efficient hill climber for constrained pseudo-boolean optimization problems

1 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016

Genetic and EvolutionaryComputation Conference 2016Conference Program

Denver, CO, USAJuly 20-24, 2016

Introduction Background Hill Climber Experiments Conclusions & Future Work

Efficient Hill Climber for ConstrainedPseudo-Boolean Optimization Problems

Francisco Chicano, Darrell Whitley, Renato Tinós





r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)

Sv(x) = f(x� v)� f(x) =mX

l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

S1,4(x) = S1(x) + S4(x)

S1(x) = f

(1)(x� 1)� f

(1)(x)

S2(x) = f

(1)(x� 2)� f

(1)(x) + f

(2)(x� 2)� f

(2)(x) + f

(3)(x� 2)� f

(3)(x)

S1,2(x) = f

(1)(x�1, 2)�f

(1)(x)+f

(2)(x�1, 2)�f

(2)(x)+f

(3)(x�1, 2)�f

(3)(x)

S1,2(x) 6= S1(x) + S2(x)

f(x) =NX

l=1

f

(l)(x)

1

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

S1,4(x) = S1(x) + S4(x)

S1(x) = f

(1)(x� 1)� f

(1)(x)

S2(x) = f

(1)(x� 2)� f

(1)(x) + f

(2)(x� 2)� f

(2)(x) + f

(3)(x� 2)� f

(3)(x)

S1,2(x) = f

(1)(x�1, 2)�f

(1)(x)+f

(2)(x�1, 2)�f

(2)(x)+f

(3)(x�1, 2)�f

(3)(x)

S1,2(x) 6= S1(x) + S2(x)

f(x) =NX

l=1

f

(l)(x)

1

• Considering binary strings of length n and Hamming distance…

Solutions in a ball of radius r

r=1r=2

r=3

Ball of radius r Previous work Research Question

How many solutions at Hamming distance r?

If r << n : Θ (nr)





• We want to find improving moves in a ball of radius r around solution x

• What is the computational cost of this exploration?

• By complete enumeration: O (nr) if the fitness evaluation is O(1)

• Previous work proposed a way to find improving moves in ball of radius r in O(1) (constant time independent of n): Szeider (DO 2011), Whitley and Chen(GECCO 2012), Chen et al. (GECCO 2013), Chicano et al. (GECCO 2014)

• The approach was extended to the multi-objective case: Chicano et al. (EvoCOP 2016)

• All the previous results are for unconstrained optimization problems

Improving moves in a ball of radius r

r

Ball of radius r Previous work Research Question





Research QuestionBall of radius r Previous work Research Question





• Definition:

• where f(i) only depends on k variables (k-bounded epistasis)

• We will also assume that the variables are arguments of at most c subfunctions

• Example (m=4, n=4, k=2):

• Are Mk Landscapes a small subset of problems? Are they interesting?

• Max-kSAT is a k-bounded pseudo-Boolean optimization problem• NK-landscapes is a (K+1)-bounded pseudo-Boolean optimization problem

• Any compressible pseudo-Boolean function can be reduced to a quadratic pseudo-Boolean function (e.g., Rosenberg, 1975)

Mk Landscape (Whitley, GECCO2015: 927-934)Pseudo-Boolean functions Scores Update Decomposition

derivative, they reduce the time needed to identify improv-ing moves from O(k22k) to O(k3). In addition, the newapproach avoids the use of the Walsh transform, making theapproach conceptually simpler.

In this paper, we generalize this result to present a localsearch algorithm that can look r moves ahead and iden-tify all improving moves. This means that moves are beingidentified in a neighborhood containing all solutions that liewithin a Hamming ball of radius r around the current so-lution. We assume that r = O(1). If r ⌧ n, the numberof solutions in such a neighborhood is ⇥(nr). New improv-ing moves located up to r moves away can be identified inconstant time. The memory required by our approach isO(n). To achieve O(1) time per move, the number of sub-functions in which any variable appears must be bounded bysome constant c. We then prove that the resulting algorithmrequires O((3kc)rn) space to track potential moves.

In order to evaluate our approach we perform an experi-mental study based on NKq-landscapes. The results revealnot only that the time required by the next ascent is inde-pendent of n, but also that increasing r we obtain a signifi-cant gain in the quality of the solutions found.

The rest of the paper is organized as follows. In the nextsection we introduce the pseudo-Boolean optimization prob-lems. Section 3 defines the“Scores”of a solution and providean algorithm to e�ciently update them during a local searchalgorithm. We propose in Section 4 a next ascent hill climberwith the ability to identify improving moves in a ball of ra-dius r in constant time. Section 5 empirically analyzes thishill climber using NKq-landscapes instances and Section 6outlines some conclusions and future work.

2. PSEUDO-BOOLEAN OPTIMIZATIONOur method for identifying improving moves in the radius

r Hamming ball can be applied to all k-bounded pseudo-Boolean Optimization problems. This makes our methodquite general: every compressible pseudo-Boolean Optimiza-tion problem can be transformed into a quadratic pseudo-Boolean Optimization problem with k = 2.

The family of k-bounded pseudo-Boolean Optimizationproblems have also been described as an embedded landscape.An embedded landscape [3] with bounded epistasis k is de-fined as a function f(x) that can be written as the sumof m subfunctions, each one depending at most on k inputvariables. That is:

f(x) =mX

i=1

f (i)(x), (1)

where the subfunctions f (i) depend only on k componentsof x. Embedded Landscapes generalize NK-landscapes andthe MAX-kSAT problem. We will consider in this paper thatthe number of subfunctions is linear in n, that is m 2 O(n).For NK-landscapes m = n and is a common assumption inMAX-kSAT that m 2 O(n).

