Joshua Brakensiek

Joshua Brakensiek, Venkatesan Guruswami

The seminal work of Benczúr and Karger demonstrated cut sparsifiers of near-linear size. Subsequent extensions have yielded sparsifiers for hypergraph cuts and more recently linear codes over Abelian groups. A decade ago, Kogan and Krauthgamer asked about the sparsifiability of arbitrary constraint satisfaction problems (CSPs). For this question, a trivial lower bound is the size of a non-redundant CSP instance, which admits, for each constraint, an assignment satisfying only that constraint (so that no constraint can be dropped by the sparsifier). For instance, for graph cuts, spanning trees are non-redundant instances. Our main result is that redundant clauses are sufficient for sparsification: for any CSP predicate R, every unweighted instance of CSP(R) has a sparsifier of size at most its non-redundancy (up to polylog and $1/ε$ factors). For weighted instances, we similarly pin down the sparsifiability to the so-called chain length of the predicate. These results precisely determine the extent to which any CSP can be sparsified. Our result is established in the general setting of non-linear codes, or equivalently set families, yielding a VC-type theorem for multiplicative error approximation. A key technical ingredient in our work is a novel application of the entropy method from Gilmer's recent breakthrough on the union-closed sets conjecture. As an immediate consequence of our main theorem, a number of results in the non-redundancy literature immediately extend to CSP sparsification. We also contribute new techniques for understanding the non-redundancy of CSP predicates. By adapting methods from the matching vector codes literature in coding theory, we are able to construct an explicit predicate whose non-redundancy lies between $Ω(n^{1.5})$ and $\widetilde{O}(n^{1.6})$, the first example with a provably non-integral exponent.

Joshua Brakensiek, Yeyuan Chen, Manik Dhar et al.

Motivated by recent developments in coding theory, particular in list-decoding, we introduce a new error model which we call semi-adversarial errors. This error model bridges between fully random errors and fully adversarial errors by allowing some symbols of a message to be corrupted by an adversary while others are replaced with uniformly random symbols. As our main quest, we seek to understand optimal efficient unique decoding algorithms in the semi-adversarial model. For interleaved Reed--Solomon (IRS), folded Reed--Solomon (FRS) and univariate multiplicity codes, we design decoding algorithms running in near-linear time for most mixtures of random and adversarial errors. Our analysis matches the information-theoretic optimum for semi-adversarial errors. Our algorithm for interleaved Reed--Solomon codes is an improved implementation of the decoding algorithm by Bleichenbacher--Kiayias--Yung (BKY) for fully random errors. We use a novel monomial-tracking technique to analyze its performance in this new semi-adversarial errors. Inspired by the BKY algorithm, we use novel interpolations to extend our approach to the settings of folded Reed--Solomon and multiplicity codes, resulting in fast algorithms for unique decoding against semi-adversarial errors. Our new decoders for FRS and multiplicity codes replace the sophisticated root-finding step in traditional algorithms, such as the Guruswami--Wang algorithm, with a straightforward polynomial long division. Analysis of these algorithms requires more robust monomial-tracking arguments than IRS codes.

Joshua Brakensiek, Venkatesan Guruswami, Aaron Putterman

Given a constraint satisfaction problem (CSP) predicate $P \subseteq D^r$, the non-redundancy (NRD) of $P$ is maximum-sized instance on $n$ variables such that for every clause of the instance, there is an assignment which satisfies all but that clause. The study of NRD for various CSPs is an active area of research which combines ideas from extremal combinatorics, logic, lattice theory, and other techniques. Complete classifications are known in the cases $r=2$ and $(|D|=2, r=3)$. In this paper, we give a near-complete classification of the case $(|D|=2, r=4)$. Of the 400 distinct non-trivial Boolean predicates of arity 4, we implement an algorithmic procedure which perfectly classifies 397 of them. Of the remaining three, we solve two by reducing to extremal combinatorics problems -- leaving the last one as an open question. Along the way, we identify the first Boolean predicate whose non-redundancy asymptotics are non-polynomial.

Joshua Brakensiek, Neng Huang, Aaron Potechin et al.

