Branch and bound with domination rules

We have designed an exact algorithm based on branch and bound method with dominance rules on partial solutions. It prunes nodes in the search tree (i.e. partial solutions) not only based on lower and upper bounds but also by reasoning about their structure.

For each partial solution it keeps track on the number of matching of each pattern and lower bound on its objective value. Based on the structure of patterns, one is able to reason

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It is the problem where we want to maximize the number of satisfied clauses in given 3-CNF propositional logical formula.

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whenever for given two partial solutions of the same length one dominates other (i.e. its objective cannot be worse than the other) or if they are incomparable. When one of the partial solutions is dominated by another one, we can prune whole subtree of that solution (see Figure 3.3). The efficiency of the algorithm depends on the power of the domination technique.

E

EE ED EN

EEE EED EEN EDE EDD EDN END ENN ENE

ENEE EENE

Figure 3.3: An illustrative example of the search tree with some dominated partial solutions. Consider a constraint where we want to have at least two E shifts in schedule. Partial schedule EDD is dominated by EEN, EDN is dominated by ENE.

However, since we are mostly dealing with soft constraints, design of such procedure is not trivial. We were able to develop following method.

Dominance rule Our dominance procedure compares two partial solutions of the same length. The domination is resolved based on the number of matches of each pattern. We divide patterns into two groups and patterns from each group are treated from two different perspectives — constraint on minimal and constraint on maximal matching. We frequently use the fact that number of pattern matches is non-decreasing. This is easy to see since adding a new shift to the end of the partial schedule cannot undo the pattern match done before, therefore it can only stay the same or increase.

N E − N · · · π1−: 0 π3−: 0 π3−: 0 π4−: 0 · · · π1E: +100 π2E: -100 π3E: +100 π4E: +100 · · · π1D: +100 π2D: -100 π3D: -100 π4D: +100 · · · π1N: -100 π2N: +100 π3N: +100 π4N: 0 · · · Constraints: a) Min 1 D, Max 3 D b) Max 1 NE

Figure 3.4: An illustrative example of resources. Partial solution NE-N is associated with resource vector [−200, 0, 1]. The first element is associated with the partially evaluated reduced cost, the other are resources associated with constraints a) and b). Dominance procedure compares vectors of corresponding partial solutions element by element.

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set Pi by a lower bound on its reduced cost. We refer to each of those numbers as a resource.

A resource in partial solution a is denoted #a (see example in Figure 3.4).

As we said, patterns are divided into two groups. The first group contains patterns with unit length and the second one contains patterns spanning across more days. We define dominance on separate resources as follows:

1. lower value of the lower bound on the reduced cost dominates the higher one 2. for patterns of unit length

(a) minimum matching:

i. if #a< lb and #b < lb then the solution with higher # dominates the other

ii. if #a< lb and #b ≥ lb then b dominates a

iii. if #a≥ lb and #b ≥ lb then equivalent

(b) maximum matching:

i. if #a≤ ub and #b ≤ ub then the solution with lower # dominates the other

ii. if #a≤ ub and #b > ub then a dominates b

iii. if #a> ub and #b > ub then the solution with lower # dominates the other

3. for longer patterns

Even though patterns of unit length are a special case of longer ones, we treat them separately due to following notion of future matching. We can reason about patterns with length greater than 1 based on the possibility they will match on current ending of the partial solution. Suppose a pattern of length k and denote ∀i ∈ {1 . . . k − 1} possibilities of overlapping the pattern with the ending of partial solution a as #Ma(i).

Then

(a) minimum matching:

i. if #a≥ lb and #b ≥ lb then equivalent

ii. else if #a< lb and #b ≥ lb then b dominates a

iii. else

A. if ∃i, j : #Ma(i) > #M_b(i) and #Ma(j) < #M_b(j) then a and b are incompa-

rable

B. if ∀i : #Ma(i) ≥ #M_b(i) and ∃j : #Ma(j)> #M_b(j) then a dominates b

C. else equivalent (b) maximum matching:

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ii. if ∀i : #Ma(i)≤ #M_b(i) and ∃j : #Ma(j)< #M_b(j) then a dominates b

iii. else equivalent 4. for periodical patterns

We denote a number of possible matching of the pattern (based on its starting day and structure) in the rest (non-allocated) of the schedule a as #Fa.

(a) minimum matching:

i. if #a< lb and #b < lb then the solution with higher # dominates the other

ii. if #a< lb and #b ≥ lb then b dominates a

iii. if #a≥ lb and #b ≥ lb then equivalent

(b) maximum matching:

i. if ub ≥ #a+ #Faand ub ≥ #b+ #Fb then

A. if #a ≤ ub and #b > ub then a dominates b

B. if #a > ub and #b ≤ ub then b dominates a

ii. if #a> ub and #b > ub then the solution with lower # dominates the other

iii. else a and b are incomparable

When the solutions are incomparable then they cannot dominate each other. However, when the procedure states equivalent it means, that both resource values do not contribute to the final resolution of the dominance procedure, therefore the particular resources can be omitted.

Finally, partial solution A dominates partial solution B if and only if A is not dominated on any resource by solution B and A dominates B on at least one resource. One can show, that dominance procedure defined as above always preserves at least one optimal solution.

Search procedure The search procedure of the algorithm works as described in (Burke and Curtois, 2014). Essentially, it is a breadth-first search with limited number of expanded solutions in a queue. The queue contains a Pareto optimal set of partial solutions of the same length. When the limit of queue size is exceeded at some depth, algorithm stores Pareto optimal partial solutions found so far and continues searching to the last depth level. Then, it updates upper bound with the best complete solution found so far, increases maximal queue size and starts again from the point where it overflowed last time. If no overflow happened, algorithm found at least one optimal solution at the last level.

The weakness of this dominance rule is that it is based on the notion of Pareto optimality (Censor, 1977). With the increasing number of resources (patterns) it is becoming more likely

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that solutions will be evaluated as incomparable since it is sufficient that a single resource in one solution dominates the resource in the other solution and for different resource vice versa. Thus, we observed, that this method is not suitable in practice even though similar method is claimed to be used in (Burke and Curtois, 2014).