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Additive Sparsification of CSPs

Eden Pelleg, Stanislav Živný

TL;DR

This work studies additive (unweighted) sparsification for CSPs with fixed predicates. It develops a reduction to additive cut sparsification via the $k$-partite $k$-fold cover and a linear-algebraic decomposition to extend from Cut to all Boolean predicates, then generalizes to non-Boolean predicates with an all-but-one sparsification framework. The results show that for any constant arity $k$, CSP$(P)$ admits an additive sparsifier with $O(n\varepsilon^{-2}\log(1/\varepsilon))$ edges for all $P:\{0,1\}^k\to\{0,1\}$, and all-but-one sparsification for predicates $P:D^k\to\{0,1\}$ with $D=[q]$, along with an optimality justification. This yields unweighted, near-linear-size sparsifiers for broad CSP classes, enabling scalable approximation and analysis on hypergraphs with fixed constraint languages.

Abstract

Multiplicative cut sparsifiers, introduced by Benczúr and Karger [STOC'96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA'17] and non-Boolean domains by Butti and Živný [SIDMA'20]. Bansal, Svensson and Trevisan [FOCS'19] introduced a weaker notion of sparsification termed "additive sparsification", which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate $P:\{0,1\}^k\to\{0,1\}$ of a fixed arity $k$, we show that CSP($P$) admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP($P$) admits an additive sparsifier for any predicate $P:D^k\to\{0,1\}$ of a fixed arity $k$ on an arbitrary finite domain $D$.

Additive Sparsification of CSPs

TL;DR

This work studies additive (unweighted) sparsification for CSPs with fixed predicates. It develops a reduction to additive cut sparsification via the -partite -fold cover and a linear-algebraic decomposition to extend from Cut to all Boolean predicates, then generalizes to non-Boolean predicates with an all-but-one sparsification framework. The results show that for any constant arity , CSP admits an additive sparsifier with edges for all , and all-but-one sparsification for predicates with , along with an optimality justification. This yields unweighted, near-linear-size sparsifiers for broad CSP classes, enabling scalable approximation and analysis on hypergraphs with fixed constraint languages.

Abstract

Multiplicative cut sparsifiers, introduced by Benczúr and Karger [STOC'96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA'17] and non-Boolean domains by Butti and Živný [SIDMA'20]. Bansal, Svensson and Trevisan [FOCS'19] introduced a weaker notion of sparsification termed "additive sparsification", which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate of a fixed arity , we show that CSP() admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP() admits an additive sparsifier for any predicate of a fixed arity on an arbitrary finite domain .

Paper Structure

This paper contains 11 sections, 9 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contributions
  4. Boolean predicates
  5. Non-Boolean predicates
  6. Comparison to previous work
  7. Related work
  8. Preliminaries
  9. Sparsification of Boolean Predicates
  10. Sparsification of Non-Boolean Predicates
  11. Optimality of All-But-One Sparsification

Key Result

Theorem 3

Let $G = (V,E)$ be an undirected $n$-vertex $k$-uniform hypergraph, and $\varepsilon>0$. Then $G$ admits additive cut sparsification with error $\varepsilon$ using $O\left(\frac{n}{k}\varepsilon^{-2}\log(\frac{k}{\varepsilon})\right)$ hyperedges.

Figures (2)

  • Figure 1: Graph from Example \ref{['example']}.
  • Figure 2: An example of a representation of an assignment on $\gamma(G)$. $\text{Z}_a^{(i)}$ consists of all vertices in $V^{(i)}$ which are a copy of a vertex $v\in V$ with $a(v) = 0$, and $\overline{\text{Z}}_a^{(i)}$ consists of the rest of $V^{(i)}$. Each hyperedge has a unique path from left to right (but a path might belong to multiple hyperedges), choosing one of $\text{Z}_a^{(i)},\overline{\text{Z}}_a^{(i)}$ for each $i$. Each such path is also in 1-1 correspondence with a coordinate in $u_T$. In this example $T = \{0,3,k-1\}$ and the shaded sets represent $a_T^{-1}(0)$. By green dotted lines we indicated a path corresponding to a hyperedge counted in $\text{Val}_{\gamma(G),\textsf{Cut}}(a_T)$, and by red dashed lines we indicated a path which does not. The green dotted path corresponds to a value of $1$ in the coordinate of $u_T$ with binary representation $(1,1,0,0,1,\hdots,0,1)$, and the red dashed path to a value $0$ in the coordinate with binary representation $(1,0,1,1,0,,\hdots,1,1)$. Note that if $T = [k]$ then any hyperedge corresponding to a path only on $\text{Z}_a^{(i)}$ is not counted.

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Theorem 3: Additive Cut Sparsification additive
  • Remark 4
  • Remark 5
  • Definition 6
  • Definition 7
  • Proposition 8
  • proof
  • Definition 9
  • ...and 20 more