3. SCORES IN THE HAMMING BALLFor v, x 2 Bn, and a pseudo-Boolean function f : Bn ! R,

we denote the Score of x with respect to move v as Sv(x),defined as follows:1

Sv(x) = f(x� v)� f(x), (2)1We omit the function f in Sv(x) to simplify the notation.

where � denotes the exclusive OR bitwise operation. TheScore Sv(x) is the change in the objective function when wemove from solution x to solution x� v, that is obtained byflipping in x all the bits that are 1 in v.All possible Scores for strings v with |v| r must be

stored as a vector. The Score vector is updated as localsearch moves from one solution to another. This makes itpossible to know where the improving moves are in a ball ofradius r around the current solution. For next ascent, all ofthe improving moves can be bu↵ered. An approximate formof steepest ascent local search can be implemented usingmultiple bu↵ers [9].If we naively use equation (2) to explicitly update this

Score vector, we will have to evaluate allPr

i=0

�ni

�neigh-

bors in the Hamming ball. Instead, if the objective functionsatisfies some requirements described below, we can designan e�cient next ascent hill climber for the radius r neigh-borhood that only stores a linear number of Score values andrequires a constant time to update them. We next explainthe theoretical foundations of this next ascent hill climber.The first requirement for the objective function is that it

must be written such that each subfunction depends only onk Boolean variables of x (k-bounded epistasis). In this case,we can write the scoring function Sv(x) as an embeddedlandscape:

Sv(x) =mX

l=1

⇣f (l)(x� v)� f (l)(x)

⌘=

mX

l=1

S(l)v (x), (3)

where we use S(l)v to represent the scoring functions of the

subfunctions f (l). Let us define wl 2 Bn as the binary stringsuch that the i-th element of wl is 1 if and only if f (l) dependson variable xi. The vector wl can be considered as a maskthat characterizes the variables that a↵ect f (l). Since f (l)

has bounded epistasis k, the number of ones in wl, denotedwith |wl|, is at most k. By the definition of wl, the nextequalities immediately follow.

f (l)(x� v) = f (l)(x) for all v 2 Bn with v ^ wl = 0, (4)

S(l)v (x) =

⇢0 if wl ^ v = 0,

S(l)v^wl

(x) otherwise.(5)

Equation (5) claims that if none of the variables thatchange in the move characterized by v is an argument off (l) the Score of this subfunction is zero, since the value ofthis subfunction will not change from f (l)(x) to f (l)(x� v).On the other hand, if f (l) depends on variables that change,we only need to consider for the evaluation of S

(l)v (x) the

changed variables that a↵ect f (l). These variables are char-acterized by the mask vector v ^ wl. With the help of (5)we can write (3) as:

Sv(x) =mX

l=1wl^v 6=0

S(l)v^wl

(x), (6)

3.1 Scores DecompositionThe Score values in a ball of radius r give more informa-

tion than just the change in the objective function for movesin that ball. Let us illustrate this idea with the moves in theballs of radius r = 1 and r = 2. Let us assume that xi andxj are two variables that do not appear together as argu-ments of any subfunction f (l). Then, the Score of the move

f = + + + f(1)(x) f(2)(x) f(3)(x) f(4)(x)

x1 x2 x3 x4





• Based on Mk Landscapes

Going Multi-Objective: Vector Mk LandscapePseudo-Boolean functions Scores Update Decomposition

x1 x2 x3 x4 x5

f

(1)1 f

(2)1 f

(3)1 f

(4)1 f

(5)1

f

(1)2 f

(2)2 f

(3)2

(a) Vector Mk Landscape

x1 x2

x3x4x5

(b) Co-occurrence graph

Fig. 1. A vector Mk Landscape with k = 2, n = 5 variables and d = 2 dimensions(top) and its corresponding co-occurrence graph (bottom).

fj(x) > fj(y). When the vector function is clear from the context, we will use �instead of �f .

Definition 3 (Pareto Optimal Set and Pareto Front). Given a vectorfunction f : Bn ! Rd, the Pareto Optimal Set is the set of solutions P that arenot dominated by any other solution in Bn. That is:

P = {x 2 Bn|@y 2 Bn, y � x} . (2)

The Pareto Front is the image by f of the Pareto Optimal Set: PF = f(P ).

Definition 4 (Set of Non-dominated Solutions). Given a vector functionf : Bn ! Rd, we say that a set X ✓ Bn is a set of non-dominated solutions whenthere is no pair of solutions x, y 2 X where y � x, that is, 8x 2 X, @y 2 X, y � x.

Definition 5 (Local Optimum [11]). Given a vector function f : Bn ! Rd,and a neighborhood function N : Bn ! 2B

n

, we say that solution x is a localoptimum if it is not dominated by any other solution in its neighborhood: @y 2N(x), y � x.

3 Moves in a Hamming Ball

We can characterize a move in Bn by a binary string v 2 Bn having 1 in allthe bits that change in the solution. Following [3] we will extend the concept ofScore5 to vector functions.5 What we call Score here is also named �-evaluation by other authors [13].

f1

f2





Going Multi-Objective & ConstrainedPseudo-Boolean functions Scores Update Decomposition

f1

f2

x1 x2 x3 x4 x5

f

(1)1

f

(1)2

f

(2)1

f

(2)2

f

(3)1

g

(1)1 g

(2)1 g

(3)1

1

g1

Feasible solutions:

2. BACKGROUNDIn constrained multi-objective optimization, there is a vec-

tor function f : Bn ! Rd to optimize, called the objectivefunction. We will assume, without loss of generality, thatall the objectives (components of the vector function) areto be maximized. The constraints of the problem will begiven in the form1 g(x) � 0, where g : Bn ! Rb is a vec-tor function, that will be called constraint function2. Thatis, a solution is feasible if all the components of the vectorfunction g are nonnegative when evaluated in that solution.This type of constraints does not represent a limitation. Anyother equality or inequality constraint can be expressed inthe form gi(x) � 0, including those that use strict inequalityconstraints (> and <)3. The set of feasible solutions of aproblem will be denoted by Xg = {x 2 Bn|g(x) � 0}.

