Computability and Complexity
Defining the Complexity Classes P and NP The complexity classes P and NP formalize the intuitive notion of “easy” and “hard” problems in computer science. The class P consists of decision problems that can be solved in polynomial time by a deterministic Turing machine. Specifically, a problem X is in P if there exists an…
Alternation hierarchies are classifications of computational problems based on their inherent reasoning complexity. They arrange problems into a hierarchy based on the number of alternations between existential and universal quantifiers needed to express them. At the lowest level are problems that can be expressed without any alternations, like satisfiability (SAT). Higher levels require more intricate…
The P vs. NP Problem The P vs. NP problem refers to the open question of whether or not the complexity classes P and NP are equal. P represents the set of problems that can be solved in polynomial time by a deterministic Turing machine. NP represents problems where solutions can be verified in polynomial…
The P vs NP Problem: A Central Question in Computational Complexity Theory The complexity classes P and NP are fundamental to the field of computational complexity theory. The class P consists of all decision problems that can be solved in polynomial time by a deterministic Turing machine. In contrast, NP encompasses all decision problems where…
The CLIQUE problem is a classic NP-hard computational task that involves finding fully connected subgraphs called cliques within a larger graph. Despite decades of research, pinpointing the precise thresholds at which CLIQUE transitions from tractable to intractable remains an open challenge. This analysis examines the problem definition, parameterized complexity analysis, tractability thresholds, instance examples, analysis…
The incompressibility method is a technique in computational complexity theory for proving lower bounds on the time complexity of computational problems. The key idea is to use randomly generated strings that are incompressible to force any algorithm to store or transmit a certain amount of information, which requires a minimum number of steps. The Concept…
Defining ETH, NEXP, and EXP The Exponential Time Hypothesis (ETH) is a conjecture in computational complexity theory that states that 3-SAT, the satisfiability problem for Boolean formulas in 3-conjunctive normal form where each clause has at most 3 literals, cannot be solved in subexponential time by a deterministic Turing machine. More formally, ETH claims that…
The Role of Randomness in Computational Complexity Randomness plays an intriguing role within the field of computational complexity theory. At the heart of this field lies the P vs NP problem – one of the central open questions in computer science and mathematics. This problem centers around the relationship between two complexity classes: P –…
Defining Counting Problems Counting problems involve determining the number of solutions for a particular computational problem. For example, considering the Boolean satisfiability problem (SAT), instead of asking whether a satisfiable assignment exists, a counting version asks “how many satisfying assignments are there?” This is the #SAT problem. More formally, counting problems take the decision version…
The Core Issue: Lack of Understanding Complexity Classes Below Polynomial Space A glaring gap in our comprehension of computational complexity theory manifests in the shortage of natural problems known to reside in sub-polynomial space classes beneath polynomial space (PSPACE). While space classes exceeding polynomial space host multifarious problems of practical relevance, few natural candidates seem…