The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many Max-CSPs remain as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance. Examples of results obtained for this restricted setting are: (1) Optimal inapproximability for Max-3-Lin and Max-3-Sat (Håstad, J. ACM 2001). (2) Approximation resistance for predicates supporting pairwise independent subgroups (Chan, J. ACM 2016). (3) Hardness of the “(2 + ϵ)-Sat” problem and other Promise CSPs (Austrin et al., SIAM J. Comput. 2017). The main technical tool used to establish these results is a new way of folding the long code which we call “functional folding”.

A Boolean predicate A is defined to be promise-useful if PCSP(A,B) is tractable for some non-trivial B and otherwise it is promise-useless. We initiate investigations of this notion and derive sufficient conditions for both promise-usefulness and promise-uselessness (assuming PNP). While we do not obtain a complete characterization, our conditions are sufficient to classify all predicates of arity at most 4 and almost all predicates of arity 5. We also derive asymptotic results to show that for large arities a vast majority of all predicates are promise-useless. Our results are primarily obtained by a thorough study of the "Promise-SAT" problem, in which we are given a k-SAT instance with the promise that there is a satisfying assignment for which the literal values of each clause satisfy some additional constraint. The algorithmic results are based on the basic LP + affine IP algorithm of Brakensiek et al. (SICOMP, 2020) while we use a number of novel criteria to establish NP-hardness.

We study Frege proofs using depth-d Boolean formulas for the Tseitin contradiction on $n\times n$ grids. We prove that if each line in the proof is of size M then the number of lines is exponential in $n/(\log M)^{O(d)}$. This strengthens a recent result of Pitassi et al. [12]. The key technical step is a multi-switching lemma extending the switching lemma of Hastad [8] for a space of restrictions related to the Tseitin contradiction. The strengthened lemma also allows us to improve the lower bound for standard proof size of bounded depth Frege refutations from exponential in $\tilde{\Omega}(n^{1/59d})$ to exponential in $\tilde{\Omega}(n^{1/(2d-1)})$.

A linearly ordered (LO) k-colouring of a hypergraph assigns to each vertex a colour from the set {0, 1,..., k - 1} in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO k-colouring of an LO 2-colourable 3-uniform hypergraph for any constant k >= 2 [STACS'21] but even the case k = 3 is still open. Nakajima and Zivny gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with O *(root n) colours [ICALP'22] and an LO colouring with O*( 3 root n) colours [ACM ToCT'23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with O*( 5 root n) colours. We present two simple polynomial-time algorithms that find an LO colouring with O(log(2)(n)) colours, which is an exponential improvement.

We study Frege proofs for the one-to-one graph Pigeon Hole Principle defined on the $n \times n$ grid where n is odd. We are interested in the case where each formula in the proof is a depth d formula in the basis given by $\wedge, \vee$, and $\neg$. We prove that in this situation the proof needs to be of size exponential in $n^{\Omega(1 / d)}$. If we restrict the size of each line in the proof to be of size M then the number of lines needed is exponential in $n /(\log M)^{O(d)}$. The main technical component of the proofs is to design a new family of random restrictions and to prove the appropriate switching lemmas.

We prove a lower bound on the size of a small depth Frege refutation of the Tseitin contradiction on the grid. We conclude that polynomial size such refutations must use formulas of almost logarithmic depth.

We give an explicit construction of length- n binary codes capable of correcting the deletion of two bits that have size 2 n / n 4+ o (1) . This matches up to lower order terms the existential result, based on an inefficient greedy choice of codewords, that guarantees such codes of size Ω(2 n / n 4 ). Our construction is based on augmenting the classic Varshamov-Tenengolts construction of single deletion codes with additional check equations. We also give an explicit construction of binary codes of size Ω( 2 n / n 3+ o (1) ) that can be list decoded from two deletions using lists of size two. Previously, even the existence of such codes was not clear.

The Galvin problem asks for the minimum size of a family $\mathcal{F} \subseteq \binom {[n]} {n/2}$ with the property that, for any set $A$ of size $\frac n 2$, there is a set $S \in \mathcal{F}$ which is balanced on $A$, meaning that $|S \cap A| = |S \cap \overline{A}|$. We consider a generalization of this question that comes from a possible approach in complexity theory. In the generalization the required property is, for any $A$, to be able to find $d$ sets from a family $\mathcal{F} \subseteq \binom {[n]} {n/d}$ that form a partition of $[n]$ and such that each part is balanced on $A$. We construct such families of size polynomial in the parameters $n$ and $d$.