Given a vector function f : Bn ! Rd, we say that solutionx 2 Bn dominates solution y 2 Bn, denoted with x �f y, ifand only if f(x) � f(y) and there exists j 2 {1, 2, . . . , d} suchthat fj(x) > fj(y). When the vector function is clear fromthe context, we will use � instead of �f . Observe that we donot restrict the definition of dominance to feasible solutionsof a problem, since we will need to consider dominance alsobetween feasible and non-feasible solutions. Furthermore,we will use the concept of dominance using as vector functionthe constraint function.The Pareto Optimal Set is the set of feasible solutions P

that are not dominated by any other feasible solution in Xg:

P = {x 2 Xg|@y 2 Xg, y � x} . (1)

The Pareto Front is the image by f of the Pareto OptimalSet: PF = f(P ).

Definition 1 (Local Optimum [10]). Given a vectorfunction f : Bn ! Rd, a constraint function g : Bn ! Rb,and a neighborhood function N : Bn ! 2B

n, we say that

solution x 2 Xg is a local optimum if it is not dominated byany other feasible solution in its neighborhood: @y 2 N(x)\Xg, y � x.

In this work we focus our attention to problems where theobjective and constraint functions have k-bounded epistasis,that is, their components areMk Landscapes [12]. These vec-tor functions have been recently called vector Mk landscapesand are defined as follows.

Definition 2 (Vector Mk Landscape [4]). Giventwo constants k and d, a vector Mk Landscape f : Bn ! Rd

is a d-dimensional vector pseudo-Boolean function definedover Bn whose components are Mk Landscapes. That is,each component fi can be written as a sum of mi subfunc-tions, each one depending at most on k input variables:

fi(x) =miX

l=1

f(l)i (x) for 1 i d, (2)

where the subfunctions f(l)i depend on k components of x.

Figure 1(a) shows a multi-objective constraint pseudo-Boolean problem with d = 2 objective functions and b = 11The relational operators applied to vectors will mean thateach component must fulfill the inequality.2We will use boldface to denote vectors in Rd or Rb, suchas f , but we will use normal weight for vectors in Bn, like x.3This is a consequence of the finite search space.

x1 x2 x3 x4 x5

f(1)1 f

(1)2 f

(2)1 f

(2)2 f

(3)1

g(1)1 g

(2)1 g

(3)1


x1 x2

x3x4x5


Figure 1: A multi-objective constrained pseudo-Boolean problem with k = 2, n = 5 variables, d = 2objectives and b = 1 constraint (top) and its corre-sponding co-occurrence graph (bottom).

constraint function. The first objective function, f1, can bewritten as the sum of 3 subfunctions, f

(1)1 to f

(3)1 , shown

with white rectangles. The second component function, f2,can be written as the sum of 2 subfunctions, f (1)

2 and f(2)2 ,

shown with blue (grey) rectangles. Finally, the constraint

g1 can be written as the sum of 3 subfunctions, g(1)1 to g(3)1 ,

and is shown with circles in the figure. All the subfunctionsdepend at most on k = 2 variables.Considering only vector Mk landscapes for the objective

and constraint functions is not a limitation, since every pseu-do-Boolean function that can be expressed in algebraic formusing polynomial space can be transformed in polynomialtime into a quadratic (k = 2) pseudo-Boolean function [11].An important tool for the forthcoming analysis is the co-

ocurrence graph.

Definition 3 (Co-ocurrence Graph [5]). Given anobjective function f : Bn ! Rd and a constraint functiong : Bn ! Rb, the co-occurrence graph is G = (V,E), whereV is the set of Boolean variables and E contains all thepairs of variables (xj1 , xj2) that co-occur (both variables arearguments) in a subfunction of the objective or the constraintfunction.

In Figure 1(b) we show the variable co-occurrence graphof the vector Mk Landscape of Figure 1(a).A move in Bn can be characterized by a binary string

v 2 Bn having 1 in all the bits that change in the solution.The Score of a move, has been previously defined as theincrement in the objective function when that move is taken.

Definition 4 (Score). For v, x 2 Bn, and a vectorfunction f : Bn ! Rd, we denote the Score of x with respectto move v for function f as S

(f)v (x), defined as follows:

S(f)v (x) = f(x� v)� f(x), (3)

where � is the exclusive OR bitwise operation.

When constraints are considered, we also need to trackwhere feasible solutions are, and Scores of the contraint

Objective

function

Constraint function





• Let us represent a potential move of the current solution with a binary vector v having 1s in the positions that should be flipped

• The score of move v for solution x is the difference in the fitness value of the neighboring and the current solution

• Scores are useful to identify improving moves: if Sv(x) > 0, v is an improving move

Scores: definition

Current solution, x Neighboring solution, y Move, v01110101010101001 01111011010101001 0000111000000000001110101010101001 00110101110101111 0100000010000011001110101010101001 01000101010101001 00110000000000000

Pseudo-Boolean functions Scores Update Decomposition

2. BACKGROUNDIn constrained multi-objective optimization, there is a vec-

tor function f : Bn ! Rd to optimize, called the objectivefunction. We will assume, without loss of generality, thatall the objectives (components of the vector function) areto be maximized. The constraints of the problem will begiven in the form1 g(x) � 0, where g : Bn ! Rb is a vec-tor function, that will be called constraint function2. Thatis, a solution is feasible if all the components of the vectorfunction g are nonnegative when evaluated in that solution.This type of constraints does not represent a limitation. Anyother equality or inequality constraint can be expressed inthe form gi(x) � 0, including those that use strict inequalityconstraints (> and <)3. The set of feasible solutions of aproblem will be denoted by Xg = {x 2 Bn|g(x) � 0}.

