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Per Austrin

Publications and source records attributed to Per Austrin.

At least 19 recordsLinked to original sources
arXiv2025

On the Usefulness of Promises

A Boolean predicate $A$ is defined to be promise-useful if $\operatorname{PCSP}(A,B)$ 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 $\text{P} \ne \text{NP}$). 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.

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arXiv2024

Algorithms for the Diverse-k-SAT problem: the geometry of satisfying assignments

Given a $k$-CNF formula and an integer $s$, we study algorithms that obtain $s$ solutions to the formula that are maximally dispersed. For $s=2$, the problem of computing the diameter of a $k$-CNF formula was initiated by Creszenzi and Rossi, who showed strong hardness results even for $k=2$. Assuming SETH, the current best upper bound [Angelsmark and Thapper '04] goes to $4^n$ as $k \rightarrow \infty$. As our first result, we give exact algorithms for using the Fast Fourier Transform and clique-finding that run in $O^*(2^{(s-1)n})$ and $O^*(s^2 |\Omega_{F}|^{\omega \lceil s/3 \rceil})$ respectively, where $|\Omega_{F}|$ is the size of the solution space of the formula $F$ and $\omega$ is the matrix multiplication exponent. As our main result, we re-analyze the popular PPZ (Paturi, Pudlak, Zane '97) and Sch\"{o}ning's ('02) algorithms (which find one solution in time $O^*(2^{\varepsilon_{k}n})$ for $\varepsilon_{k} \approx 1-\Theta(1/k)$), and show that in the same time, they can be used to approximate the diameter as well as the dispersion ($s>2$) problems. While we need to modify Sch\"{o}ning's original algorithm, we show that the PPZ algorithm, without any modification, samples solutions in a geometric sense. We believe that this property may be of independent interest. Finally, we present algorithms to output approximately diverse, approximately optimal solutions to NP-complete optimization problems running in time $\text{poly}(s)O^*(2^{\varepsilon n})$ with $\varepsilon<1$ for several problems such as Minimum Hitting Set and Feedback Vertex Set. For these problems, all existing exact methods for finding optimal diverse solutions have a runtime with at least an exponential dependence on the number of solutions $s$. Our methods find bi-approximations with polynomial dependence on $s$.

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arXiv2023

Sum-of-Squares Lower Bounds for the Minimum Circuit Size Problem

We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$, SoS requires degree $\Omega(s^{1-\epsilon})$ to prove that $f$ does not have circuits of size $s$ (for any $s > \mathrm{poly}(n)$). As a corollary we obtain that there are no low degree SoS proofs of the statement NP $\not \subseteq $ P/poly. We also show that for any $0 < \alpha < 1$ there are Boolean functions with circuit complexity larger than $2^{n^{\alpha}}$ but SoS requires size $2^{2^{\Omega(n^{\alpha})}}$ to prove this. In addition we prove analogous results on the minimum \emph{monotone} circuit size for monotone Boolean slice functions. Our approach is quite general. Namely, we show that if a proof system $Q$ has strong enough constraint satisfaction problem lower bounds that only depend on good expansion of the constraint-variable incidence graph and, furthermore, $Q$ is expressive enough that variables can be substituted by local Boolean functions, then the MCSP problem is hard for $Q$.

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arXiv2022

Perfect Matching in Random Graphs is as Hard as Tseitin

We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $\Omega(n / \log n)$ in the Polynomial Calculus (over fields of characteristic $\ne 2$) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lov\'asz-Schrijver proof system requires $n^\delta$ rounds to refute these formulas for some $\delta > 0$. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.

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arXiv2019

Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder

Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within \approx 0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within \approx 0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (\approx 0.878 for Max-Cut, and \approx 0.940 for Max-2-Sat). The hardness obtained for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem.