For a test T ⊆ {0, 1}n, define k*(T) to be the maximum k such that there exists a k-wise uniform distribution over {0, 1}n whose support is a subset of T. For Ht = {x ∈ {0, 1}n : | ∑ixi − n/2| ≤ t}, we prove k*(Ht) = Θ (t2/n + 1). For Sm, c = {x ∈ {0, 1}n : ∑ixi ≡ c (mod m)}, we prove that k*(Sm, c) = Θ (n/m2). For some k = O(n/m) we also show that any k-wise uniform distribution puts probability mass at most 1/m + 1/100 over Sm, c. Finally, for any fixed odd m we show that there is an integer k = (1 − Ω(1))n such that any k-wise uniform distribution lands in T with probability exponentially close to |Sm, c|/2n; and this result is false for any even m.

This is the Knuth Prize lecture. We discuss the approximability of Boolean Constraint Satisfaction Problems (CSPs). In this situation we are given a large number of constraints, each of the form of a fixed predicate P applied to a sequence of literals. The goal is to find an assignment that satisfies the maximum number of constrains.

We generalize the quantum algorithm for computing short discrete logarithms previously introduced by Ekerå [2] so as to allow for various tradeoffs between the number of times that the algorithm need be executed on the one hand, and the complexity of the algorithm and the requirements it imposes on the quantum computer on the other hand. Furthermore, we describe applications of algorithms for computing short discrete logarithms. In particular, we show how other important problems such as those of factoring RSA integers and of finding the order of groups under side information may be recast as short discrete logarithm problems. This gives rise to an algorithm for factoring RSA integers that is less complex than Shor’s general factoring algorithm in the sense that it imposes smaller requirements on the quantum computer. In both our algorithm and Shor’s algorithm, the main hurdle is to compute a modular exponentiation in superposition. When factoring an n bit integer, the exponent is of length 2n bits in Shor’s algorithm, compared to slightly more than n/2 bits in our algorithm.

We prove the following hardness result for a natural promise variant of the classical CNF-satisfiability problem: Given a CNF-formula where each clause has width w and the guarantee that there exis ...

We prove the following hardness result for anatural promise variant of the classical CNF-satisfiabilityproblem: Given a CNF-formula where each clause has widthw and the guarantee that there exists an assignment satisfyingat least g = [w/2] - 1 literals in each clause, it is NP-hard tofind a satisfying assignment to the formula (that sets at leastone literal to true in each clause). On the other hand, when g = [w/2], it is easy to find a satisfying assignment via simplegeneralizations of the algorithms for 2-SAT. Viewing 2-SAT ∈ P as easiness of SAT when 1-in-2 literals are true in every clause, and NP-hardness of 3-SAT as intractability of SAT when 1-in-3 literals are true, our resultshows, for any fixed ε > 0, the hardness of finding a satisfyingassignment to instances of "(2 + ε)-SAT" where the density ofsatisfied literals in each clause is promised to exceed 1/(2+ε). We also strengthen the results to prove that given a (2k + 1)-uniform hypergraph that can be 2-colored such that each edgehas perfect balance (at most k + 1 vertices of either color), itis NP-hard to find a 2-coloring that avoids a monochromaticedge. In other words, a set system with discrepancy 1 is hard todistinguish from a set system with worst possible discrepancy.

We consider codes over fixed alphabets against worst-case symbol deletions. For any fixed $k \ge 2$, we construct a family of codes over alphabet of size $k$ with positive rate, which allow efficient recovery from a worst-case deletion fraction approaching $1-\frac{2}{k+\sqrt k}$. In particular, for binary codes, we are able to recover a fraction of deletions approaching $1/(\sqrt 2 +1)=\sqrt 2-1 \approx 0.414$. Previously, even non-constructively the largest deletion fraction known to be correctable with positive rate was $1-\Theta(1/\sqrt{k})$, and around $0.17$ for the binary case. Our result pins down the largest fraction of correctable deletions for $k$-ary codes as $1-\Theta(1/k)$, since $1-1/k$ is an upper bound even for the simpler model of erasures where the locations of the missing symbols are known. Closing the gap between $(\sqrt 2 -1)$ and $1/2$ for the limit of worst-case deletions correctable by binary codes remains a tantalizing open question.

An instance of the 2-Lin(2) problem is a system of equations of the form xi + xj = b (mod 2). Given such a system in which it’s possible to satisfy all but an C e fraction of the equations, we sh ...

We consider systems of homogeneous linear equations modulo 2 with three variables in each equation and study balanced assignments as solutions to such equations. We prove that it is hard to distinguish systems where there is a balanced assignment that satisfies a fraction 1− ε of the equations from systems where the best balanced assignment satisfies a fraction 2 + ε of the equations assuming that NP is not contained in quasipolynomial time. This improves on a similar result by Holmerin and Khot who relied on the assumption that NP is not contained in subexponential time. The key for the improvement is to replace long codes used by Holmerin and Khot by the low-degree long code. ACM Classification: F.2.2, F.1.3 AMS Classification: 68Q17, 68Q25