Johan Håstad’s research while affiliated with KTH Royal Institute of Technology and other places

What is this page?

This page lists works of an author who doesn't have a ResearchGate profile or hasn't added the works to their profile yet. It is automatically generated from public (personal) data to further our legitimate goal of comprehensive and accurate scientific recordkeeping. If you are this author and want this page removed, please let us know.

Publications (158)

Optimal Inapproximability with Universal Factor Graphs
  • Article

December 2023

·

11 Reads

·

4 Citations

ACM Transactions on Algorithms

Per Austrin

·

Jonah Brown-Cohen

·

Johan Håstad

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”.

On bounded depth proofs for Tseitin formulas on the grid; revisited
  • Preprint
  • File available

September 2022

·

30 Reads

Johan Håstad

·

Kilian Risse

We study Frege proofs using depth-d Boolean formulas for the Tseitin contradiction on n×nn \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)n/(\log M)^{O(d)}. This strengthens a recent result of Pitassi et al. [PRT22]. The key technical step is a multi-switching lemma extending the switching lemma of H\r{a}stad [H\r{a}s20] 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 Ω~(n1/59d)\tilde \Omega (n^{1/59d}) to exponential in Ω~(n1/(2d−1))\tilde \Omega (n^{1/(2d-1)}).

Download
Explicit Two-Deletion Codes With Redundancy Matching the Existential Bound

March 2021

·

39 Reads

·

53 Citations

IEEE Transactions on Information Theory

·

Johan Hastad

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.

On Small-depth Frege Proofs for Tseitin for Grids

December 2020

·

3 Reads

·

10 Citations

Journal of the ACM

Johan Håstad

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

Explicit two-deletion codes with redundancy matching the existential bound

July 2020

·

25 Reads

·

Johan Håstad

We give an explicit construction of length-n binary codes capable of correcting the deletion of two bits that have size 2n/n4+o(1)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(2n/n4)\Omega(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(2n/n3+o(1))\Omega(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.

d-Galvin Families

February 2020

·

11 Reads

The Electronic Journal of Combinatorics

Johan Håstad

·

Guillaume Lagarde

·

Joseph Swernofsky

The Galvin problem asks for the minimum size of a family F⊆([n]n/2)\mathcal{F} \subseteq \binom {[n]} {n/2} with the property that, for any set A of size n2\frac n 2, there is a set S∈FS \in \mathcal{F} which is balanced on A, meaning that ∣S∩A∣=∣S∩A‾∣|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 F⊆([n]n/d)\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.

Citations (82)

... Recently, Dantchev, Galesi, Ghani, and Martin [Dan+24] exhibited a 2 Ω(n/ log n) lower bound on the size of any general resolution refutation of BPHP m n for all m > n. In fact, they showed that BPHP m n requires proofs of size 2 Ω(n 1−ε ) for a more powerful class of proof systems that extend resolution by operating on k-DNFs (known as Res(k) proofs) for k ≤ log 1/2−ε ′ n. (Note that any sound proof system operating on DNFs requires size at least 2 n Ω(1) to refute of PHP n+1 n [PBI93; KPW95;Hås23].) In addition, [Dan+24] showed that BPHP m n has no refutations in the Sherali-Adams proof system [SA90] of size smaller than 2 Ω(n/ log 2 n) . Finally, just as PHP m n has polynomial-size Sum-of-Squares refutations [GHP01], Dantchev et al. showed that BPHP m n has polynomial-sized Sum-of-Squares refutations. ...

Reference:

Multiparty Communication Complexity of Collision Finding
On small-depth Frege proofs for PHP
  • Citing Conference Paper

  • November 2023

Johan Håstad

... According to existing literature, there are essentially three main approaches for constructing low-redundancy codes capable of correcting deletion-related errors. The first approach employs Varshamov-Tenengolts (VT) syndromes [4], [10], [13]- [16], [20]- [22], [26]. This method is particularly attractive due to its support for linear-time encoding, making it highly efficient for practical applications. ...

Reference:

Correcting Errors Through Partitioning and Burst-Deletion Correction
Explicit Two-Deletion Codes With Redundancy Matching the Existential Bound
  • Citing Article

  • March 2021

IEEE Transactions on Information Theory

·

Johan Hastad

... It has been a long-standing difficult problem to deal efficiently with synchronization errors, i.e., insertion and deletion eroros, see [56,57,59,79,77,58,76,51,28,73,74,4,35]. The insertion-deletion codes correcting synchronization errors have wide applications in racetrack memory errorcorrections, language processing, data analysis and DNA storage, see [68,8,6,82,49,55,16,7]. ...

Reference:

Explicit Good Subspace-metric and Subset-metric Codes as Insertion-deletion Codes
Explicit two-deletion codes with redundancy matching the existential bound
  • Citing Chapter

  • January 2021

·

Johan Håstad

... Assuming the Unique Games Conjecture, there is an NP-hardness result for Boolean unique games that matches the performance of the Goemans-Williamson algorithm [20]. The best NP-hardness we can currently prove (without assumptions) falls short of this, but the case of Boolean alphabet is where the community has come the furthest towards proving hardness of unique games for completeness 1 − δ for small δ > 0: there is NP-hardness for a very narrow gap 1 − δ vs. 1 − 2δ [17,23], and there is an approach to prove NP-hardness for a much wider gap 1 − δ vs. 1 − C · δ for any constant C ≥ 1 and sufficiently small δ > 0 [24,12]. A natural question is: ...

Reference:

Strong Parallel Repetition for Unique Games on Small Set Expanders
Improved NP-inapproximability for 2-variable linear equations
  • Citing Article

  • December 2017

Theory of Computing

J. Håstad

·

Sangxia Huang

·

Rajsekar Manokaran

·

[...]

·

John Wright

... In a very recent major breakthrough [35] showed that Tseitin formulas over a 3-expander graph on n nodes requires super-polynomial bounded-depth Frege refutations at depth O( √ log n). This result was later improved to depth up to C log n log log n by Håstad in [28] but for Tseitin formulas defined only on the 2-dimensional grid, where C is a positive constant. ...

Reference:

Bounded-depth Frege complexity of Tseitin formulas for all graphs
On Small-Depth Frege Proofs for Tseitin for Grids
  • Citing Conference Paper

  • October 2017

Johan Hastad

... A seminal result of Håstad [14] gives optimal inapproximability results for many CSPs including Max-k-SAT, Max-k-LIN over Abelian groups, Set Splitting, etc. Once we understand the opitmal worst-case complexity of a Max-CSP, it is interesting to understand how the complexity of the problem changes under certain restrictions on the instances. One such example of restrictions is the study of promise CSPs [4,7] in which it is guaranteed that a richer solution exists (e.g., a given graph has a proper 3-coloring) and the goal is to find or even approximate a weaker solution (e.g., find a coloring with 10 colors that maximizes the number of non-monochromatic edges). Another example is the study of complexity of Max-CSP when the instance is generated using a certain random process [11,16]. ...

Reference:

Max-3-Lin Over Non-abelian Groups with Universal Factor Graphs
(2+\varepsilon)$-Sat Is NP-hard
  • Citing Article

  • January 2017

SIAM Journal on Computing

Per Austrin

·

·

Johan Håstad