Streaming approximation resistance of every ordering CSP

TL;DR

The work proves that for any finite family of ordering predicates , Max-OCSP() cannot be nontrivially approximated in the single-pass streaming model unless linear space is used, extending approximation-resistance phenomena to streaming. The authors construct YES/NO instance distributions and relate OCSP values to standard CSP values through q-coarsening, SPHE/SSHE properties, and small-partitionexpansion, then apply IRMD-based hardness to obtain Omega(n) space lower bounds. They provide a tight characterization up to polylog factors, with MAS not approximable beyond 1/2 and Max-Btwn beyond 1/3 in o(n) space, and show near-linear space solvability for sparse instances. This framework bridges CSP hardness results to OCSPs in streaming, offering fundamental limits on sublinear-space streaming algorithms for these problems.

Abstract

An ordering constraint satisfaction problem (OCSP) is defined by a family $\mathcal{F}$ of predicates mapping permutations on $\{1,\ldots,k\}$ to $\{0,1\}$. An instance of Max-OCSP($\mathcal{F}$) on $n$ variables consists of a list of constraints, each consisting of a predicate from $\mathcal{F}$ applied on $k$ distinct variables. The goal is to find an ordering of the $n$ variables that maximizes the number of constraints for which the induced ordering on the $k$ variables satisfies the predicate. OCSPs capture well-studied problems including `maximum acyclic subgraph' (MAS) and "maximum betweenness". In this work, we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, when an instance is presented as a stream of constraints. We show that for every $\mathcal{F}$, Max-OCSP($\mathcal{F}$) is approximation-resistant to $o(n)$-space streaming algorithms, i.e., algorithms using $o(n)$ space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of MAS our result shows that for every $ε>0$, MAS is not $(1/2+ε)$-approximable in $o(n)$ space. The previous best inapproximability result, due to Guruswami and Tao (APPROX'19), only ruled out $3/4$-approximations in $o(\sqrt n)$ space. Our results build on a recent work of Chou, Golovnev, Sudan, Velingker, and Velusamy (STOC'22), who provide a tight, linear-space inapproximability theorem for a broad class of "standard" (i.e., non-ordering) constraint satisfaction problems (CSPs) over arbitrary (finite) alphabets. We construct a family of appropriate standard CSPs from any given OCSP, apply their hardness result to this family of CSPs, and then convert back to our OCSP.

Figures (1)

• Figure 1: The constraint graphs of $\textsf{MAS}$ instances which could plausibly be drawn from $\mathcal{Y}$ and $\mathcal{N}$, respectively, for $q=5$ and $n=12$. Recall that $\textsf{MAS}$ is a binary $\textsf{Max-OCSP}$ with ordering constraint function $\Pi_{\textsf{MAS}}$ supported only on $(1,2)$. According to the definition of $\mathcal{Y}$ (see \ref{['def:YES_NO_MaxOCSP_dist']}, with $\boldsymbol{\pi} = (1,2)$), instances are sampled by first sampling a $q$-partition $\mathbf{b} = (b_1,\ldots,b_n) \in [q]^n$, and then sampling some constraints; every sampled constraint $(j_1,j_2)$ must satisfy $b_{j_2} \equiv b_{j_1}+1 \pmod q$. On the other hand, there are no requirements on $(b_{j_1},b_{j_2})$ for instances sampled from $\mathcal{N}$. Above, the blocks of the partition $\mathbf{b}$ are labeled $1,\ldots,5$, and the reader can verify that the edges satisfy the appropriate requirements. We also color the edges in a specific way: We color an edge $(j_1,j_2)$ green, orange, or red if $b_{j_2} > b_{j_1}$, $b_{j_2} = b_{j_1}$, or $b_{j_2} < b_{j_1}$, respectively. This visually suggests important elements of our proofs that $\mathcal{Y}$ has $\textsf{MAS}$ values close to $1$ and $\mathcal{N}$ has $\textsf{MAS}$ values close to $\frac{1}{2}$ (for formal statements, see \ref{['lemma:y-lower-bound']} and \ref{['lemma:n-upper-bound']}, respectively). Specifically, in the case of $\mathcal{Y}$, if we arbitrarily arrange the vertices in each block, we will get an ordering in which every green edge is satisfied, and we expect all but $\frac{1}{q}$ fraction of the edges to be green (i.e., all but those which go from block $q$ to block $1$). On the other hand, if we executed a similar process in $\mathcal{N}$, the resulting ordering would satisfy all green edges and some subset of the orange edges; however, in expectation, these account only for $\frac{q(q+1)}{2q^2} = \frac{q+1}{2q} \approx \frac{1}{2}$ fraction of the edges.

Theorems & Definitions (41)

• Remark
• theorem 1.1: Main theorem
• theorem 1.2: $\widetilde{O}(n)$-space algorithm
• lemma 2.1: KK19
• lemma 2.2: Chernoff bounds
• lemma 2.3: Stirling approximation
• theorem 3.1: Main theorem (single-predicate case)
• proof : Proof of \ref{['thm:main']}
• definition 3.2: $\mathcal{Y}^{\Pi,\boldsymbol{\pi}}_{q,\alpha,T}(n)$ and $\mathcal{N}^{\Pi}_{q,\alpha,T}(n)$
• lemma 3.3: $\mathcal{Y}$ has high $\textsf{Max-OCSP}(\Pi)$ values