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We introduce a notion of code sparsification that generalizes the notion of cut sparsification in graphs. For a (linear) code C ⊆ 𝔽nq of dimension k a (1 ± ɛ)-sparsification of size s is given by a weighted set S ⊆ [n] with |S| ≤ s such that for every codeword c ∈ C the projection c|s of c to the set S has (weighted) hamming weight which is a (1 ± ɛ) approximation of the hamming weight of c. We show that for every code there exists a (1 ± ɛ)-sparsification of size s = Õ(k log(q)/ɛ2). This immediately implies known results on graph and hypergraph cut sparsification up to polylogarithmic factors (with a simple unified proof) — the former follows from the well-known fact that cuts in a graph form a linear code over 𝔽2, while the latter is obtained by a simple encoding of hypergraph cuts. Further, by connections between the eigenvalues of the Laplacians of Cayley graphs over to the weights of codewords, we also give the first proof of the existence of spectral Cayley graph sparsifiers over by Cayley graphs, i.e., where we sparsify the set of generators to nearly-optimal size. Additionally, this work can be viewed as a continuation of a line of works on building sparsifiers for constraint satisfaction problems (CSPs); this result shows that there exist near-linear size sparsifiers for CSPs over 𝔽p-valued variables whose unsatisfying assignments can be expressed as the zeros of a linear equation modulo a prime p. As an application we give a full characterization of ternary Boolean CSPs (CSPs where the underlying predicate acts on three Boolean variables) that allow for near-linear size sparsification. This makes progress on a question posed by Kogan and Krauthgamer (ITCS 2015) asking which CSPs allow for near-linear size sparsifiers (in the number of variables). At the heart of our result is a codeword counting bound that we believe is of independent interest. Indeed, extending Karger's cut-counting bound (SODA 1993), we show a novel decomposition theorem of linear codes: we show that every linear code has a (relatively) small subset of coordinates such that after deleting those coordinates, the code on the remaining coordinates has a smooth upper bound on the number of codewords of small weight. Using the deleted coordinates in addition to a (weighted) random sample of the remaining coordinates now allows us to sparsify the whole code. The proof of this decomposition theorem extends Karger's proof (and the contraction method) in a clean way, while enabling the extensions listed above without any additional complexity in the proofs.
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Efficient Algorithms and New Characterizations for CSP Sparsification
CSP sparsification, introduced by Kogan and Krauthgamer (ITCS 2015), considers the following question: how much can an instance of a constraint satisfaction problem be sparsified (by retaining a reweighted subset of the constraints) while still roughly capturing the weight of constraints satisfied by {\em every} assignment. CSP sparsification captures as a special case several well-studied problems including graph cut-sparsification, hypergraph cut-sparsification, hypergraph XOR-sparsification, and corresponds to a general class of hypergraph sparsification problems where an arbitrary 0/1-valued {\em splitting function} is used to define the notion of cutting a hyperedge (see, for instance, Veldt-Benson-Kleinberg SIAM Review 2022). The main question here is to understand, for a given constraint predicate P:Σr→{0,1} (where variables are assigned values in Σ), the smallest constant c such that O˜(nc) sized sparsifiers exist for every instance of a constraint satisfaction problem over P. A recent work of Khanna, Putterman and Sudan (SODA 2024) [KPS24] showed {\em existence} of near-linear size sparsifiers for new classes of CSPs. In this work (1) we significantly extend the class of CSPs for which nearly linear-size sparsifications can be shown to exist while also extending the scope to settings with non-linear-sized sparsifications; (2) we give a polynomial-time algorithm to extract such sparsifications for all the problems we study including the first efficient sparsification algorithms for the problems studied in [KPS24].
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- Award ID(s):
- 2152413
- PAR ID:
- 10574633
- Publisher / Repository:
- ACM Corr
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We introduce a notion of code sparsification that generalizes the notion of cut sparsification in graphs. For a (linear) code C ⊆ 𝔽nq of dimension k a (1 ± ɛ)-sparsification of size s is given by a weighted set S ⊆ [n] with |S| ≤ s such that for every codeword c ∈ C the projection c|s of c to the set S has (weighted) hamming weight which is a (1 ± ɛ) approximation of the hamming weight of c. We show that for every code there exists a (1 ± ɛ)-sparsification of size s = Õ(k log(q)/ɛ2). This immediately implies known results on graph and hypergraph cut sparsification up to polylogarithmic factors (with a simple unified proof) — the former follows from the well-known fact that cuts in a graph form a linear code over 𝔽2, while the latter is obtained by a simple encoding of hypergraph cuts. Further, by connections between the eigenvalues of the Laplacians of Cayley graphs over to the weights of codewords, we also give the first proof of the existence of spectral Cayley graph sparsifiers over by Cayley graphs, i.e., where we sparsify the set of generators to nearly-optimal size. Additionally, this work can be viewed as a continuation of a line of works on building sparsifiers for constraint satisfaction problems (CSPs); this result shows that there exist near-linear size sparsifiers for CSPs over 𝔽p-valued variables whose unsatisfying assignments can be expressed as the zeros of a linear equation modulo a prime p. As an application we give a full characterization of ternary Boolean CSPs (CSPs where the underlying predicate acts on three Boolean variables) that allow for near-linear size sparsification. This makes progress on a question posed by Kogan and Krauthgamer (ITCS 2015) asking which CSPs allow for near-linear size sparsifiers (in the number of variables). At the heart of our result is a codeword counting bound that we believe is of independent interest. Indeed, extending Karger's cut-counting bound (SODA 1993), we show a novel decomposition theorem of linear codes: we show that every linear code has a (relatively) small subset of coordinates such that after deleting those coordinates, the code on the remaining coordinates has a smooth upper bound on the number of codewords of small weight. Using the deleted coordinates in addition to a (weighted) random sample of the remaining coordinates now allows us to sparsify the whole code. The proof of this decomposition theorem extends Karger's proof (and the contraction method) in a clean way, while enabling the extensions listed above without any additional complexity in the proofs.more » « less
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