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arXiv2019

Simplified inpproximability of hypergraph coloring via t-agreeing families

We reprove the results on the hardness of approximating hypergraph coloring using a different technique based on bounds on the size of extremal $t$-agreeing families of $[q]^n$. Specifically, using theorems of Frankl-Tokushige [FT99], Ahlswede-Khachatrian [AK98] and Frankl [F76] on the size of such families, we give simple and unified proofs of quasi NP-hardness of the following problems: $\bullet$ coloring a $3$ colorable $4$-uniform hypergraph with $(\log n)^\delta$ many colors $\bullet$ coloring a $3$ colorable $3$-uniform hypergraph with $\tilde{O}(\sqrt{\log \log n})$ many colors $\bullet$ coloring a $2$ colorable $6$-uniform hypergraph with $(\log n)^\delta$ many colors $\bullet$ coloring a $2$ colorable $4$-uniform hypergraph with $\tilde{O}(\sqrt{\log \log n})$ many colors where $n$ is the number of vertices of the hypergraph and $\delta>0$ is a universal constant.

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arXiv2018

Improved Inapproximability of Rainbow Coloring

A rainbow $q$-coloring of a $k$-uniform hypergraph is a $q$-coloring of the vertex set such that every hyperedge contains all $q$ colors. We prove that given a rainbow $(k - 2\lfloor \sqrt{k}\rfloor)$-colorable $k$-uniform hypergraph, it is NP-hard to find a normal $2$-coloring. Previously, this was only known for rainbow $\lfloor k/2 \rfloor$-colorable hypergraphs (Guruswami and Lee, SODA 2015). We also study a generalization which we call rainbow $(q, p)$-coloring, defined as a coloring using $q$ colors such that every hyperedge contains at least $p$ colors. We prove that given a rainbow $(k - \lfloor \sqrt{kc} \rfloor, k- \lfloor3\sqrt{kc} \rfloor)$-colorable $k$ uniform hypergraph, it is NP-hard to find a normal $c$-coloring for any $c = o(k)$. The proof of our second result relies on two combinatorial theorems. One of the theorems was proved by Sarkaria (J. Comb. Theory. 1990) using topological methods and the other theorem we prove using a generalized Borsuk-Ulam theorem.

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arXiv2017

Tensor network complexity of multilinear maps

We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low border rank tensor decompositions, such as $O(n^{\omega+\epsilon})$ time matrix multiplication, and in addition many other algorithms such as $O(n \log n)$ time discrete Fourier transform and $O^*(2^n)$ time for computing the permanent of a matrix. However tensor networks sometimes yield faster algorithms than those that follow from low-rank decompositions. For instance the fastest known $O(n^{(\omega +\epsilon)t})$ time algorithms for counting $3t$-cliques can be implemented with tensor networks, even though the underlying tensor has border rank $n^{3t}$ for all $t \ge 2$. For counting homomorphisms of a general pattern graph $P$ into a host graph on $n$ vertices we obtain an upper bound of $O(n^{(\omega+\epsilon)\operatorname{bw}(P)/2})$ where $\operatorname{bw}(P)$ is the branchwidth of $P$. This essentially matches the bound for counting cliques, and yields small improvements over previous algorithms for many choices of $P$. While powerful, the model still has limitations, and we are able to show a number of unconditional lower bounds for various multilinear maps, including: (a) an $\Omega(n^{\operatorname{bw}(P)})$ time lower bound for counting homomorphisms from $P$ to an $n$-vertex graph, matching the upper bound if $\omega = 2$. In particular for $P$ a $v$-clique this yields an $\Omega(n^{\lceil 2v/3 \rceil})$ time lower bound for counting $v$-cliques, and for $P$ a $k$-uniform $v$-hyperclique we obtain an $\Omega(n^v)$ time lower bound for $k \ge 3$, ruling out tensor networks as an approach to obtaining non-trivial algorithms for hyperclique counting and the Max-$3$-CSP problem. (b) an $\Omega(2^{0.918n})$ time lower bound for the permanent of an $n \times n$ matrix.

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arXiv2016

Sharper Upper Bounds for Unbalanced Uniquely Decodable Code Pairs

Two sets $A, B \subseteq \{0, 1\}^n$ form a Uniquely Decodable Code Pair (UDCP) if every pair $a \in A$, $b \in B$ yields a distinct sum $a+b$, where the addition is over $\mathbb{Z}^n$. We show that every UDCP $A, B$, with $|A| = 2^{(1-\epsilon)n}$ and $|B| = 2^{\beta n}$, satisfies $\beta \leq 0.4228 +\sqrt{\epsilon}$. For sufficiently small $\epsilon$, this bound significantly improves previous bounds by Urbanke and Li~[Information Theory Workshop '98] and Ordentlich and Shayevitz~[2014, arXiv:1412.8415], which upper bound $\beta$ by $0.4921$ and $0.4798$, respectively, as $\epsilon$ approaches $0$.

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arXiv2015

Dense Subset Sum may be the hardest

The Subset Sum problem asks whether a given set of $n$ positive integers contains a subset of elements that sum up to a given target $t$. It is an outstanding open question whether the $O^*(2^{n/2})$-time algorithm for Subset Sum by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a "truly faster", $O^*(2^{(0.5-\delta)n})$-time algorithm, with some constant $\delta > 0$. Continuing an earlier work [STACS 2015], we study Subset Sum parameterized by the maximum bin size $\beta$, defined as the largest number of subsets of the $n$ input integers that yield the same sum. For every $\epsilon > 0$ we give a truly faster algorithm for instances with $\beta \leq 2^{(0.5-\epsilon)n}$, as well as instances with $\beta \geq 2^{0.661n}$. Consequently, we also obtain a characterization in terms of the popular density parameter $n/\log_2 t$: if all instances of density at least $1.003$ admit a truly faster algorithm, then so does every instance. This goes against the current intuition that instances of density 1 are the hardest, and therefore is a step toward answering the open question in the affirmative. Our results stem from novel combinations of earlier algorithms for Subset Sum and a study of an extremal question in additive combinatorics connected to the problem of Uniquely Decodable Code Pairs in information theory.

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arXiv2013

On the NP-Hardness of Approximating Ordering Constraint Satisfaction Problems

We show improved NP-hardness of approximating Ordering Constraint Satisfaction Problems (OCSPs). For the two most well-studied OCSPs, Maximum Acyclic Subgraph and Maximum Betweenness, we prove inapproximability of $14/15+\epsilon$ and $1/2+\epsilon$. An OCSP is said to be approximation resistant if it is hard to approximate better than taking a uniformly random ordering. We prove that the Maximum Non-Betweenness Problem is approximation resistant and that there are width-$m$ approximation-resistant OCSPs accepting only a fraction $1 / (m/2)!$ of assignments. These results provide the first examples of approximation-resistant OCSPs subject only to P $\neq$ \NP.

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arXiv2013

Space--Time Tradeoffs for Subset Sum: An Improved Worst Case Algorithm

The technique of Schroeppel and Shamir (SICOMP, 1981) has long been the most efficient way to trade space against time for the SUBSET SUM problem. In the random-instance setting, however, improved tradeoffs exist. In particular, the recently discovered dissection method of Dinur et al. (CRYPTO 2012) yields a significantly improved space--time tradeoff curve for instances with strong randomness properties. Our main result is that these strong randomness assumptions can be removed, obtaining the same space--time tradeoffs in the worst case. We also show that for small space usage the dissection algorithm can be almost fully parallelized. Our strategy for dealing with arbitrary instances is to instead inject the randomness into the dissection process itself by working over a carefully selected but random composite modulus, and to introduce explicit space--time controls into the algorithm by means of a "bailout mechanism".

