The Shortest Common Parameterized Supersequence Problem

In this paper, we consider the problem of the shortest common parameterized supersequence. In particular, we consider an explicit reduction from the problem to the satisfiability problem.

Investigation of various problems on sequences of symbols forms a fundamental part of computer science (see e.g.[1] - [6]).One of the most important problems in analysis of sequences is the shortest common supersequence problem.There are a large number of different variants of this problem.In particular, we can mention the shortest common parameterized supersequence problem [7].In this paper, we consider an approach to solve the shortest common parameterized supersequence problem (SCPS).Note that encoding various hard problems as instances of the satisfiability problem and solving them with efficient satisfiability algorithms has caused considerable interest (see e.g.[8] - [23]).An explicit reduction from SCPS to the satisfiability problem was proposed in [7].In this paper, we consider an explicit reduction from SCPS to the 3-satisfiability problem and the maximum 2-satisfiability problem (MAX2SAT).
Note that where l > 4.
Using relations (1) -( 4) we can obtain an explicit transformation of function ξ from [7] in τ such that ξ ⇔ τ and τ is a 3-CNF.Clearly, τ give us an explicit reduction from SCPS to 3SAT.Now we consider an explicit reduction from SCPS to MAX2SAT.In the decision version MAX2SAT can be formulated as following.
MAX2SAT: Instance: Given a set U of variables, a collection C of clauses over U such that each clause C ∈ C has at most two literals, and a positive integer k.
Question: Is there a truth assignment for U that simultaneously satisfies at least k of the clauses in C?
Now, consider the following boolean function: It is easy to check that there is no way to satisfy all these clauses.Note that these ten clauses have the following interesting property: Any truth assignment that satisfies x∨y ∨z can be extended to satisfy seven of them and no more, while the remaining truth assignment can be extended to satisfy only six of them (see e.g.[24], proof of theorem 9.2.).
Using this property we can construct η by simply replacing in τ every occurrence of x ∨ y ∨ z where x, y, and z are some literals by (5) where w is a new variable.It is easy to see that τ ⇔ η.It is clear that η give us explicit reduction from SCPS to MAX2SAT.
There is a well known site on which posted solvers for SAT [25].In addition to the solvers the site also represented a large set of test problems.This set includes a randomly generated problems of 3SAT.We create a generator of natural instances for SCPS.Also we use test problems from [25].
We use algorithms fgrasp and posit from [25].Also we use a simple genetic algorithm (SGA) and design our own genetic algorithm (OA) for SAT which based on algorithms from [25].
Consider a boolean function g(x 1 , x 2 , . . ., x n ) = ∧ m i=1 C i , where m ≥ 1, and each of the C i is the disjunction of one or more literals.Let |C i | be a number of literals in C i .Let occ(x i , g) be a number of occurrences of x i in g.Respectively, let occ(¬x i , g) be a number of occurrences of x i in g.For example, if We consider a number of natural principles that define importance of a variable x i for satisfiability of boolean function g.These principles suggest us correct values of variables.

If min
4. Given positive integers p 1 , p 2 , . . ., p m , q 1 , q 2 , . . ., q m .If 5. Given positive integers p 1 , p 2 , . . ., p m , q 1 , q 2 , . . ., q m and a set of rational numbers Based on these principles, we can consider the following five types of commands: P 1 , P 2 , P 3 [p 1 , p 2 , . . ., p m ], P 4 [p 1 , p 2 , . . ., p m , q 1 , q 2 , . . ., q m ], Also we consider the following three commands for run algorithms: Try fgrasp, Try posit, and Try ga, where Try ga runs a simple genetic algorithm.Denote by R the set of commands of these eight types.Arbitrary element of R * it is possible to consider as a program for finding values of variables of a boolean function.We assume that such programs are executed on a cluster.Execution of each of commands of type P i reduces the number of variables of a boolean function by one.Execution of each of commands Try fgrasp, Try posit, and Try ga consists in the run of corresponding algorithm for current boolean function on a separate set of calculation nodes and the transition to the next command.Algorithms fgrasp [25] and posit [25] we run only on one calculation node.Genetic algorithms can be used in parallel execution.We use auxiliary genetic algorithms which determine the number of calculation nodes and, if necessary, scale commands P 3 , P 4 , P 5 .
Initially, we selected a random subset of R * .We use a genetic algorithm to select a program from the current subset of R * and a genetic algorithm for evolving the current subset of R * .The evolution of the current subset of R * implemented on a separate set of calculation nodes.For every subsequent boolean functions it is used the current subset of R * which is obtained by taking into account the results of previous runs.
We use heterogeneous cluster based on three clusters (Cluster USU, Linux, 8 calculation nodes; umt, Linux, 256 calculation nodes; um64, Linux, 124 calculation nodes) [26].Each test was run on a cluster of at least 100 nodes.
Algorithms fgrasp and posit used only for 3SAT.Respectively, for SAT and MAX2SAT we use only simple genetic algorithm (SGA), and our algorithm (OA).Selected experimental results for 3SAT, SAT, and MAX2SAT are given in Tables 1 and 2 It is interesting to note that the performance of OA depends essentially on the number of letters from Π (see e.g.Table 3 and 4).
In experiments presented in Tables 1 -4 OA used after 10 000 runs.The dependence of the performance of OA on the number of runs is shown in Table 5.
In this paper we describe an approach to solve the problem SCPS.This approach is based on constructing logical models for SCPS.Also we consider

Table 2 :
Experimental results for SAT and MAX2SAT

Table 3 :
Experimental results for 3SAT runs of OA depending on the number of letters from Π some genetic and local search algorithms for the problem.Our experimental results showed that our reduction to 3SAT with SAT-solver can be used as a test bed for intelligent algorithms for SCPS.The main question that is of interest for future research is to study the dependence of complexity of solution on the distribution of letters from Π.ACKNOWLEDGEMENTS.The work was partially supported by Analytical Departmental Program "Developing the scientific potential of high school", RFBR, research project No. 13-01-00048 a, and Ural Federal University development program with the financial support of young scientists.

Table 4 :
Experimental results for SAT runs of OA depending on the number of letters from Π

Table 5 :
Experimental results for 3SAT runs of OA depending on the number of runs