On the NP-Hardness Approximation Curve for Max-2Lin(2)

TL;DR

The paper advances NP-hardness inapproximability for Max-2Lin(2) by constructing a curve $s(c)$ over $c\in[0.5,1]$ through Hadamard-based gadget reductions and a lifting technique that lets small gadgets yield better hardness as $k\to\infty$. A central innovation is the infinity-relaxed soundness $\mathrm{rs}_{\infty}$, formalized via LPs and leaky-flow concepts, enabling analysis of large, lifted gadgets. The authors compute explicit numerical gadgets for $k=2,3,4$, define the curve $s(c)$ via a restricted compressed $\mathrm{rs}_{\infty}$ LP, and identify a key point at $c\approx0.9232$, $s\approx0.8856$ which yields an inapproximability factor of about $1.48969$ for Min-2Lin(2)-deletion. This work complements UGC-based Gap$^{\mathrm{SDP}}(c)$ results by providing an NP-hardness curve that does not rely on UGC, while acknowledging intrinsic limitations of Had$k$-to-2Lin(2) gadget reductions in matching Gap$^{\mathrm{SDP}}(c)$ as $c\to1$.

Abstract

In the Max-2Lin(2) problem you are given a system of equations on the form $x_i + x_j \equiv b \pmod{2}$, and your objective is to find an assignment that satisfies as many equations as possible. Let $c \in [0.5, 1]$ denote the maximum fraction of satisfiable equations. In this paper we construct a curve $s (c)$ such that it is NP-hard to find a solution satisfying at least a fraction $s$ of equations. This curve either matches or improves all of the previously known inapproximability NP-hardness results for Max-2Lin(2). In particular, we show that if $c \geqslant 0.9232$ then $\frac{1 - s (c)}{1 - c} > 1.48969$, which improves the NP-hardness inapproximability constant for the min deletion version of Max-2Lin(2). Our work complements the work of O'Donnell and Wu that studied the same question assuming the Unique Games Conjecture. Similar to earlier inapproximability results for Max-2Lin(2), we use a gadget reduction from the $(2^k - 1)$-ary Hadamard predicate. Previous works used $k$ ranging from $2$ to $4$. Our main result is a procedure for taking a gadget for some fixed $k$, and use it as a building block to construct better and better gadgets as $k$ tends to infinity. Our method can be used to boost the result of both smaller gadgets created by hand $(k = 3)$ or larger gadgets constructed using a computer $(k = 4)$.

Figures (3)

• Figure 1: The $y$-axis shows the soundness $s$ and the $x$-axis the completeness $c$. The blue filled curve is our NP-hardness curve $s (c)$. The red dashed curve is the $\mathrm{Gap}_{\mathrm{SDP}} (c)$ by O'Donnell and Wu's OW. The points marked with arrows are prior inapproximability results of Max-2Lin(2). The blue cross on the curve marks our best inapproximability result for Min-2Lin(2)-deletion, see Figure \ref{['fig3']}. Note that both of the curves in this figure are convex functions.
• Figure 4: The matrix $M_k$ for $k = 3$. It is an $8 \times 7$ matrix. Note that prepending a zero column to $M_k$ and switching $0/1$ to $1/-1$ would make it into a Hadamard matrix, which is symmetric.
• Figure 5: This plot shows two types of Had$2$-to-2Lin(2) gadgets. The filled curve describes the minimisation of $\mathop{\mathrm{rs}}\nolimits$ and the striped curve describes the minimisation of $\mathop{\mathrm{rs_\infty}}\nolimits$. The completeness value is on the $x$-axis, and either $\frac{1 - \mathop{\mathrm{rs}}\nolimits (G)}{1 - c (G)}$ or $\frac{1 - \mathop{\mathrm{rs_\infty}}\nolimits (G)}{1 - c (G)}$ on the $y$-axis. In this particular case, the case of $k = 2$, it turns out that these two curves are identical.

Theorems & Definitions (79)

• Theorem 1
• Corollary 2
• Definition 3
• Definition 4
• Proposition 5
• Definition 6
• Definition 7
• Remark 8
• Definition 9
• Definition 10