Time-Series Constraints: Improvements and Application in CP and MIP Contexts

A checker for a constraint on a variable sequence can often be compactly specified by an automaton, possibly with accumulators, that consumes the sequence of values taken by the variables; such an automaton can also be used to decompose its specified constraint into a conjunction of logical constraints. The inference achieved by this decomposition in a CP solver can be boosted by automatically generated implied constraints on the accumulators, provided the latter are updated in the automaton transitions by linear expressions. Automata with nonlinear accumulator updates can be automatically synthesised for a large family of time-series constraints. In this paper, we describe and evaluate extensions to those techniques. First, we improve the automaton synthesis to generate automata with fewer accumulators. Second, we decompose a constraint specified by an automaton with accumulators into a conjunction of linear inequalities, for use by a MIP solver. Third, we generalise the implied constraint generation to cover the entire family of time-series constraints. The newly synthesised automata for time-series constraints outperform the old ones, for both the CP and MIP decompositions, and the generated implied constraints boost the inference, again for both the CP and MIP decompositions. We evaluate CP and MIP solvers on a prototypical application modelled using time-series constraints. 1 Context and Motivation Frameworks are given in [4,14] for specifying a constraint on a sequence of variables in a high-level way by means of a finite automaton, possibly augmented with accumulators in the framework of [4]. An automaton can be seen as a checker for ground instances of the specified constraint. For example, in a nonogram puzzle, a row constrained to contain two stretches of black cells, of lengths 4 and 3 in this order, separated by at least one white cell but preceded and followed by any amounts of white cells, can be checked by an automaton equivalent to the regular expression w∗bw+bw∗, where the row is represented by a sequence of variables whose domain value ‘w’ stands for white and ‘b’ for black. Accumulators enable the specification of a constraint γ on a variable sequence X by an automaton whose size does not depend on the length of X: accumulators are initialised at the start state and are updated through the transitions; upon acceptance, the accumulators are linked to another variable of γ via an arithmetic constraint. For example, one could constrain the number of white cells between the two black stretches in the nonogram constraint above to be at most half the length of the row. The framework of [14] lifts an automaton without accumulators into a propagator for the specified constraint; it maintains domain consistency in polynomial time. The more general framework of [4] lifts an automaton, possibly with accumulators, into a decomposition of the specified constraint in terms of constraints with existing propagators; in the presence of accumulators, this decomposition does not maintain domain consistency in general [2]. Encoding the potential accumulator values in the states of the automaton may lead to an exponentially large automaton. In this paper, we focus on automata with accumulators. The propagation achieved by the automaton decomposition of [4] in a CP solver can be boosted by invariants, seen as implied constraints, on the accumulators. If the latter are updated in the automaton transitions by linear expressions on the accumulators — such as increments and decrements by constant amounts (as in c := c + 1) or by other accumulators (as in c := c + r), or resets (as in c := 0) — then such implied constraints can be automatically generated [11]. Automata with non-linear accumulator updates can be automatically synthesised for a large family of structural time-series constraints [3]. A time series is here a sequence of integers, corresponding to measurements taken over a time interval. Time series are common in many application areas, such as the power output of electric power stations over multiple days, or environmental data (temperature, humidity, CO2 level) in buildings. Time series are constrained by physical or organisational limits, which restrict the evolution of the series. After a summary of the background material in Section 2, the contributions and impact of this paper are as follows: – We improve the automated automaton synthesis of [3] so as to synthesise automata with fewer accumulators and simpler accumulator updates, using fewer ‘min’ and ‘max’ operators, say (Section 3). – We decompose a constraint specified by an automaton with accumulators into a linear-sized conjunction of linear inequalities, for use by a mixedinteger programming (MIP) solver (Section 4). – We generalise the implied constraint generation of [11] so as to cover the entire family of time-series constraints of [3] and to rank the generated implied constraints by decreasing propagation strength, thereby easing the human selection of which implied constraints actually to use (Section 5). – We show that the newly synthesised automata for time-series constraints outperform the automata of [3], for both the CP and MIP decompositions, and that the newly generated implied constraints boost the inference, again for both the CP and MIP decompositions (Section 6). – We evaluate CP and MIP solvers on a prototypical application modelled with the help of time-series constraints (Section 7). 2 Specifying (Time-Series) Constraints using Automata We showed in [3] that many constraints γ(N, 〈X0, . . . , Xn−1〉) on an unknown time series 〈X0, . . . , Xn−1〉 of given length n can be specified as a triple 〈p, f, g〉, where p is a regular expression over the alphabet {<,=, >} and is called the pattern; f ∈ {max, min, one, range, surface, width} is called the feature; and g ∈ {Max, Min, Sum} is called the aggregator. The semantics is that integer variable N is required to be the aggregation, computed using g, of the list of features f of all maximal words matching p within the sequence 〈S0, . . . , Sn−2〉 of variables, called the signature sequence, which is linked to the time series via the signature constraints (Xi < Xi+1 ⇔ Si = ‘<’) ∧ (Xi = Xi+1 ⇔ Si = ‘=’) ∧ (Xi > Xi+1 ⇔ Si = ‘>’) for all i ∈ [0, n − 2]. A list of 23 patterns was identified, giving 266 constraints. We now introduce our running example. Example 1. The MaxWidthStrictlyDecreasingSequence(N,X) constraint, requiring N to be the maximum width of the maximal strictly decreasing sequences within the time series X, is specified by the pattern >, the feature width, and the aggregator Max. The time series 〈4, 4, 3, 2, 2, 6, 3, 5〉 contains two maximal strictly decreasing sequences, namely 4 > 3 > 2 and 6 > 3, of widths 3 and 2, so their maximum width isN = 3. The following figure shows how to check MaxWidthStrictlyDecreasingSequence(3, 〈4, 4, 3, 2, 2, 6, 3, 5〉) by (I) building the signature sequence by comparing adjacent time-series values; (II) finding all maximal words matching the regular expression >; (III) computing the feature width of each such strictly decreasing sequence; and (IV) aggregating the feature values using the Max aggregator: