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Parameterized Complexity and Approximability of Coverability Problems in Weighted Petri Nets

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Application and Theory of Petri Nets and Concurrency (PETRI NETS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10258))

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Abstract

Many databases have been filled with the chemical reactions found in scientific publications and the associated information (efficiency, chemical products involved...). They can be used to define functions representing costs such as the ecoligical impact of the reactions. A major challenge is to use computer driven optimization in order to improve synthesis process and to provide algorithms to help determining a minimum cost pathway (series of reactions) for the synthesis of a molecule.

As, the classical Petri nets do not allows us to consider the optimization component, a weighted model has to be defined and the complexity of the associated problems studied. In this paper we introduce the weighted Petri nets in which each transition is associated with a weight. We define the Minimum Weight Synthesis Problem: find a minimum weight series of transitions to fire to produce a given target component. It mainly differ from classical coverability as it is an optimization problem.

We prove that this problem is EXPSPACE-Complete and that there is no polynomial approximation even when both in and outdegree are fixed to two and the target state is a single component. We also consider a more constraint version of the problem limiting the number of fired transitions. We prove this problem falls into PSPACE and the parametrized versions into XP but it remains not approximable.

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Acknowledgment

We thank the three anonymous reviewers for their constructive comments that helped to improve the quality of this paper and of future work.

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Correspondence to Dimitri Watel .

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A Proof of Theorem 12

A Proof of Theorem 12

Theorem 11. FS–MLWSP is W[1]-complete with respect to l and \(\pi \).

Proof

By Lemma 3 and Theorem 10, we have to prove that FS–MLWSP is W[1] with respect to l, \(\rho \), \(\pi \) and C. To do so, we describe a reduction to the partitioned clique problem. Let \(\mathcal {I} = ((\mathcal {P}, \mathcal {T}, \emptyset ), \omega , \mathcal {C}, l)\) be an instance of FS–MLWSP. We assume that \(\mathcal {I}\) contains a fake transition \(t_0\) such that \(t^-_0 = t^+_0 = \emptyset \) so that \(\mathcal {I}\) contains a feasible solution of size lower than l if and only if it contains a feasible solution of size exactly l. We now create an instance \(\mathcal {J} = (G, V_1, V_2, \dots , V_l, V_{l+1})\) of the partitioned clique problem such that \(\mathcal {J}\) contains a feasible solution if and only if \(\mathcal {I}\) does. An example is given in Fig. 7.

In the remaining of this proof, given a state M, the ordered set of states \(\{\alpha _1, \alpha _2, \dots , \alpha _k\}\) is called a decomposition of M if and only if \(\sum _{j = 1}^{k} \alpha _j = M\). The state \(\alpha _j\) may be empty.

Let t be a transition of \(\mathcal {I}\). For each \(i \in [\![2;l ]\!]\), we create at most \((i-1)^\rho \cdot (l-(i+1))^\pi \) nodes in \(V_i\), one per decomposition of \(t^-\) in \(i-1\) states and per decomposition of \(t^+\) in \(l-i+1\) states. If \(t^- = \emptyset \), we also create at most \(l^\pi \) nodes in \(V_1\), one per decomposition of \(t^+\) into l states. Finally, we add at most \(l^C\) nodes to \(V_{l+1}\), one for each decomposition of \(\mathcal {C}\) into l sets. For example, in Fig. 7, \(t_1^- = \emptyset \) and \(t_1^+ = x + 2y\). We add six nodes to \(V_1\) as there are six possible decompositions of \(t_i^+\) into 2 states : \(\{\emptyset , x + 2y\}, \{x, 2y\}, \{x+y, y\}\) and the symmetrical decompositions. Note that the number of nodes in G is FPT with respect to the parameters l, \(\rho \), \(\pi \) and C.

Let \(i < j \in [\![1; l ]\!]\), and \(v_i \in V_i\) and \(v_j \in V_j\) be two nodes of G. The node \(v_i\) is associated with a transition \(t_i\), a decomposition of \(t^-_i\) and a decomposition \(\{\alpha _{i+1}, \alpha _{i+2}, \dots , \alpha _{l+1}\}\) of \(t_i^+\). Similarly, the node \(v_j\) is associated with a transition \(t_j\), a decomposition \(\{\beta _{1}, \beta _{2}, \dots , \beta _{j-1}\}\) of \(t_j^-\) and a decomposition of \(t_j^+\). We add to G the edge \((v_i, v_j)\) if and only if \(\alpha _{j}\) covers \(\beta _{i}\).

Similarly, let \(i \in [\![1; l ]\!]\), and \(v_i \in V_i\) and \(v_{l+1} \in V_{l+1}\) be two nodes of G. The node \(v_i\) (respectively \(v_{l+1}\)) is associated with a transition \(t_i\), a decomposition of \(t^-_i\) and a decomposition \(\{\alpha _{i+1}, \alpha _{i+2}, \dots , \alpha _{l+1}\}\) of \(t_i^+\) (resp. a decomposition \(\{\beta _1, \beta _2, \dots , \beta _l\}\) of \(\mathcal {C}\)). We add to G the edge \((v_i, v_{l+1})\) if and only if \(\alpha _{l+1}\) covers \(\beta _i\).

Fig. 7.
figure 7

Example of reduction from an instance \(\mathcal {I}\) of FS–MLWSP with \(l=2\) on the left to an instance \(\mathcal {J}\) of Partitioned Clique on the right. Due to lack of space, the fake transition is not part of \(\mathcal {I}\). Each node of the clique instance is either a transition t with decompositions of \(t^-\) (on the left) and \(t^+\) (on the right) or an decomposition of C into 2 states. The sequence \((t_1,t_2)\) is a feasible solution of \(\mathcal {I}\). The bold clique on the right is a feasible solution of \(\mathcal {J}\) associated with that sequence.

For example, in Fig. 7, the last node of \(V_1\) and the second node of \(V_2\) are respectively associated with the decomposition \(\{x+2y, \emptyset \}\) of \(t_1^+\) and the decomposition \(\{x+y\}\) of \(t_2^-\). The two nodes are linked as \(x+2y\) covers \(x+y\).

We now prove that \(\mathcal {J}\) contains a feasible solution if and only if \(\mathcal {I}\) has an enabled sequence at \(\emptyset \) with l transitions such that the resulting state covers \(\mathcal {C}\).

We first assume there is a clique of G containing a node \(v_i\) of \(V_i\) for every \(i \in [\![1;l+1 ]\!]\). Let \(T = (t_1, t_2, \dots , t_l)\) be the sequence such that \(t_i\) is the transition associated with \(v_i\) for \(i \le l\). Let \(\{\beta _{i,1}, \beta _{i,2}, \dots , \beta _{i,i-1}\}\) and \(\{\alpha _{i,i+1}, \alpha _{i,i+2}, \dots , \alpha _{i,l+1}\}\) be respectively the decompositions of \(t_i^-\) and \(t_i^+\) associated with \(v_i\) and let \(\{\mathcal {C}_{1}, \mathcal {C}_{2}, \dots , \mathcal {C}_{l}\}\) be the decomposition of \(\mathcal {C}\) associated with \(v_{l+1}\).

We now show that T is enabled at \(\emptyset \). As \(v_1 \in V_1\), then \(t_1^- = \emptyset \) and \(t_1\) is enabled at \(\emptyset \). Let \(i \in [\![1;l-1 ]\!]\). As \((v_1, v_2 \dots v_i, v_{i+1})\) is a clique of G, then, by construction, \(\alpha _{j,i+1}\) covers \(\beta _{j,i+1}\) for every \(j \in [\![1;i ]\!]\). For each place x of \(\mathcal {P}\):

$$\begin{aligned} \sum \limits _{j = 1}^{i} (t_j^+(x) - t_j^-(x))&\ge \sum \limits _{j = 1}^{i} \left( \alpha _{j,i+1}(x) + \sum \limits _{k=j+1}^i \alpha _{j,k}(x) + \sum \limits _{k=1}^{j-1} \beta _{k,j}(x) \right) \\&\ge \sum \limits _{j = 1}^{i} \alpha _{j,i+1}(x) + \sum \limits _{j < k \le i} (\alpha _{j,k}(x) - \beta _{j,k})(x) \\&\ge \sum \limits _{j = 1}^{i} \alpha _{j,i+1}(x) = t_{i+1}^-(x) \end{aligned}$$

