Related papers: Computations with polynomial evaluation oracle: ru…
This paper depicts algorithms for solving the decision Boolean Satisfiability Problem. An extreme problem is formulated to analyze the complexity of algorithms and the complexity for solving it. A novel and easy reformulation as a lottery…
We orchestrate a randomized algorithm for #$k$-SAT which counts the exact number of satisfying assignments in $2^{o(n)}$ time. The existence of such algorithm signifies that the #ETH is hereby refuted, and so are $\oplus$ETH, ETH, #SETH,…
A large number of NP-hard graph problems can be solved in $f(w)n^{O(1)}$ time and space when the input graph is provided together with a tree decomposition of width $w$, in many cases with a modest exponential dependence $f(w)$ on $w$.…
We revisit the satisfiability problem for two-variable logic, denoted by SAT(FO2), which is known to be NEXP-complete. The upper bound is usually derived from its well known Exponential Size Model (ESM) property. Whether it can be…
We investigate fine-grained algorithmic aspects of identification problems in graphs and set systems, with a focus on Locating-Dominating Set and Test Cover. We prove the (tight) conditional lower bounds for these problems when…
What is the minimum amount of information and time needed to solve 2SAT? When the instance is known, it can be solved in polynomial time, but is this also possible without knowing the instance? Bei, Chen and Zhang (STOC '13) considered a…
In this paper, we begin the exploration of vertex-ordering problems through the lens of exponential-time approximation algorithms. In particular, we ask the following question: Can we simultaneously beat the running times of the fastest…
We prove the first unconditional consistency result for superpolynomial circuit lower bounds with a relatively strong theory of bounded arithmetic. Namely, we show that the theory V$^0_2$ is consistent with the conjecture that NEXP…
We establish new hardness results for decision tree optimization problems, adding to a line of work that dates back to Hyafil and Rivest in 1976. We prove, under randomized ETH, superpolynomial lower bounds for two basic problems: given an…
The Subset Sum problem is a classical NP-complete problem with a long-standing $O^*(2^{n/2})$ deterministic bound due to Horowitz and Sahni. We present results at two distinct levels of generality. First (instance-sensitive bound), we…
We present a simple deterministic reduction which, assuming the Exponential Time Hypothesis ($\mathsf{ETH}$), yields tight lower bounds for approximating the parameterized Maximum Likelihood Decoding problem ($\mathsf{MLD}$) and the…
We show that any submodular minimization (SM) problem defined on a linear constraint set with constraints having up to two variables per inequality, are 2-approximable in polynomial time. If the constraints are monotone (the two variables…
Most state-of-the-art satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field…
We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called $k$-LCIS: Given $k$ integer sequences $X_1,\dots,X_k$ of length at most $n$, the task is to determine the…
The Boolean satisfiability problem (SAT) is a well-known example of monotonic reasoning, of intense practical interest due to fast solvers, complemented by rigorous fine-grained complexity results. However, for non-monotonic reasoning,…
We investigate the question whether Subset Sum can be solved by a polynomial-time algorithm with access to a certificate of length poly(k) where k is the maximal number of bits in an input number. In other words, can it be solved using only…
In recent years, significant progress has been made in explaining the apparent hardness of improving upon the naive solutions for many fundamental polynomially solvable problems. This progress has come in the form of conditional lower…
Vertex Subset Problems (VSPs) are a class of combinatorial optimization problems on graphs where the goal is to find a subset of vertices satisfying a predefined condition. Two prominent approaches for solving VSPs are dynamic programming…
This paper introduces new notions of asymptotic proofs, PT(polynomial-time)-extensions, PTM(polynomial-time Turing machine)-omega-consistency, etc. on formal theories of arithmetic including PA (Peano Arithmetic). This paper shows that P…
Here, we give an algorithm for deciding if the nonnegative rank of a matrix $M$ of dimension $m \times n$ is at most $r$ which runs in time $(nm)^{O(r^2)}$. This is the first exact algorithm that runs in time singly-exponential in $r$. This…