x) are removed from solution x to create a neighboring nd p2 = /d is the rate at which components in the set C reate a neighboring solution y 2 N(x). By simple algebra, Avg(f(y)) y2N(x) = f(x) p1f(x) + p2 X w2C w(c) f(x) ! = f(x where = ↵ + , and ¯f = /(↵ + ) P c2C w(c) [16]. ↵ d d ↵ d d ↵ d d ¯f = ↵ + X c2C w(c) Does not change with the neighborhood IN OUT Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Component Model Elementariness
15. 15. 15 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Elementariness of HPO with reversals: IN • How many times is a component IN the solution removed in the neighborhood? α = n-2 Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Component Model Elementariness
16. 16. 16 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Elementariness of HPO with reversals: IN • How many times is a component OUT of the solution included in the neighborhood? • HPO with reversals is elementary and the wave equation is: β = 2 and all feasible locations j < i. When all possible values of j are considered, this causes all of the vertices in the permutation left of vertex vi to come into a position adjacent to vi except for vi 1 which is already adjacent. Next consider a cut at location i+1 (i is still ﬁxed) and all feasible locations m > i+1. When all of the possible of value of m are considered all of the vertices in the permutation to the right of vertex vi are moved into a position adjacent to vi except vi+1. Thus, in these cases vi does not move, but every edge not in the solution x that is incident on vertex i is sampled once. Since this is true for all vertices, it follows that every edge not in the current solution is sampled twice ( = 2): once for each of the vertices in which it is incident. Therefore, summing over all the neighbors: d · Avg(f(y))y2N(x) = d · f(x) (n 2)f(x) + 2 P c2C w(c) f(x) . Computing the average over the neighborhood and taking into account the result of Lemma 2: Avg(f(y)) y2N(x) = f(x) n 2 n(n 1)/2 1 f(x) + 2 n(n 1)/2 1 X c2C w(c) f(x) ! = f(x) + n n(n 1)/2 1 ( ¯f f(x)). ut Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Component Model Elementariness
17. 17. 17 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Quadratic Assignment Problem: Definition Previously on QAP Decomposition of HPO rij • A QAP instance is composed of n facilities and n locations • A distance rij is specified between each pair of locations • A flow wpq is specified between each pair of facilities • The problem consists in assigning the facilities to the locations minimizing cost wpq Facilities Locations generalization Kronecker’s delta Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work
18. 18. 18 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 QAP Decomposition (Chicano, Whitley, Alba) • Auxiliary ϕ functions • Three elementary components for the swap neighborhood p … q i jπ q … p Eigenvalues Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Previously on QAP Decomposition of HPO
19. 19. 19 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 • The three components of QAP Kronecker’s delta QAP Decomposition (Chicano, Whitley, Alba) Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Previously on QAP Decomposition of HPO
20. 20. 20 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 • If any of the matrices w or r is symmetric, then f2n is a constant • If any of the matrices w or r is antisymmetric, then f2(n-1) is a constant Properties of the decomposition Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Previously on QAP Decomposition of HPO
21. 21. 21 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 • TSP is a particular case of QAP with two elementary components: f2n and f2(n-1) • HPO is another subproblem of QAP Subproblems of QAP: TSP and HPO d d ¯f = ↵ + X c2C w(c) f(⇡) = n 1X i=1 w⇡(i),⇡(i+1) + w⇡(n),⇡(1) Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Previously on QAP Decomposition of HPO
22. 22. 22 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 • If distance matrix in HPO is symmetric then f2n is a constant and the landscape has two elementary components: Subproblems of QAP: TSP and HPO Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work Previously on QAP Decomposition of HPO
23. 23. 23 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 • HPO with the reduced reversals neighborhood is an elementary landscape • HPO with the swap neighborhood is the sum of two elementary landscapes • The analysis of the problem can be used to compute statistics that could help in the search • Spectral theory of quasi-Abelian Cayley graph cannot be easily applied to the reversals Conclusions Future Work • Analysis of partial neighborhoods • Search for additional applications of the elementary landscape decomposition • How do the elementary components of HPO relate to the equivalent TSP problem? Conclusions & Future Work Introduction to Landscapes Hamiltonian Path Optimization Result for Reversals Result for Swaps Conclusions & Future Work
24. 24. 24 / 24Granada, Spain, April 2014EvoCOP 2014 UNIVERSITY*OF*GRANADA* 23225*APRIL*2014 Thanks for your attention !!! Elementary Landscape Decomposition of the Hamiltonian Path Optimization Problem