Madhukar N. Thakur

Logical definability of NP optimization problems

We investigate here NP optimization problems from a logical definability standpoint. We show that the class of optimization problems whose optimum is definable using first-order formulae coincides with the class of polynomially bounded NP optimization problems on finite structures. After this, we analyze the relative expressive power of various classes of optimization problems that arise in this framework. Some of our results show that logical definiability has different implications for NP maximization problems than it has for NP minimization problems, in terms of both expressive power and approximation properties.

Approximation Properties of NP Minimization Classes

In this paper we study NP optimization problems from the perspective of descriptive complexity theory. We introduce a new approach to the logical definability of NP optimization problems by focusing on the ways in which the feasible solutions of the problems can be described using first-order formulae. We delineate the expressive power of this new framework and determine the relations between the various classes of optimization problems arising in this context. We also show that, assuming that P ? NP, it is an undecidable problem to tell if a given first-order formula defines an approximable NP optimization problem. After this, we investigate the impact that syntactic or structural properties of formulae have on the approximation properties of optimization problems defined by the formulae. Our main finding is that the approximation properties of many natural NP minimization problems are due to the fact that they are definable in the new framework using positive first-order formulae. In particular, we isolate a syntactically defined class of NP minimization problems that contains the MIN SET COVER problem and has the property that all its members have a polynomial-time O(log n)-approximation algorithm. Finally, we pursue results in a different direction and obtain a machine-independent characterization of the NP = ? co-NP problem in terms of logical definability of the MAX CLIOUE problem.

On the complexity of the containment problem for conjunctive queries with built-in predicates

When the inputs are conjunctive queries with #, 5, or < as built-in predicates, the query containment problem ‘is Qr 5 Qz?,, is I

Approximation properties of NP minimization classes

-complete and, thus, highly intractable. In this paper, we investigate the impact of syntactic and structural conditions on the computational complexity of tho query containment problem for safe conjunctive queries with discquation # as a built-in predicate. In the case of Np~r~‘, conjunctive queries (no built-in predicates), it is known that the boundary between polynomial-time solvability and NP-completeness is crossed, when the number of occurrcnccs of any database predicate in Qr increases from two to three, We show here that, as regards safe conjunctivc qucrics with disequations, the same syntactic condition dolincatcs the boundary between membership in coNP and II?

Scheduling real-time disk transfers for continuous media applications

completencss, Moreover, it is also known that the “pure” conjunctive query containment problem is solvable in polynomial time, if the hypergraph associated with the database predicates of Qz is acyclic. In contrast, we show that the vory samo structural condition does not lower the computational complexity of the containment problem for safe conjunctivc queries with disequations, that is, the problem remnins II~completc. We also analyze the computational complexity of the quary equivalence problem for conjunctive queries with disequations, when one of the two queries is fixed. We show that this problem can be DP-complete, where DP is the class of nil decision problems that are the conjunction of a problem in NP and a problem in coNP. It follows that, as regards conjunctive queries with disequations, the complexity of the query cquivalcncc problem may be higher than the complexity of the query containment problem, when one of the two qucrics is fixed.

Integer Programming as a Framework for Optimization and Approximability

The authors introduce a novel approach to the logical definability of NP optimization problems by focusing on the expressibility of feasible solutions. They show that in this framework first-order sentences capture exactly all polynomially bounded optimization problems. They also show that, assuming P not=NP, it is an undecidable problem to determine whether a given first-order sentence defines an approximable optimization problem. They then isolate a syntactically defined class of NP minimization problems that contains the min set cover problem and has the property that every problem in it has a logarithmic approximation algorithm. They conclude by giving a machine-independent characterization of the NP=co-NP problem in terms of logical expressibility of the max clique problem.<<ETX>>

Integer programming as a framework for optimization and approximability

The authors study how continuous media data can be stored and accessed in the Swift distributed input/output (IO) architecture. They provide a scheme for scheduling real-time data transfers that satisfies the strict requirements of continuous-media applications. This scheme allows large data objects to be stored and retrieved concurrently from multiple disks to satisfy the high data rate requirements typical of real-time video and audio data. To do this, data transfer requests are split into smaller requests, which are then handled by the various components by Swift. On-line algorithms are studied that respond to a data request by promising to either satisfy or reject it. Each response must be made before the next request is seen by the algorithm. The authors discuss two different performance measures to evaluate such algorithms and show that no on-line algorithm can optimize these criteria to less than a constant fraction of the optimal. Finally, they propose an algorithm for handling such requests on-line and the related data structures.<<ETX>>

Descriptive complexity of Hash P functions

Structural approximation theory seeks to provide a framework for expressing optimization problems and isolating structural or syntactic conditions that explain the apparent difference in the approximation properties of different NP-optimization problems. In this paper, we initiate a study of structural approximation using integer programming (an optimization problem in its own right) as a general framework for expressing optimization problems. We isolate three classes of constant-approximable maximization problems, based on restricting appropriately the syntactic form of the integer programs expressing them. The first of these classes subsumes Max?1, which is the syntactic version of the well-studied class MaxNP. Moreover, by allowing variables to take on not just 0/1 values but rather values in a polylogarithmic or polynomial range, we obtain syntactic maximization classes that are polylog-approximable and poly-approximable, respectively. The other two classes contain problems, such as MaxMatching, for which no previous structural explanation of approximability has been found. We also investigate structurally defined classes of integer programs for minimization problems and show a difference between their maximization counterparts.

Polynomial-time optimization, parallel approximation, and fixpoint logic

Structural approximation theory seeks to provide a framework for expressing optimization problems, and isolating structural or syntactic conditions that explain the (apparent) difference in the approximation properties of different NP-optimization problems. In this paper, we initiate a study of structural approximation using integer programming (an optimization problem in its own right) as a general framework for expressing optimization problems. We isolate three classes of constant-approximable maximization problems, based on restricting appropriately the syntactic form of the integer programs expressing them. The first of these classes subsumes MAX /spl Sigma//sub 1/, which is the syntactic version of the well-studied class MAX NP. Moreover, by allowing variables to take on not just 0/1 values but rather values in a polylogarithmic or polynomial range, we obtain syntactic maximization classes that are polylog-approximable and poly-approximable, respectively. The other two classes contain problems, such as MAX MATCHING, for which no previous structural explanation of approximability has been found.

Polynomial-time Optimization, Parallel Approximation, and Fixpoint Logic (Extended Abstract).

A logic-based framework for defining counting problems is given, and it is shown that it exactly captures the problems in Valiants counting class Hash P. The expressive power of the framework is studied under natural syntactic restrictions, and it is shown that some of the subclasses obtained in this way contain problems in Hash P with interesting computational properties. In particular, using syntactic conditions, a class of polynomial-time-computable Hash P problems is isolated, as well as a class in which every problem is approximable by a polynomial-time randomized algorithm. These results set the foundation for further study of the descriptive complexity of the class Hash P. In contrast, it is shown, under reasonable complexity theoretic assumptions, that it is an undecidable problem to tell if a counting problem expressed in the framework is polynomial-time computable or is approximable by a randomized polynomial-time algorithm. Some open problems are discussed.<<ETX>>