This paper identiies the complexity class of statically constructed partially deceptive problems by proving a turing reduction of 3-SAT problem to partially deceptive problem. This essentially proves that the class of partially deceptive problems is NP-turing complete and one may not be able to nd a polynomial complexity algorithm for solving deceptive problems unless P = NP. This paper also ad...

Randomized search heuristics have widely been applied to complex engineering problems as well as to problems from combinatorial optimization. We investigate the runtime behavior of randomized search heuristics and present runtime bounds for these heuristics on some well-known combinatorial optimization problems. Such analyses can help to understand better the working principle of these algorith...

Multi-criteria shortest path problems (MSPP) are called as NP-Hard. For MSPPs, a unique solution for optimizing all the criteria simultaneously will rarely exist in reality. Algorithmic and approximation schemes are available to solve these problems; however, the complexity of these approaches often prohibits their implementation on real-world applications. This paper describes the development ...

Much of modern cryptography, starting from public-key encryption and going beyond, is based on the hardness of structured (mostly algebraic) problems like factoring, discrete log, or finding short lattice vectors. While structure is perhaps what enables advanced applications, it also puts the hardness of these problems in question. In particular, this structure often puts them in low (and so ca...

An instance of Max CSP is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satisfied constraints. Max CSP captures many well-known problems (such as Max k-SAT and Max Cut) and is consequently NP-hard. Thus, it is natural to study how restrictions on the allowed constraint types (or constraint languages) affe...

An instance of the maximum constraint satisfaction problem (Max CSP) is a nite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satis ed constraints. Max CSP captures many well-known problems (such as Max k-SAT and Max Cut) and is consequently NP-hard. Thus, it is natural to study how restrictions on the allowed cons...

So far in this course, we have been proving upper bounds on the approximation factors achievable for certain NP -hard problems by giving approximation algorithms for them. In this lecture, we shift gears and prove lower bounds for some of these problems, under the assumption that P 6= NP . For some problems, essentially matching upper and lower bounds are known, indicating that the approximabil...

The original Extremal Optimization (EO) algorithm and its modified versions have been successfully applied to a variety of NP-hard optimization problems. However, there exists a problem that almost all existing EO-based algorithms have overlooked the inherent structural properties behind the optimization problems, e.g., the backbone information. This paper proposes a novel stochastic search met...

We prove that all of Karp’s 21 original NP-complete problems have a version that is hard to approximate. These versions are obtained from the original problems by adding essentially the same simple constraint. We further show that these problems are absurdly hard to approximate. In fact, no polynomial-time algorithm can even approximate log(k) of the magnitude of these problems to within any co...

We introduce and study the properties of Boolean autoencoder circuits. In particular, we show that the Boolean autoencoder circuit problem is equivalent to a clustering problem on the hypercube. We show that clustering m binary vectors on the n-dimensional hypercube into k clusters is NP-hard, as soon as the number of clusters scales like m (ε > 0), and thus the general Boolean autoencoder prob...