Given a vector function f : Bn ! Rd, we say that solutionx 2 Bn dominates solution y 2 Bn, denoted with x �f y, ifand only if f(x) � f(y) and there exists j 2 {1, 2, . . . , d} suchthat fj(x) > fj(y). When the vector function is clear fromthe context, we will use � instead of �f . Observe that we donot restrict the definition of dominance to feasible solutionsof a problem, since we will need to consider dominance alsobetween feasible and non-feasible solutions. Furthermore,we will use the concept of dominance using as vector functionthe constraint function.The Pareto Optimal Set is the set of feasible solutions P

that are not dominated by any other feasible solution in Xg:

P = {x 2 Xg|@y 2 Xg, y � x} . (1)

The Pareto Front is the image by f of the Pareto OptimalSet: PF = f(P ).

Definition 1 (Local Optimum [10]). Given a vectorfunction f : Bn ! Rd, a constraint function g : Bn ! Rb,and a neighborhood function N : Bn ! 2B

n, we say that

solution x 2 Xg is a local optimum if it is not dominated byany other feasible solution in its neighborhood: @y 2 N(x)\Xg, y � x.

In this work we focus our attention to problems where theobjective and constraint functions have k-bounded epistasis,that is, their components areMk Landscapes [12]. These vec-tor functions have been recently called vector Mk landscapesand are defined as follows.

Definition 2 (Vector Mk Landscape [4]). Giventwo constants k and d, a vector Mk Landscape f : Bn ! Rd

is a d-dimensional vector pseudo-Boolean function definedover Bn whose components are Mk Landscapes. That is,each component fi can be written as a sum of mi subfunc-tions, each one depending at most on k input variables:

fi(x) =miX

l=1

f(l)i (x) for 1 i d, (2)

where the subfunctions f(l)i depend on k components of x.

Figure 1(a) shows a multi-objective constraint pseudo-Boolean problem with d = 2 objective functions and b = 11The relational operators applied to vectors will mean thateach component must fulfill the inequality.2We will use boldface to denote vectors in Rd or Rb, suchas f , but we will use normal weight for vectors in Bn, like x.3This is a consequence of the finite search space.

x1 x2 x3 x4 x5

f(1)1 f

(1)2 f

(2)1 f

(2)2 f

(3)1

g(1)1 g

(2)1 g

(3)1


x1 x2

x3x4x5


Figure 1: A multi-objective constrained pseudo-Boolean problem with k = 2, n = 5 variables, d = 2objectives and b = 1 constraint (top) and its corre-sponding co-occurrence graph (bottom).

constraint function. The first objective function, f1, can bewritten as the sum of 3 subfunctions, f

(1)1 to f

(3)1 , shown

with white rectangles. The second component function, f2,can be written as the sum of 2 subfunctions, f (1)

2 and f(2)2 ,

shown with blue (grey) rectangles. Finally, the constraint

g1 can be written as the sum of 3 subfunctions, g(1)1 to g(3)1 ,

and is shown with circles in the figure. All the subfunctionsdepend at most on k = 2 variables.Considering only vector Mk landscapes for the objective

and constraint functions is not a limitation, since every pseu-do-Boolean function that can be expressed in algebraic formusing polynomial space can be transformed in polynomialtime into a quadratic (k = 2) pseudo-Boolean function [11].An important tool for the forthcoming analysis is the co-

ocurrence graph.

Definition 3 (Co-ocurrence Graph [5]). Given anobjective function f : Bn ! Rd and a constraint functiong : Bn ! Rb, the co-occurrence graph is G = (V,E), whereV is the set of Boolean variables and E contains all thepairs of variables (xj1 , xj2) that co-occur (both variables arearguments) in a subfunction of the objective or the constraintfunction.

In Figure 1(b) we show the variable co-occurrence graphof the vector Mk Landscape of Figure 1(a).A move in Bn can be characterized by a binary string

v 2 Bn having 1 in all the bits that change in the solution.The Score of a move, has been previously defined as theincrement in the objective function when that move is taken.

Definition 4 (Score). For v, x 2 Bn, and a vectorfunction f : Bn ! Rd, we denote the Score of x with respectto move v for function f as S

(f)v (x), defined as follows:

S(f)v (x) = f(x� v)� f(x), (3)

where � is the exclusive OR bitwise operation.

When constraints are considered, we also need to trackwhere feasible solutions are, and Scores of the contraint





• The key idea is to compute the scores from scratch once at the beginning and update their values as the solution moves (less expensive)

Scores update

r

Selected improving move

Update the scores






• The key idea is to compute the scores from scratch once at the beginning and update their values as the solution moves (less expensive)

• How can we do it less expensive?

• We have still O(nr) scores to update!

• … thanks to two key facts:

• We don’t need all the O(nr) scores to have complete information of the influence of a move in the objective or constraint (vector) function, only O(n) scores

• From the ones we need, we only have to update a constant number of them and we can do each update in constant time

Key facts for efficient scores update

r






What is an improving move in MO?• An improving move is one that provides a solution dominating the current one

• We define strong and weak improving moves.

• We are interested in strong improving moves

Taking improving moves Taking feasible moves Algorithm

f2

f1 Assuming maximization

Can we safely take onlystrong improving moves?





We need to take weak improving moves

• We could miss some higher order strong improving moves if we don’t take weak improving moves

S2

S1

stored

stored

not stored

current solution

x1 x2 x3 x4 x5

f

(1)1

f

(1)2

f

(2)1

f

(2)2

f

(3)1

g

(1)1 g

(2)1 g

(3)1

S(f)v1[v2(x) = S(f)

v1 (x) + S(f)v2 (x) (1)

1






Cycling

• We can make the hill climber to cycle if we take weak improving moves

S2

S1

stored

current solution

S2

S1

stored

current solution






Solution: weights for the scores• Given a weight vector w with wi > 0

S2

S1

w

1st strong improving2nd w-improving

2nd w-improving

A w-improving move isone with w · Sv(x) > 0






Feasibility does not only depends on the scores

• In order to classify the moves as feasible or unfeasible we have to check all the scores, even if they have not changed

