We show simple constant-round interactive proof systems for
problems capturing the approximability, to within a factor of $\sqrt{n}$,
of optimization problems in integer lattices; specifically,
the closest vector problem (CVP), and the shortest vector problem (SVP).
These interactive proofs are for the ``coNP direction'';
that is, ... more >>>

We study the question of whether the class DisNP of
disjoint pairs (A, B) of NP-sets contains a complete pair.
The question relates to the question of whether optimal
proof systems exist, and we relate it to the previously
studied question of whether there exists ... more >>>

We show that for each k > 0, MA/1 (MA with 1 bit of advice) does not have circuits of size n^k. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP_{||}^{NP}.

We extend our main result in several ways. For ... more >>>

Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by ... more >>>

A classic result due to Schaefer (1978) classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in $\mathsf{P}$ or NP-hard. This paper considers a promise-problem variant of CSPs called PCSPs. A PCSP over a finite set of pairs of constraints $\Gamma$ consists of a pair $(\Psi_P, ... more >>>

Promise CSPs are a relaxation of constraint satisfaction problems where the goal is to find an assignment satisfying a relaxed version of the constraints. Several well known problems can be cast as promise CSPs including approximate graph and hypergraph coloring, discrepancy minimization, and interesting variants of satisfiability. Similar to CSPs, ... more >>>

We study the complexity of approximation on satisfiable instances for graph homomorphism problems. For a fixed graph $H$, the $H$-colouring problem is to decide whether a given graph has a homomorphism to $H$. By a result of Hell and Nešet?il, this problem is NP-hard for any non-bipartite graph $H$. In ... more >>>

A $k$-uniform hypergraph is said to be $r$-rainbow colorable if there is an $r$-coloring of its vertices such that every hyperedge intersects all $r$ color classes. Given as input such a hypergraph, finding a $r$-rainbow coloring of it is NP-hard for all $k \ge 3$ and $r \ge 2$. ... more >>>

In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not ... more >>>

The approximate graph colouring problem concerns colouring a $k$-colourable
graph with $c$ colours, where $c\geq k$. This problem naturally generalises
to promise graph homomorphism and further to promise constraint satisfaction
problems. Complexity analysis of all these problems is notoriously difficult.
In this paper, we introduce ... more >>>

The Acceptance Probability Estimation Problem (APEP) is to additively approximate the acceptance probability of a Boolean circuit. This problem admits a probabilistic approximation scheme. A central question is whether we can design a pseudodeterministic approximation algorithm for this problem: a probabilistic polynomial-time algorithm that outputs a canonical approximation with high ... more >>>

The study of the complexity of the constraint satisfaction problem (CSP), centred around the Feder-Vardi Dichotomy Conjecture, has been very prominent in the last two decades. After a long concerted effort and many partial results, the Dichotomy Conjecture has been proved in 2017 independently by Bulatov and Zhuk.

At about ... more >>>

We propose a new definition of the class of search problems that correspond to BPP.
Specifically, a problem in this class is specified by a polynomial-time approximable function $q:\{0,1\}^*\times\{0,1\}^*\to[0,1]$ that associates, with each possible solution $y$ to an instance $x$, a quality $q(x,y)$.
Intuitively, quality 1 corresponds to perfectly ... more >>>