We introduce the Nondeterministic Strong Exponential Time Hypothesis
(NSETH) as a natural extension of the Strong Exponential Time
Hypothesis (SETH). We show that both refuting and proving
NSETH would have interesting consequences.

In particular we show that disproving NSETH would ... more >>>

For every polynomial $q$, we present worst-case to average-case (almost-linear-time) reductions for a class of problems in $\cal P$ that are widely conjectured not to be solvable in time $q$.
These classes contain, for example, the problems of counting the number of $k$-cliques in a graph, for any fixed $k\geq3$.
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We show that popular hardness conjectures about problems from the field of fine-grained complexity theory imply structural results for resource-based complexity classes. Namely, we show that if either k-Orthogonal Vectors or k-CLIQUE requires $n^{\epsilon k}$ time, for some constant $\epsilon > 1/2$, to count (note that these conjectures are significantly ... more >>>

Given a set of $n$ points in $\mathbb R^d$, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the $\ell_p$-metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its fine-grained complexity in the Euclidean metric when ... more >>>

Under the Strong Exponential Time Hypothesis, an integer linear program with $n$ Boolean-valued variables and $m$ equations cannot be solved in $c^n$ time for any constant $c < 2$. If the domain of the variables is relaxed to $[0,1]$, the associated linear program can of course be solved in polynomial ... more >>>

In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products.

We say that an optimization problem $\Pi$ is direct product feasible if it is possible to efficiently aggregate any $k$ instances of $\Pi$ and form one large instance ... more >>>

In [20] Goldwasser, Grossman and Holden introduced pseudo-deterministic interactive proofs for search problems where a powerful prover can convince a probabilistic polynomial time verifier that a solution to a search problem is canonical. They studied search problems for which polynomial time algorithms are not known and for which many solutions ... more >>>

We consider the problem of counting the number of copies of a fixed graph $H$ within an input graph $G$. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input $G$ has {\em bounded degeneracy}. ... more >>>

In this paper, we show how one may (efficiently) construct two types of extremal combinatorial objects whose existence was previously conjectural.

(*) Panchromatic Graphs: For fixed integer k, a k-panchromatic graph is, roughly speaking, a balanced bipartite graph with one partition class equipartitioned into k colour classes in ... more >>>

Statistics of small subgraph counts such as triangles, four-cycles, and $s$-$t$ paths of short lengths reveal important structural properties of the underlying graph. These problems have been widely studied in social network analysis. In most relevant applications, the graphs are not only massive but also change dynamically over time. Most ... more >>>

In a Merlin-Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability $1$, and rejects invalid proofs with probability arbitrarily close to $1$. The running time of such a system is defined to be the length of Merlin's proof plus the running time of Arthur. We ... more >>>

Computational problems on point lattices play a central role in many areas of computer science including integer programming, coding theory, cryptanalysis, and especially the design of secure cryptosystems. In this survey, we present known results and open questions related to the complexity of the most important of these problems, the ... more >>>

Many results in fine-grained complexity reveal intriguing consequences from solving various SAT problems even slightly faster than exhaustive search. We prove a ``self-improving'' (or ``bootstrapping'') theorem for Circuit-SAT, $\#$Circuit-SAT, and its fully-quantified version: solving one of these problems faster for ``large'' circuit sizes implies a significant speed-up for ``smaller'' circuit ... more >>>

A compression problem is defined with respect to an efficient encoding function $f$; given a string $x$, our task is to find the shortest $y$ such that $f(y) = x$. The obvious brute-force algorithm for solving this compression task on $n$-bit strings runs in time $O(2^{\ell} \cdot t(n))$, where $\ell$ ... more >>>

A line of work has shown how nontrivial uniform algorithms for analyzing circuits can be used to derive non-uniform circuit lower bounds. We show how the non-existence of nontrivial circuit-analysis algorithms can also imply non-uniform circuit lower bounds. Our connections yield new win-win circuit lower bounds, and suggest a potential ... more >>>