• This can be done in O(n), but not in O(1) in general

g2

g1

Unfeasiblesolution

current solution

v1

v1

v2

Feasible region






Feasibility in the Hamming Ball

x1 x2 x3 x4 x5

f

(1)1

f

(1)2

f

(2)1

f

(2)2

f

(3)1

g

(1)1 g

(2)1 g

(3)1

S(f)v1[v2(x) = S(f)

v1 (x) + S(f)v2 (x) (1)

S(g)v1[v2(x) = S(g)

v1 (x) + S(g)v2

(x) (2)

1

g2

g1

Unfeasiblestored solution current solution

v1

v1

v2

Feasible region

v2

Unfeasiblestored solution

Feasible notstored solution






Feasibility in the Hamming Ball

g2

g1

current solution

v1

v1

v2

Feasible region

v2

w*

w*-feasible

• Bad news: this does not work when the current solution is unfeasible• Worse news: there is no “efficient” way to identify a feasible solution when the

current solution is unfeasible






Unfeasible regionFeasible region

No w-improvingor w*-feasiblemoves ▶ stop

strongimproving in g

w*-improvingin g

solution y

otherwise

Hill ClimberTaking improving moves Taking feasible moves Algorithm





MNK LandscapeProblem Results Source Code

Why NKq and not NK?Floating point precision

• An MNK Landscape is a multi-objective pseudo-Boolean problem where each objective is an NK Landscape (Aguirre, Tanaka, CEC 2004: 196-203)

• An NK-landscape is a pseudo-Boolean optimization problem with objective function:where each subfunction f(l) depends on variable xl and Kother variables

• The subfunctions are randomly generated and the values are taken in the range [0,1]

• In NKq-landscapes the subfunctions take integer values in the range [0,q-1]• We used shifted NKq-landscapes in the experiments with values in two ranges:

• [-49, 50] slightly constrained problems (2% unfeasible)

• [-50, 49] highly constrained problems (80% unfeasible)

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

S1,4(x) = S1(x) + S4(x)

S1(x) = f

(1)(x� 1)� f

(1)(x)

S2(x) = f

(1)(x� 2)� f

(1)(x) + f

(2)(x� 2)� f

(2)(x) + f

(3)(x� 2)� f

(3)(x)

S1,2(x) = f

(1)(x�1, 2)�f

(1)(x)+f

(2)(x�1, 2)�f

(2)(x)+f

(3)(x�1, 2)�f

(3)(x)

S1,2(x) 6= S1(x) + S2(x)

f(x) =NX

l=1

f

(l)(x)

1

• In the adjacent model the variables are consecutive

f = + + + f(1)(x) f(3)(x) f(2)(x) f(4)(x)

x1 x2 x3 x4





account to compute the time per move. In our experimentsthis initialization procedure required between 138 and 6,308milliseconds. The rest of the time required by the multi-start hill climber is divided by the number of moves takenand reported as average time per move.

Figure 3 shows the average time per move in microsecondsfor the multi-start hill climber solving constrained MNKLandscapes, where N varies from 10, 000 to 100, 000, therange of values for the subfunctions codomain is [�49, 50],the number of objectives is d = 1 and d = 2, the number ofconstraints is b = 1 and b = 2, and the exploration radius rvaries from 1 to 3. These instances are slightly constrainedwith around 2% of the search space being unfeasible10. Weperformed 30 independent runs of the algorithm for eachconfiguration, and the results are the average of these 30runs.

0 200 400 600 800

1000 1200 1400 1600

10 20 30 40 50 60 70 80 90 100Aver

age

time

per m

ove

(mic

rose

cond

s)

N (number of variables in thousands)

r=1, d=1, b=1r=1, d=2, b=1r=1, d=1, b=2r=1, d=2, b=2r=2, d=1, b=1r=2, d=2, b=1


Figure 3: Average time per move in microsec-onds for the Multi-Start Hill Climber based on Al-gorithm 1 for constrained MNK Landscapes withd = 1, 2, b = 1, 2, N = 10, 000 to 100, 000, subfunctionscodomain [�49, 50], and r = 1 to 3.

We can observe an almost constant time per move as N in-creases. Since this is a slightly constrained instance, the hillclimber explores the feasible region most of the time, wherethe implementation trick mentioned above prevents the re-classification of all the moves, which takes linear time. Wealso observe that the radius of exploration has the high-est influence on the time. This time is proportional to(3ck)r (see Section 2.1), but k is fixed in this experiment(k = K+1 = 4). Thus, only the number of objectives d, thenumber of constraints b, and the radius r externally a↵ectthe time. The objectives and constraints a↵ect the value ofc, in particular, c = (K+1)(d+ b). This expression appearsin the base of (3ck)r, but r is in the exponent. This explainswhy r has the highest influence. On the other hand, sincethe sum d+b influences c, the di↵erence between d = 1, c = 2and d = 2, c = 1 is small in Figure 3.In Figure 4 we show the average time per move (in mi-

croseconds) obtained by the multi-start hill climber solv-ing a highly constrained MNK Landscape. The subfunc-tions codomain here is [�50, 49] and N varies from 1, 000to 10, 000. There is only one constraint and two objectives(b = 1, d = 2). The fraction of unfeasible solutions in thiscase is around 80%.

10This is an estimation computed as the fraction of unfeasiblesolutions obtained after generating random solutions.

0

2000

4000

6000

8000

10000

12000

14000

16000

1 2 3 4 5 6 7 8 9 10

Aver

age

time

per m

ove

(mic

rose

cond

s)


r=1 r=2 r=3

Figure 4: Average time per move in microsecondsfor the Multi-Start Hill Climber based on Algo-rithm 1 for constrained MNK Landscapes with d = 2,b = 1, N = 1, 000 to 10, 000, subfunctions codomain[�50, 49], and r = 1 to 3.

