arXiv subjects

Prabhu Manyem

Publications and source records attributed to Prabhu Manyem.

8 recordsLinked to original sources

arXiv2022

Clique and Vertex Cover are solvable in polynomial time if the input structure is ordered and contains a successor predicate

In this manuscript, assuming that Graedel's 1991 results are correct (which implies that bounds on the solution values for optimization problems can be expressed in existential second order logic where the first order part is universal Horn), I will show that Clique and Vertex Cover can be solved in polynomial time if the input structure is ordered and contains a successor predicate. In the last section, we will argue about the validity of Graedel's 1991 results. Update: Manuscript withdrawn, because results are incorrect. If phi = phi_1 AND phi_2, and phi is a Horn formula, it does NOT mean that both phi_1 and phi_2 are Horn formulae. Furthermore, the cardinality constraint CANNOT be expressed as a universal Horn sentence in ESO (NOT even when the structure is ordered).

cs.CC↗

arXiv2022

Maximum independent set (stable set) problem: Computational testing with binary search and convex programming using a bin packing approach

This paper deals with the maximum independent set (M.I.S.) problem, also known as the stable set problem. The basic mathematical programming model that captures this problem is an Integer Program (I.P.) with zero-one variables $x_j$ and only the \textit{edge inequalities} with an objective function value of the form $~\textstyle \sum_{j=1}^N x_j~$ where $N$ is the number of vertices in the input. We consider $LP(k)$, which is the Linear programming (LP) relaxation of the I.P. with an additional constraint $\textstyle \sum_{j=1}^N x_j = k ~~ (0 \le k \le N). ~~ $ We then consider a convex programming variant $CP(k)$ of $LP(k)$, which is the same as $LP(k)$, except that the objective function is a nonlinear convex function (which we minimise). $~$The M.I.S. problem can be solved by solving $CP(k)$ for every value of $k$ in the interval $~0 \le k \le N~$ where the convex function is minimised using a \it{bin packing} type of approach. In this paper, we present efforts to developing a convex function for $CP(k)$.. However, in the latest version, in the absence of a convex function, we have introduced a new function; and for a certain instance, when we provide partial solutions (that is, for 5 vertices out of 150), the frequency of hitting an optimal complete integer solution increases significantly.

cs.DS↗

arXiv2014

Decision versions of optimization problems: cardinality constraint (lower bound) as a CNF Horn formula for Maximum Matching

We provide a formula for the lower bound in the form of $|F| \ge K$, in such a way that the decision version of unweighted non-bipartite matching can be solved in polynomial time. ~The parameter $K$ can vary from instance to instance. We assume that the domains, the set of vertices and the set of edges, are ordered. To our knowledge, no polynomially solvable satisfiability expression has been developed for this problem so far, or for that matter, for any decision problem derived from optimization. Hence for such problems, this opens up a new approach to solving them.

cs.LO↗

arXiv2012

Expressibility at the machine level versus structure level: ESO universal Horn Logic and the class P

We show that ESO universal Horn logic (existential second logic where the first order part is a universal Horn formula) is insufficient to capture P, the class of problems decidable in polynomial time. This statement is true in the presence of a successor relation in the input vocabulary. We provide two proofs --- one based on reduced products of two structures, and another based on approximability theory (the second proof is under the assumption that P is not the same as NP). We show that the difference between the results here and those in Grädel (1991), is due to the fact that the expressions this paper deals with are at the "structure level", whereas the expressions in Grädel (1991) are at the "machine level" --- a case of Easier done than said.

cs.CC↗

arXiv2011

Duality Gap, Computational Complexity and NP Completeness: A Survey

We survey research that studies the connection between the computational complexity of optimization problems on the one hand, and the duality gap between the primal and dual optimization problems on the other. To our knowledge, this is the first survey that connects the two very important areas. We further look at a similar phenomenon in finite model theory relating to complexity and optimization.

math.OC↗

arXiv2011

Lower Bounds on Syntactic Logic Expressions for Optimization Problems and Duality using Lagrangian Dual to characterize optimality conditions

We show that simple syntactic expressions such as existential second order (ESO) universal Horn formulae can express NP-hard optimisation problems. There is a significant difference between the expressibilities of decision problems and optimisation problems. This is similar to the difference in computation times for the two classes of problems; for example, a 2SAT Horn formula can be satisfied in polynomial time, whereas the optimisation version in NP-hard. It is known that all polynomially solvable decision problems can be expressed as ESO universal ($Π_1$) Horn sentences in the presence of a successor relation. We show here that, on the other hand, if $P \neq NP$, optimisation problems defy such a characterisation, by demonstrating that even a $Π_0$ (quantifier free) Horn formula is unable to guarantee polynomial time solvability. Finally, by connecting concepts in optimisation duality with those in descriptive complexity, we will show a method by which optimisation problems can be solved by a single call to a "decision" Turing machine, as opposed to multiple calls using a classical binary search setting.

cs.LO↗

arXiv2010

Existential Second Order Logic Expression With Horn First Order for Maximum Clique (Decision Version)

We show that the maximum clique problem (decision version) can be expressed in existential second order (ESO) logic, where the first order part is a Horn formula in second-order quantified predicates. Without ordering, the first order part is $Π_2$ Horn; if ordering is used, then it is universal Horn (in which case, the second order variables can be determined in polynomial time). UPDATE: Manuscript withdrawn, because results are incorrect. If phi = phi_1 AND phi_2, and phi is a Horn formula, it does NOT mean that both phi_1 and phi_2 are Horn formulae. Furthermore, the cardinality constraint CANNOT be expressed as a universal Horn sentence in ESO (NOT even when the structure is ordered). Graedel's theorem is valid at a lower (machine) level, but probably NOT at a higher level.

cs.CC↗

arXiv2006

Syntactic Characterisations of Polynomial-Time Optimisation Classes (Syntactic Characterizations of Polynomial-Time Optimization Classes)

In Descriptive Complexity, there is a vast amount of literature on decision problems, and their classes such as \textbf{P, NP, L and NL}. ~ However, research on the descriptive complexity of optimisation problems has been limited. Optimisation problems corresponding to the \textbf{NP} class have been characterised in terms of logic expressions by Papadimitriou and Yannakakis, Panconesi and Ranjan, Kolaitis and Thakur, Khanna et al, and by Zimand. Grädel characterised the polynomial class \textbf{P} of decision problems. In this paper, we attempt to characterise the optimisation versions of \textbf{P} via expressions in second order logic, many of them using universal Horn formulae with successor relations. The polynomially bound versions of maximisation (maximization) and minimisation (minimization) problems are treated first, and then the maximisation problems in the "not necessarily polynomially bound" class.

cs.CC↗