The input to the Multiway Cut problem is a weighted undirected graph, with nonnegative edge weights, and $k$ designated terminals. The goal is to partition the vertices of the graph into $k$ parts, each containing exactly one of the terminals, such that the sum of weights of the edges connecting vertices in different parts of the partition is minimized. The problem is APX-hard for $k\ge3$. The currently best known approximation algorithm for the problem for arbitrary $k$, obtained by Sharma and Vondrák [STOC 2014] more than a decade ago, has an approximation ratio of 1.2965. We present an algorithm with an improved approximation ratio of 1.2787. Also, for small values of $k \ge 4$ we obtain the first improvements in 25 years over the currently best approximation ratios obtained by Karger et al. [STOC 1999]. (For $k=3$ an optimal approximation algorithm is known.) Our main technical contributions are new insights on rounding the LP relaxation of Călinescu, Karloff, and Rabani [STOC 1998], whose integrality ratio matches Multiway Cut's approximability ratio, assuming the Unique Games Conjecture [Manokaran et al., STOC 2008]. First, we introduce a generalized form of a rounding scheme suggested by Kleinberg and Tardos [FOCS 1999] and use it to replace the Exponential Clocks rounding scheme used by Buchbinder et al. [STOC 2013] and by Sharma and Vondrák. Second, while previous algorithms use a mixture of two, three, or four basic rounding schemes, each from a different family of rounding schemes, our algorithm uses a computationally-discovered mixture of hundreds of basic rounding schemes, each parametrized by a random variable with a distinct probability distribution, including in particular many different rounding schemes from the same family. We give a completely rigorous analysis of our improved algorithms using a combination of analytical techniques and interval arithmetic.

Joshua Brakensiek, Venkatesan Guruswami, Bart M. P. Jansen et al.

The non-redundancy (NRD) of a constraint satisfaction problem (CSP) is a combinatorial quantity closely tied to the behavior of CSPs in various computational models including their sparsification, kernelization, and streaming complexity. A primary open question in the study of non-redundancy is the identification of which CSP predicates have near-linear NRD. Recent works by Carbonnel [CP 2022], Khanna, Putterman and Sudan [STOC 2025], Brakensiek and Guruswami [STOC 2025] and Brakensiek, Guruswami, Jansen, Lagerkvist, and Wahlström [2025] have introduced various forms of gadget reductions between CSPs to relate their non-redundancy. The primary contribution of this work is to recontextualize many of these gadget reductions in a framework which we call hypergraph projections. By studying a quantity we call the shrinking factor of these hypergraph projections, we can more precisely predict when a gadget reduction between predicates can yield a super-linear NRD lower bound, greatly improving on the analysis of previous works. To illustrate the power of our framework, we identify some concrete CSP predicates whose non-redundancy is at the cusp of our understanding and show how our methods give lower bounds that could not have been achieved with these previous methods. We also demonstrate how these gadget reductions can be automatically deduced using SAT solvers, thereby opening up novel computational avenues for discovering further relationships between the non-redundancy of various CSPs.

Arpon Basu, Joshua Brakensiek, Aaron Putterman

We study the problem of Hamiltonian sparsification: given a parameter $\varepsilon \in (0,1)$ and an $n$-qubit Hamiltonian $H$ which is the sum of $r$-local positive semi-definite (PSD) terms $H_1, \dots H_m$, our goal is to compute a sparse set $L \subseteq [m]$, along with weights $w: L \rightarrow \mathbb{R}_{\geq 0}$ such that for every state $|ψ\rangle\in \mathbb{C}^{2^n}$, $$ \sum_{i \in L} w(i) \langle ψ| H_i | ψ\rangle \in (1 \pm ε) \sum_{i = 1}^m \langle ψ| H_i | ψ\rangle $$. When the set $L$ is significantly smaller than $m$, this reduces the number of terms in the underlying system, while still ensuring that the behavior of the system is essentially unchanged. We show that many Hamiltonians indeed are sparsifiable to a number of terms much smaller than $n^r$, including: (a) Hamiltonians where each term is an $r$-local Pauli string, (b) Hamiltonians where each term is an $r$-local random operator of rank $R$, for $R \geq 2^{r-1}+1$, and (c) Hamiltonians where each term is an arbitrary $r$-local operator of rank $\geq 2^r -1$ (a.k.a. Quantum SAT). Taken together, our results show that the sparsifiability of Hamiltonians is a robust phenomenon, contrary to prevailing belief (see for instance, Aharonov-Zhou ITCS 2019, QIP 2019). Our results find applications, for instance, to better (semi-)streaming algorithms for quantum Max-Cut, answering a question left open by Kallaugher and Parekh (FOCS 2022). In fact, our results even codify that quantum systems are often easier to sparsify than their classical counterparts.