In this case, we can appreciate a clear linear increase inthe time per move as N increases. Observe that the timeper move in this figure is higher than the time per move inFigure 3, even when the search space is smaller (N is 10times smaller). The reason is that the hill climber entersthe unfeasible region very often and it needs to re-classifyall the moves almost after every move, which takes lineartime.

5.2 Quality of the SolutionsExploring larger neighborhoods allows the hill climber to

jump to better solutions but it also requires more time permove. Thus, increasing r does not necessarily improve thequality of the solutions when time is the stopping criterion.For each problem instance, there is a value of r for whichthe quality of the solutions is maximum. We will investigatehere this maximum for the problems we are solving.In Figure 5 we show the average value of the best solution

found for the single-objective slightly constrained instances(d = 1). We can observe that the quality of the solutionsincreases as r does. This means that the optimum value forr is 3 or higher in this case. We can also observe how thenumber of constraints seems not to a↵ect the quality of thesolutions in this case and the two lines are overlapped. Thenumber of constraints must certainly have an influence inthe quality of the solutions found by the hill climber. Butthis influence is negligible in slightly constrained problemslike these ones.We now repeat the analysis for the bi-objective instances

(d = 2) using the 50%-empirical attainment surfaces11 tocompare the fronts obtained when r is changed. In Fig-ures 6(a) and 6(b) we show 50%-EAS for instances withN = 10, 000 and N = 50, 000, constrained by b = 1 andb = 2 functions, respectively. In this case we notice thatthe 50%-EAS obtained with r = 2 always dominates theone obtained with r = 1 (this happens also in the instancesthat are not shown). However, the 50%-EAS obtained for

11The 50%-empirical attainment surface (50%-EAS) limitsthe region in the objective space that is dominated by halfthe runs of the algorithm. It generalizes the concept of me-dian to the multi-objective case (see [9] for more details).

Runtime: highly constrained MNK LandscapesNeighborhood size: 166 billion

Problem Results Source Code







0 200 400 600 800

1000 1200 1400 1600

10 20 30 40 50 60 70 80 90 100Aver

age

time

per m

ove

(mic

rose

cond

s)








0

2000

4000

6000

8000

10000

12000

14000

16000

1 2 3 4 5 6 7 8 9 10

Aver

age

time

per m

ove

(mic

rose

cond

s)


r=1 r=2 r=3








Runtime: slightly constrained MNK Landscapes

Neighborhood size: 166 trillion




0 200 400 600 800

1000 1200 1400 1600

10 20 30 40 50 60 70 80 90 100Aver

age

time

per m

ove

(mic

rose

cond

s)








0

2000

4000

6000

8000

10000

12000

14000

16000

1 2 3 4 5 6 7 8 9 10

Aver

age

time

per m

ove

(mic

rose

cond

s)


r=1 r=2 r=3












Source Codehttps://github.com/jfrchicanog/EfficientHillClimbers/tree/constrained-multiobjective






Conclusions and Future WorkConclusions & Future Work

• Adding constrains to the MK Landscapes has a cost in terms of efficiency: from O(1) to O(n)• The space required to store the information isstill linear in the size of the problem n

Conclusions

• Generalize to other search spaces• Combine with high-level algorithms (MOEA/D)

Future Work



Denver, CO, USAJuly 20-24, 2016Acknowledgements

Efficient Hill Climber for Constrained Pseudo-Boolean Optimization Problems





Examples: 1 and 4

f(1) f(2) f(3) f(4)

x1 x2 x3 x4

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

1

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

1

f(1) f(2) f(3) f(4)

x1 x2 x3 x4

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

1






Example: 1,4

f(1) f(2) f(3) f(4)

x1 x2 x3 x4

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

1

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

1

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

1

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

1

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

S1,4(x) = S1(x) + S4(x)

1

We don’t need to store S1,4(x) since can be computed from othersIf none of 1 and 4 are improving moves, 1,4 will not be an improving move






Quality

• 50% Empirical Attainment Functions (Knowles, ISDA 2005: 552-557)


0

500000

1e+06

1.5e+06

2e+06

2.5e+06

3e+06

10 20 30 40 50 60 70 80 90 100

Aver

age

qual

ity o

f bes

t fou

nd s

olut

ion


r=1, b=1r=1, b=2r=2, b=1r=2, b=2r=3, b=1r=3, b=2

Figure 5: Average (over 30 runs) solution qualityof the best solution found by the Multi-Start HillClimber based on Algorithm 1 for a MNK Land-scape with d = 1, b = 1, 2, subfunctions codomain[�49, 50], N = 10, 000 to 100, 000, and r = 1 to 3.

0

50000

100000

150000

200000

250000

300000

0 50000 100000 150000 200000 250000 300000

f 2

f1

r=1r=2r=3

(a) N = 10, 000, b = 1

0

200000

400000

600000

800000

1e+06

1.2e+06

1.4e+06

0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06

f 2

f1

r=1r=2r=3

(b) N = 50, 000, b = 2

Figure 6: 50%-empirical attainment surfaces of the30 runs of the Multi-Start Hill Climber based onAlgorithm 1 for a MNK Landscape with d = 2, sub-functions codomain [�49, 50], and r = 1 to 3.

r = 3 does not always dominate the one with r = 2. AsN increases, the 50%-EAS obtained with r = 3 shrinks andis usually dominated by the 50%-EAS obtained with r = 2.Thus, we can say that r = 2 is the optimal value for theseinstances.

6. CONCLUSIONS AND FUTURE WORKThe use of problem knowledge is always beneficial to de-

sign more e↵ective search procedures and operators. In thecase of pseudo-Boolean optimization the formal structureof Mk Landscapes has been previously used to develop ef-ficient hill climbers that explore a Hamming ball of radiusr in constant time if the number of subfunctions each vari-able appears in is bounded by a constant. We conclude inthis work that this O(1) time cannot be guaranteed if wewant to solve constrained problems and we proposed a newhill climber for multi-objective constrained problems whoseworst case runtime is O(n), which is still low compared tothe complete exploration of the Hamming ball when r > 1.

