For some computational problems (e.g., product configuration, planning, diagnosis, query answering, phylogeny reconstruction) computing a set of similar diverse solutions may be desirable for better decision-making. With this motivation, we studied several decision optimization versions of this problem in the context of Answer Set Programming (ASP), analyzed their computational complexity, and introduced offline online methods to compute similar diverse solutions of such computational problems with respect to a given distance function. All these methods rely on the idea of computing solutions to a problem by means of finding the answer sets for an ASP program that describes the problem. The offline methods compute all solutions in advance using the ASP formulation of the problem with an ASP solver, like Clasp, and then identify similar diverse solutions using clustering methods. The online methods compute similar diverse solutions following one of the three approaches: by reformulating the ASP representation of the problem to compute similar diverse solutions at once using an ASP solver; by computing similar diverse solutions iteratively (one after other) using an ASP solver; by modifying the search algorithm of an ASP solver to compute similar diverse solutions incrementally. We modified Clasp to implement the last online method and called it Clasp-NK. In the first two online methods, the given distance function is represented in ASP; in the last one it is implemented in C++. We showed the applicability and the effectiveness of these methods on reconstruction of similar diverse phylogenies for Indo-European languages, and on several planning problems in Blocks World. We observed that in terms of computational efficiency the last online method outperforms the others; also it allows us to compute similar diverse solutions when the distance function cannot be represented in ASP.
which are similar to @math -distant solution and @math -close solution . On the other hand, @cite_31 asks for a solution @math -close to a partial solution rather than a set of solutions. Also, the distance measure considered in these problems is computable in polynomial time. However, with @math containing a single solution and @math being computable in polynomial time, @math -distant solution (resp. @math -close solution ) becomes essentially the same as @math Distant (resp. @math Close ).
@cite_31
2132302072
We prove the following about the Nearest Lattice Vector Problem (in any l sub p norm), the Nearest Code-word Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NP-hard. 2. If for some spl epsiv >0 there exists a polynomial time algorithm that approximates the optimum within a factor of 2 sup log(0.5- spl epsiv ) sup n then NP is in quasi-polynomial deterministic time: NP spl sube DTIME(n sup poly(log sup n) ). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the l sub spl infin norm. Improving the factor 2 sup log(0.5- spl epsiv ) sup n to spl radic (dim) for either of the lattice problems would imply the hardness of the Shortest Vector Problem in l sub 2 norm; an old open problem. Our proofs use reductions from few-prover, one-round interactive proof systems, either directly, or through a set-cover problem. >