Related papers: On the Complexity of Finding Narrow Proofs
We study optimization problems that are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. Specifically, we focus on Maximum Independent…
We systematically study the computational complexity of a broad class of computational problems in phylogenetic reconstruction. The class contains for example the rooted triple consistency problem, forbidden subtree problems, the quartet…
In the paper where he first defined Communication Complexity, Yao asks: \emph{Is computing $CC(f)$ (the 2-way communication complexity of a given function $f$) NP-complete?} The problem of deciding whether $CC(f) \le k$, when given the…
The 1-in-3 and Not-All-Equal satisfiability problems for Boolean CNF formulas are two well-known NP-hard problems. In contrast, the promise 1-in-3 vs. Not-All-Equal problem can be solved in polynomial time. In the present work, we…
We show that the CNF satisfiability problem (SAT) can be solved in time $O^*(1.1199^{(d-2)n})$, where $d$ is either the maximum number of occurrences of any variable or the average number of occurrences of all variables if no variable…
In this paper, we introduce a methodology, called decomposition-based reductions, for showing the equivalence among various problems of bounded-width. First, we show that the following are equivalent for any $\alpha > 0$: * SAT can be…
In this paper, we give a conditional lower bound of $n^{\Omega(k)}$ on running time for the classic k-median and k-means clustering objectives (where n is the size of the input), even in low-dimensional Euclidean space of dimension four,…
We first give an $\O(2^{n/3})$ quantum algorithm for the 0-1 Knapsack problem with $n$ variables. More generally, for 0-1 Integer Linear Programs with $n$ variables and $d$ inequalities we give an $\O(2^{n/3}n^d)$ quantum algorithm. For $d…
We present a general method for converting any family of unsatisfiable CNF formulas that is hard for one of the simplest proof systems, tree resolution, into formulas that require large rank in any proof system that manipulates polynomials…
We determine the exact threshold of satisfiability for random instances of a particular NP-complete constraint satisfaction problem (CSP). This is the first random CSP model for which we have determined a precise linear satisfiability…
Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The…
This dissertation proves lower bounds on the inherent difficulty of deciding flow analysis problems in higher-order programming languages. We give exact characterizations of the computational complexity of 0CFA, the $k$CFA hierarchy, and…
The Optimal Morse Matching (OMM) problem asks for a discrete gradient vector field on a simplicial complex that minimizes the number of critical simplices. It is NP-hard and has been studied extensively in heuristic, approximation, and…
We investigate the space complexity of refuting $3$-CNFs in Resolution and algebraic systems. No lower bound for refuting any family of $3$-CNFs was previously known for the total space in resolution or for the monomial space in algebraic…
In the context of proving lower bounds on proof space in k-DNF resolution, [Ben-Sasson and Nordstrom 2009] introduced the concept of minimally unsatisfiable sets of k-DNF formulas and proved that a minimally unsatisfiable k-DNF set with m…
Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such that every hyperedge includes one vertex from each part, the k-dimensional matching problem asks whether there is a disjoint collection of the hyperedges which…
We consider sets $\Gamma(n,s,k)$ of narrow clauses expressing that no definition of a size $s$ circuit with $n$ inputs is refutable in resolution R in $k$ steps. We show that every CNF shortly refutable in Extended R, ER, can be easily…
We exhibit families of $4$-CNF formulas over $n$ variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) $d$ but require SOS proofs of size $n^{\Omega(d)}$ for values of $d = d(n)$ from constant all the…
We analyse how the standard reductions between constraint satisfaction problems affect their proof complexity. We show that, for the most studied propositional, algebraic, and semi-algebraic proof systems, the classical constructions of…
The question of whether the complexity class P is equal to the complexity class NP has been a seemingly intractable problem for over 4 decades. It has been clear that if an algorithm existed that would solve the problems in the NP class in…