Deriving differential approximation results for $k\,$CSPs from combinatorial designs

TL;DR

The paper investigates differential approximability of k-CSP-q using combinatorial designs, notably orthogonal arrays and difference schemes, to relate average solution quality to design-based frequency properties. It introduces alphabet-reduction pairs of arrays (ARPAs) and cover pairs of arrays to enable reductions from k-CSP-q to smaller alphabets and to analyze local neighborhoods around solutions. The authors derive new differential-approximation bounds across several CSP families, including k-CSP-q, Ek-CSP(I_q^t), Ek-CSP(E_q), and CSP(O_q), and connect these bounds to OA/DS-based constructions and Nesterov-type reductions. They also develop a neighborhood-based perspective, showing how Hamming balls and partition-based multisets can transfer average-differential guarantees to local optima and instance diameter, with explicit bounds like Ω(1/n^k) for diameter approximations. Overall, the work highlights how combinatorial designs illuminate structural differential approximation guarantees for CSPs and provides new constant-factor reductions, with several open questions left for tightening these bounds and extending the framework.

Abstract

Inapproximability results for $\mathsf{Max\,k\,CSP\!-\!q}$ have been traditionally established using balanced $t$-wise independent distributions, which are closely related to orthogonal arrays, a famous family of combinatorial designs. In this work, we investigate the role of these combinatorial structures in the context of the differential approximability of $\mathsf{k\,CSP\!-\!q}$, providing new structural insights and approximation bounds. We first establish a direct connection between the average differential ratio on $\mathsf{k\,CSP\!-\!q}$ instances and orthogonal arrays. This allows us to derive the new differential approximability bounds of $1/q^k$ for $(k +1)$-partite instances, $Ω(1/n^{\lfloor k/2\rfloor})$ for Boolean instances, $Ω(1/n)$ when $k =2$, and $Ω(1/n^{k -\lceil\log_{Θ(q)}k\rceil})$ when $k, q\geq 3$. We then introduce families of array pairs, called {\em alphabet reduction pairs of arrays}, that are still related to balanced $k$-wise independence. Using these pairs of arrays, we establish a reduction from $\mathsf{k\,CSP\!-\!q}$ to $\mathsf{k\,CSP\!-\!k}$ (where $q >k$), with an expansion factor of $1/(q -k/2)^k$ on the differential approximation guarantee. Combining this with a 1998 result by Yuri Nesterov, we conclude that $\mathsf{2\,CSP\!-\!q}$ is approximable within a differential factor of $0.429/(q -1)^2$. Finally, using similar Boolean array pairs, {\em called cover pairs of arrays}, we prove that every Hamming ball of radius $k$ provides a $Ω(1/n^k)$-approximation of the instance diameter. Thus, our work highlights the relevance of combinatorial designs for establishing structural differential approximation guarantees for CSPs.

Figures (3)

• Figure 1: Illustration of an instance $I$ of $\mathsf{CSP\!-\!q}$ and the notations used to describe such an instance, where $q =3$, $n =4$ and $m =4$.
• Figure 2: Quantities involved in the gain ratio achieved by a given solution $x$ (in dotted lines) and the average differential ratio (in dashed lines), on an instance $I$ where the goal is to maximize.
• Figure 3: The $6$-gadget from H97 that transforms each constraint $(x_{i_1} +x_{i_2} +x_{i_3}\equiv a_i\bmod{2})$ of an instance $I$ of $\mathsf{Max\,E3\,Lin\!-\!2}$ into a set of $XNOR^2$ (shown as solid lines) and $XOR^2$ (shown as dashed lines) constraints.

Theorems & Definitions (65)

• Theorem 1.1: C13
• Definition 2.1: see e.g. HSS99
• Definition 2.2: see e.g. HSS99
• Definition 2.3
• Theorem 2.1
• Proposition 2.1
• Theorem 2.2: B52
• Theorem 2.3: CSV19
• Corollary 2.1
• Definition 2.4