Optimization, approximation, and complexity classes

Extending and implementing the stable model semantics

Complexity and Approximation

http://compalg.inf.elte.hu/~tony/Kutatas/BinPacking/Johnson-BinPacking-NP-Column-JAlg-2002.pdf

The NP-completeness column: An ongoing guide

In order to deal with over-constrained Constraint Satisfaction Problems, various extensions of the CSP framework have been considered by taking into account costs, uncertainties, preferences, priorities...Each extension uses a specific mathematical operator (+, max...) to aggregate constraint violations.

In this paper, we consider a simple algebraic framework, related to Partial Constraint Satisfaction, which subsumes most of these proposals and use it to characterize existing proposals in terms of rationality and computational complexity. We exhibit simple relationships between these proposals, try to extend some traditional CSP algorithms and prove that some of these extensions may be computationally expensive.

https://ijcai.org/Proceedings/95-1/Papers/083.pdf

Valued constraint satisfaction problems: hard and easy problems

We consider NP-complete optimization problems at the level of computing their optimal value, and define a class of functions called OptP to capture this level of structure. We show that TRAVELING SALESPERSON and KNAPSACK are complete for OptP, and that CLIQUE and COLORING are complete for a subclass of OptP. These results show a deeper level of structure in these problems than was previously known. We also show that OptP is closely related to FPSAT, the class of functions computable in polynomial time with an oracle for NP. This allows us to quantify exactly “how much” NP-completeness is in these problems. In particular, in this measure, we show that TRAVELING SALESPERSON is strictly harder than CLIQUE and that CLIQUE is strictly harder than BIN PACKING . A further result is that an OptP-completeness result implies NP-, Dp-, and Δ2P-completeness results, thus tying these four classes closely together.

The complexity of optimization problems

We study computational complexity theory and define a class of optimization problems called OptP (Optimization Polynomial Time), and we show that TRAVELLING SALESPERSON, KNAPSACK and 0-1 INTEGER LINEAR PROGRAMMING are complete for OptP. OptP is a natural generalization of NP (Nondeterministic Polynomial Time), but while NP only considers problems at the level of their yes/no question, the value of an OptP function is the optimal value of the problem. This approach enables us to show a deeper level of structure in these problems than is possible in NP.
OptP is a subset of FPSAT, the class of functions computable in polynomial time with an oracle for NP. Our central result is that any FPSAT function decomposes into an OptP function followed by polynomial-time computation. The significance of this result is that it quantifies "how much" NP-completeness is in a problem, i.e., the number of NP queries it takes to compute the function. It also allows us to unify the classes NP, DP and DELTA-2 in a natural way. For example, we prove that an OptP-completeness result implies, as corollaries, NP-, DP- and DELTA-2- completeness results.
We also prove separation results on subclasses of FPSAT by restricting the number of calls to the NP oracle. For example, TRAVELLING SALESPERSON is complete for O(n) queries, CLIQUE is complete for O(log n) queries and BIN PACKING can be solved in O(log log n) queries. We prove these classes distinct under the assumption that P does not equal NP.
Finally, we consider generalizations of OptP to higher levels in the Polynomial-Time Hierarchy. We define the DOUBLE KNAPSACK problem and prove that it is complete for DELTA-3, the first example of a natural complete problem for this class, and the highest level in the Polynomial Hierarchy with a known natural complete problem.

/pdf/the-complexity-of-optimization-problems-3ixjjcrt2g.pdf

Mark W. Krentel

Papers

The complexity of optimization problems

The complexity of optimization problems

A Note on the Transaction Backout Problem