Combining the proposed hill climber with high-level searchalgorithms seems a promising line of work that could pro-duce new very e�cient optimization algorithms. Anotherfuture line of research is the adaptaion of the key ingredientsof this hill climber to other search spaces, like permutationsor integer vectors.

AcknowledgementsThis research was partially funded by the Fulbright pro-gram, the Spanish Ministry of Education, Culture and Sport(CAS12/00274), the Spanish Ministry of Economy and Com-petitiveness and FEDER (TIN2014-57341-R), the Univer-sity of Malaga, Andalucıa Tech, the Air Force O�ce ofScientific Research, Air Force Materiel Command, USAF(FA9550-11-1-0088), the FAPESP (2015/06462-1) and CNPq.

7. REFERENCES[1] Hernan E. Aguirre and Kiyoshi Tanaka. Insights on

properties of multiobjective MNK-landscapes. InProceedings of CEC, volume 1, pages 196–203, 2004.

[2] Wenxiang Chen, Darrell Whitley, Doug Hains, andAdele Howe. Second order partial derivatives forNK-landscapes. In Proceeding of GECCO, pages503–510, New York, NY, USA, 2013. ACM.

[3] Francisco Chicano, Darrell Whitley, and Andrew M.Sutton. E�cient identification of improving moves in aball for pseudo-boolean problems. In Proceedings ofGECCO, pages 437–444. ACM, 2014.

[4] Francisco Chicano, Darrell Whitley, and RenatoTinos. E�cient hill climber for multi-objectivepseudo-boolean optimization. In Proceedings ofEvoCOP, pages 88–103, 2016.

[5] Yves Crama, Pierre Hansen, and Brigitte Jaumard.The basic algorithm for pseudo-boolean programmingrevisited. Discrete Applied Mathematics,29(2-3):171–185, 1990.

[6] Brian W. Goldman and William F. Punch. Gray-boxoptimization using the parameter-less populationpyramid. In Proceedings of GECCO, pages 855–862,New York, NY, USA, 2015. ACM.

[7] Holger H. Hoos and Thomas Stutzle. Stochastic LocalSearch: Foundations and Applications. MorganKaufman, 2004.

[8] Hans Kellerer, Ulrich Pferschy, and David Pisinger.Knapsack Problems. Springer-Verlag, 2004.

[9] J. Knowles. A summary-attainment-surface plottingmethod for visualizing the performance of stochasticmultiobjective optimizers. In Proceedings of ISDA,pages 552–557, 2005.

[10] Luis Paquete, Tommaso Schiavinotto, and ThomasStutzle. On local optima in multiobjectivecombinatorial optimization problems. Annals ofOperations Research, 156(1):83–97, 2007.

[11] Ivo G. Rosenberg. Reduction of bivalent maximizationto the quadratic case. Cahiers Centre Etudes Rech.Oper., 17:71–74, 1975.

[12] Darrell Whitley. Mk landscapes, NK landscapes,MAX-kSAT: A proof that the only challengingproblems are deceptive. In Proceedings of GECCO,pages 927–934, New York, NY, USA, 2015. ACM.

[13] Darrell Whitley and Wenxiang Chen. Constant timesteepest descent local search with lookahead forNK-landscapes and MAX-kSAT. In Proceedings ofGECCO, pages 1357–1364, 2012.

[14] Darrell Whitley, Wenxiang Chen, and Adele E. Howe.An empirical evaluation of O(1) steepest descent forNK-landscapes. In Proceedings of PPSN, pages92–101, 2012.

0

500000

1e+06

1.5e+06

2e+06

2.5e+06

3e+06

10 20 30 40 50 60 70 80 90 100

Aver

age

qual

ity o

f bes

t fou

nd s

olut

ion


r=1, b=1r=1, b=2r=2, b=1r=2, b=2r=3, b=1r=3, b=2


0

50000

100000

150000

200000

250000

300000

0 50000 100000 150000 200000 250000 300000

f 2

f1

r=1r=2r=3

(a) N = 10, 000, b = 1

0

200000

400000

600000

800000

1e+06

1.2e+06

1.4e+06

0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06

f 2

f1

r=1r=2r=3

(b) N = 50, 000, b = 2






















0

500000

1e+06

1.5e+06

2e+06

2.5e+06

3e+06

10 20 30 40 50 60 70 80 90 100Av

erag

e qu

ality

of b

est f

ound

sol

utio

nN (number of variables in thousands)

r=1, b=1r=1, b=2r=2, b=1r=2, b=2r=3, b=1r=3, b=2


0

50000

100000

150000

200000

250000

300000

0 50000 100000 150000 200000 250000 300000

f 2

f1

r=1r=2r=3

(a) N = 10, 000, b = 1

0

200000

400000

600000

800000

1e+06

1.2e+06

1.4e+06

0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06

f 2

f1

r=1r=2r=3

(b) N = 50, 000, b = 2






















Single-objectiveconstrained problem

Bi-objective

constrainedproblem

s

1 constraint 2 constraints





Example: 1,2

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

S1,4(x) = S1(x) + S4(x)

S1(x) = f

(1)(x� 1)� f

(1)(x)

1

f(1) f(2) f(3)

x1 x2 x3 x4

f(1) f(2) f(3)

x1 x2 x3 x4

f(1)

x1 x2

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

S1,4(x) = S1(x) + S4(x)

S1(x) = f

(1)(x� 1)� f

(1)(x)

S2(x) = f

(1)(x� 2)� f

(1)(x) + f

(2)(x� 2)� f

(2)(x) + f

(3)(x� 2)� f

(3)(x)

S1,2(x) = f

(1)(x�1, 2)�f

(1)(x)+f

(2)(x�1, 2)�f

(2)(x)+f

(3)(x�1, 2)�f

(3)(x)

S1,2(x) 6= S1(x) + S2(x)

1

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

S1,4(x) = S1(x) + S4(x)