Consequently, for every \(i \in [\![1;l ]\!]\), \((t_1, t_2, \dots , t_i)\) is enabled at \(\emptyset \). Finally, we can similarly show that \(\sum _{j = 1}^{l} (t_j^+(x) - t_j^-(x)) \ge \sum _{j = 1}^{l} \alpha _{j,l+1}(x)\) and, as \((v_i, v_{l+1})\) is an edge of G for \(i \le l+1\), then, by construction, \(\sum _{j = 1}^{l} \alpha _{j,l+1}(x) \ge \sum _{j = 1}^{l} \mathcal {C}_j(x) = \mathcal {C}(x)\) for every place x. Thus, when firing T, the resulting state covers \(\mathcal {C}\) and T is a feasible solution of \(\mathcal {I}\).

We now assume there exists such a feasible solution \(T = (t_1, t_2, \dots , t_l)\) of \(\mathcal {I}\) and prove \(\mathcal {J}\) contains a clique of size \(l+1\). Let \(\gamma _{i,j}\), for \(i < j \in [\![1;l ]\!]\) be the states describing the tokens produced by \(t_i\) and consumed by \(t_j\) ; let \(\gamma _{i,l+1}\) be the tokens produced by \(t_i\), not consumed by any next transition and used to cover \(\mathcal {C}\) ; and finally let \(\gamma _{i,l+1}'\) be the not consumed tokens of \(t_i^+\) that are not used to cover \(\mathcal {C}\). Thus, we have firstly \(t^+_i = \gamma _{i,l+1}' + \sum _{j=i+1}^{l+1} \gamma _{i,j}\), secondly \(t^-_j = \sum \limits _{i=1}^{j-1} \gamma _{i,j}\) and thirdly \(\sum \limits _{i=1}^{l} \gamma _{i,l+1} = \mathcal {C}\). For each set \(V_i\) with \(i \le l\), we select the node \(v_i\) associated with \(t_i\), the decomposition \(\{\gamma _{j,i}, j \le i-1\}\) of \(t_i^-\) and the decomposition \(\{\gamma _{i,i+1}, \dots , \gamma _{i,l}, \gamma _{i,l+1} + \gamma _{i,l+1}'\}\) of \(t_i^+\). We finally choose the node \(v_{l+1}\) of \(V_{i+1}\) associated with the decomposition \(\{\gamma _{j,l+1}, j \le l \}\) of \(\mathcal {C}\). If \(i < j \le l\), then \(\gamma _{i,j}\) is the j-th state of the decomposition of \(t_i^+\) and the i-th state of the decomposition of \(t_j^-\). By construction the edge \((v_i, v_j)\) belongs to G. Moreover, if \(i \le l\), the last state \(\gamma _{i,l+1} + \gamma _{i,l+1}'\) of the decomposition of \(t_i^+\) covers the i-th state \(\gamma _{i,l+1}\) of the decomposition of \(\mathcal {C}\). By construction the edge \((v_i, v_{l+1})\) belongs to G. Consequently, \(\{v_1, v_2, \dots , v_{l+1}\}\) is a clique of size of \(l+1\) in \(\mathcal {J}\).

As a result and by Theorem 10, FS–MLWSP is W[1]-Complete in l and \(\pi \).    \(\Box \)

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Watel, D., Weisser, MA., Barth, D. (2017). Parameterized Complexity and Approximability of Coverability Problems in Weighted Petri Nets. In: van der Aalst, W., Best, E. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2017. Lecture Notes in Computer Science(), vol 10258. Springer, Cham. https://doi.org/10.1007/978-3-319-57861-3_19

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