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arXiv2013

On the Power of Many One-Bit Provers

We study the class of languages, denoted by $\MIP[k, 1-\epsilon, s]$, which have $k$-prover games where each prover just sends a \emph{single} bit, with completeness $1-\epsilon$ and soundness error $s$. For the case that $k=1$ (i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson ({\em Computational Complexity'02}) demonstrate that $\SZK$ exactly characterizes languages having 1-bit proof systems with"non-trivial" soundness (i.e., $1/2 < s \leq 1-2\epsilon$). We demonstrate that for the case that $k\geq 2$, 1-bit $k$-prover games exhibit a significantly richer structure: + (Folklore) When $s \leq \frac{1}{2^k} - \epsilon$, $\MIP[k, 1-\epsilon, s] = \BPP$; + When $\frac{1}{2^k} + \epsilon \leq s < \frac{2}{2^k}-\epsilon$, $\MIP[k, 1-\epsilon, s] = \SZK$; + When $s \ge \frac{2}{2^k} + \epsilon$, $\AM \subseteq \MIP[k, 1-\epsilon, s]$; + For $s \le 0.62 k/2^k$ and sufficiently large $k$, $\MIP[k, 1-\epsilon, s] \subseteq \EXP$; + For $s \ge 2k/2^{k}$, $\MIP[k, 1, 1-\epsilon, s] = \NEXP$. As such, 1-bit $k$-prover games yield a natural "quantitative" approach to relating complexity classes such as $\BPP$,$\SZK$,$\AM$, $\EXP$, and $\NEXP$. We leave open the question of whether a more fine-grained hierarchy (between $\AM$ and $\NEXP$) can be established for the case when $s \geq \frac{2}{2^k} + \epsilon$.

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arXiv2013

A Characterization of Approximation Resistance for Even $k$-Partite CSPs

A constraint satisfaction problem (CSP) is said to be \emph{approximation resistant} if it is hard to approximate better than the trivial algorithm which picks a uniformly random assignment. Assuming the Unique Games Conjecture, we give a characterization of approximation resistance for $k$-partite CSPs defined by an even predicate.

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arXiv2012

New NP-hardness results for 3-Coloring and 2-to-1 Label Cover

We show that given a 3-colorable graph, it is NP-hard to find a 3-coloring with $(16/17 + \eps)$ of the edges bichromatic. In a related result, we show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to find a $(23/24 + \eps)$-satisfying assignment.

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arXiv2012

Better Balance by Being Biased: A 0.8776-Approximation for Max Bisection

Recently Raghavendra and Tan (SODA 2012) gave a 0.85-approximation algorithm for the Max Bisection problem. We improve their algorithm to a 0.8776-approximation. As Max Bisection is hard to approximate within $\alpha_{GW} + \epsilon \approx 0.8786$ under the Unique Games Conjecture (UGC), our algorithm is nearly optimal. We conjecture that Max Bisection is approximable within $\alpha_{GW}-\epsilon$, i.e., the bisection constraint (essentially) does not make Max Cut harder. We also obtain an optimal algorithm (assuming the UGC) for the analogous variant of Max 2-Sat. Our approximation ratio for this problem exactly matches the optimal approximation ratio for Max 2-Sat, i.e., $\alpha_{LLZ} + \epsilon \approx 0.9401$, showing that the bisection constraint does not make Max 2-Sat harder. This improves on a 0.93-approximation for this problem due to Raghavendra and Tan.

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arXiv2012

On the Usefulness of Predicates

Motivated by the pervasiveness of strong inapproximability results for Max-CSPs, we introduce a relaxed notion of an approximate solution of a Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly real-valued) constraints, and then only needs to satisfy as many of the new constraints as possible. To be more precise, we introduce the following notion of a predicate $P$ being \emph{useful} for a (real-valued) objective $Q$: given an almost satisfiable Max-$P$ instance, there is an algorithm that beats a random assignment on the corresponding Max-$Q$ instance applied to the same sets of literals. The standard notion of a nontrivial approximation algorithm for a Max-CSP with predicate $P$ is exactly the same as saying that $P$ is useful for $P$ itself. We say that $P$ is useless if it is not useful for any $Q$. This turns out to be equivalent to the following pseudo-randomness property: given an almost satisfiable instance of Max-$P$ it is hard to find an assignment such that the induced distribution on $k$-bit strings defined by the instance is not essentially uniform. Under the Unique Games Conjecture, we give a complete and simple characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is useless if and only if there is a pairwise independent distribution supported on the satisfying assignments of the predicate. It is natural to also consider the case when no negations are allowed in the CSP instance, and we derive a similar complete characterization (under the UGC) there as well. Finally, we also include some results and examples shedding additional light on the approximability of certain Max-CSPs.

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