S1(x) = f

(1)(x� 1)� f

(1)(x)

S2(x) = f

(1)(x� 2)� f

(1)(x) + f

(2)(x� 2)� f

(2)(x) + f

(3)(x� 2)� f

(3)(x)

S1,2(x) = f

(1)(x�1, 2)�f

(1)(x)+f

(2)(x�1, 2)�f

(2)(x)+f

(3)(x�1, 2)�f

(3)(x)

S1,2(x) 6= S1(x) + S2(x)

1

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

S1,4(x) = S1(x) + S4(x)

S1(x) = f

(1)(x� 1)� f

(1)(x)

S2(x) = f

(1)(x� 2)� f

(1)(x) + f

(2)(x� 2)� f

(2)(x) + f

(3)(x� 2)� f

(3)(x)

S1,2(x) = f

(1)(x�1, 2)�f

(1)(x)+f

(2)(x�1, 2)�f

(2)(x)+f

(3)(x�1, 2)�f

(3)(x)

S1,2(x) 6= S1(x) + S2(x)

1

x1 and x2“interact”






Decomposition rule for scores• When can we decompose a score as the sum of lower order scores?

• … when the variables in the move can be partitioned in subsets of variables that DON’T interact

• Let us define the Co-occurrence Graph

f(1) f(2) f(3) f(4)

x1 x2 x3 x4

There is an edge between two variables if there exists a function that depends

on both variables (they “interact”)

x4 x3

x1 x2






• Whitley and Chen proposed an O(1) approximated steepest descent for MAX-kSATand NK-landscapes based on Walsh decomposition

• For k-bounded pseudo-Boolean functions its complexity is O(k2 2k)

• Chen, Whitley, Hains and Howe reduced the time required to identify improving moves to O(k3) using partial derivatives

• Szeider proved that the exploration of a ball of radius r in MAX-kSAT and kSAT can be done in O(n) if each variable appears in a bounded number of clauses

Previous workBall of radius r Improving moves Previous work Research Question

D. Whitley and W. Chen. Constant time steepest descent local search withlookahead for NK-landscapes and MAX-kSAT. GECCO 2012: 1357–1364

W. Chen, D. Whitley, D. Hains, and A. Howe. Second order partial derivativesfor NK-landscapes. GECCO 2013: 503–510

S. Szeider. The parameterized complexity of k-flip local search for SAT and MAX SAT. Discrete Optimization, 8(1):139–145, 2011





Scores to store• In terms of the VIG a score can be decomposed if the subgraph containing the

variables in the move is NOT connected

• The number of these scores (up to radius r) is O((3kc)r n)

• Details of the proof in the paper

• With a linear amount of information we can explore a ball of radius r containing O(nr) solutions

x4 x3

x1 x2

x4 x3

x1 x2

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

S1,4(x) = S1(x) + S4(x)

S1(x) = f

(1)(x� 1)� f

(1)(x)

S2(x) = f

(1)(x� 2)� f

(1)(x) + f

(2)(x� 2)� f

(2)(x) + f

(3)(x� 2)� f

(3)(x)

S1,2(x) = f

(1)(x�1, 2)�f

(1)(x)+f

(2)(x�1, 2)�f

(2)(x)+f

(3)(x�1, 2)�f

(3)(x)

S1,2(x) 6= S1(x) + S2(x)

1

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

S1,4(x) = S1(x) + S4(x)

1

We need to store the scores of moves whose variables form a connected subgraph of the VIG






Scores to update• Let us assume that x4 is flipped

• Which scores do we need to update?

• Those that need to evaluate f(3) and f(4)

f(1) f(2) f(3) f(4)

x1 x2 x3 x4

x4 x3

x1 x2• The scores of moves containing variables adjacent

or equal to x4 in the VIG

Main idea Decomposition of scores Constant time update





Scores to update and time required• The number of neighbors of a variable in the VIG is bounded by c k

• The number of stored scores in which a variable appears is the number of spanning trees of size less than or equal to r with the variable at the root

• This number is constant

• The update of each score implies evaluating a constant number of functions that depend on at most k variables (constant), so it requires constant time

x4 x3

x1 x2

f(1) f(2) f(3) f(4)

x1 x2 x3 x4

O( b(k) (3kc)r |v| ) b(k) is a bound for the time toevaluate any subfunction

Main idea Decomposition of scores Constant time update





Results: checking the time in the random model• Random model: the number of subfunctions in which a variable appears, c, is not

bounded by a constant

NKq-landscapes• Random model• N=1,000 to 12,000• K=1 to 4• q=2K+1

• r=1 to 4• 30 instances per conf.

K=3

r=1r=2

r=3

0 2000 4000 6000 8000 10000 120000

50

100

150

200

n

TimeHsL

r=1

r=2

r=3

0 2000 4000 6000 8000 10000 120000

100000

200000

300000

400000

N

Scoresstoredinmemory

NKq-landscapes Sanity check Random model Next improvement





ScoresProblem Formulation Landscape Theory Decomposition SAT Transf. Results

f(1) f(2) f(3)

x1 x2 x3 x4

f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x)

x1 x2 x3 x4

f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x)

x1 x2 x3 x4





ScoresProblem Formulation Landscape Theory Decomposition SAT Transf. Results

f(1)

x1 x2

f(1) f(2) f(3)

x1 x2 x3 x4

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

1

r = 1 n

r = 2�n2

�

r = 3�n3

�

r

�nr

�

BallPr

i=1

�ni

�

S1(x) = f(x� 1)� f(x)


l=1

(f (l)(x� v)� f

(l)(x)) =mX

l=1

S

(l)(x)

S4(x) = f(x� 4)� f(x)

S1,4(x) = f(x� 1, 4)� f(x)

S1,4(x) = S1(x) + S4(x)

S1(x) = f

(1)(x� 1)� f

(1)(x)

1

efficient hill climber for constrained pseudo-boolean optimization problems

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