Approximating E3-LIN is NP-Hard

This lecture, which I gave for my 18.434 seminar, focuses on the MAX-E3LIN problem. We prove that approximating it is NP-hard by a reduction from LABEL-COVER.

1. Introducing MAX-E3LIN

In the MAX-E3LIN problem, our input is a series of linear equations ${\pmod 2}$ in ${n}$ binary variables, each with three terms. Equivalently, one can think of this as ${\pm 1}$ variables and ternary products. The objective is to maximize the fraction of satisfied equations.

Example 1 (Example of MAX-E3LIN instance)

\displaystyle \begin{aligned} x_1 + x_3 + x_4 &\equiv 1 \pmod 2 \\ x_1 + x_2 + x_4 &\equiv 0 \pmod 2 \\ x_1 + x_2 + x_5 &\equiv 1 \pmod 2 \\ x_1 + x_3 + x_5 &\equiv 1 \pmod 2 \end{aligned}

\displaystyle \begin{aligned} x_1 x_3 x_4 &= -1 \\ x_1 x_2 x_4 &= +1 \\ x_1 x_2 x_5 &= -1 \\ x_1 x_3 x_5 &= -1 \end{aligned}

A diligent reader can check that we may obtain ${\frac34}$ but not ${1}$.

Remark 2

We immediately notice that

• If there’s a solution with value ${1}$, we can find it easily with ${\mathbb F_2}$ linear algebra.
• It is always possible to get at least ${\frac{1}{2}}$ by selecting all-zero or all-one.

The theorem we will prove today is that these “obvious” observations are essentially the best ones possible! Our main result is that improving the above constants to 51% and 99%, say, is NP-hard.

Theorem 3 (Hardness of MAX-E3LIN)

The ${\frac{1}{2}+\varepsilon}$ vs. ${1-\delta}$ decision problem for MAX-E3LIN is NP-hard.

This means it is NP-hard to decide whether an MAX-E3LIN instance has value ${\le \frac{1}{2}+\varepsilon}$ or ${\ge 1-\delta}$ (given it is one or the other). A direct corollary of this is approximating MAX-SAT is also NP-hard.

Corollary 4

The ${\frac78+\varepsilon}$ vs. ${1-\delta}$ decision problem for MAX-SAT is NP-hard.

Remark 5

The constant ${\frac78}$ is optimal in light of a random assignment. In fact, one can replace ${1-\delta}$ with ${\delta}$, but we don’t do so here.

Proof: Given an equation ${a+b+c=1}$ in MAX-E3LIN, we consider four formulas ${a \lor \neg b \lor \neg c}$, ${\neg a \lor b \lor \neg c}$, ${a \lor \neg b \lor \neg c}$, ${a \lor b \lor c}$. Either three or four of them are satisfied, with four occurring exactly when ${a+b+c=0}$. One does a similar construction for ${a+b+c=1}$. $\Box$

The hardness of MAX-E3LIN is relevant to the PCP theorem: using MAX-E3LIN gadgets, Ha}stad was able to prove a very strong version of the PCP theorem, in which the verifier merely reads just three bits of a proof!

Let ${\varepsilon, \delta > 0}$. We have

$\displaystyle \mathbf{NP} \subseteq \mathbf{PCP}_{\frac{1}{2}+\varepsilon, 1-\delta}(3, O(\log n)).$

In other words, any ${L \in \mathbf{NP}}$ has a (non-adaptive) verifier with the following properties.

• The verifier uses ${O(\log n)}$ random bits, and queries just three (!) bits.
• The acceptance condition is either ${a+b+c=1}$ or ${a+b+c=0}$.
• If ${x \in L}$, then there is a proof ${\Pi}$ which is accepted with probability at least ${1-\delta}$.
• If ${x \notin L}$, then every proof is accepted with probability at most ${\frac{1}{2} + \varepsilon}$.

2. Label Cover

We will prove our main result by reducing from the LABEL-COVER. Recall LABEL-COVER is played as follows: we have a bipartite graph ${G = U \cup V}$, a set of keys ${K}$ for vertices of ${U}$ and a set of labels ${L}$ for ${V}$. For every edge ${e = \{u,v\}}$ there is a function ${\pi_e : L \rightarrow K}$ specifying a key ${k = \pi_e(\ell) \in K}$ for every label ${\ell \in L}$. The goal is to label the graph ${G}$ while maximizing the number of edges ${e}$ with compatible key-label pairs.

Approximating LABEL-COVER is NP-hard:

Theorem 7 (Hardness of LABEL-COVER)

The ${\eta}$ vs. ${1}$ decision problem for LABEL-COVER is NP-hard for every ${\eta > 0}$, given ${|K|}$ and ${|L|}$ are sufficiently large in ${\eta}$.

So for any ${\eta > 0}$, it is NP-hard to decide whether one can satisfy all edges or fewer than ${\eta}$ of them.

3. Setup

We are going to make a reduction of the following shape:

In words this means that

• “Completeness”: If the LABEL-COVER instance is completely satisfiable, then we get a solution of value ${\ge 1 - \delta}$ in the resulting MAX-E3LIN.
• “Soundness”: If the LABEL-COVER instance has value ${\le \eta}$, then we get a solution of value ${\le \frac{1}{2} + \varepsilon}$ in the resulting MAX-E3LIN.

Thus given an oracle for MAX-E3LIN decision, we can obtain ${\eta}$ vs. ${1}$ decision for LABEL-COVER, which we know is hard.

The setup for this is quite involved, using a huge number of variables. Just to agree on some conventions:

Definition 8 (“Long Code”)

A ${K}$-indexed binary string ${x = (x_k)_k}$ is a ${\pm 1}$ sequence indexed by ${K}$. We can think of it as an element of ${\{\pm 1\}^K}$. An ${L}$-binary string ${y = (y_\ell)_\ell}$ is defined similarly.

Now we initialize ${|U| \cdot 2^{|K|} + |V| \cdot 2^{|L|}}$ variables:

• At every vertex ${u \in U}$, we will create ${2^{|K|}}$ binary variables, one for every ${K}$-indexed binary string. It is better to collect these variables into a function

$\displaystyle f_u : \{\pm1\}^K \rightarrow \{\pm1\}.$

• Similarly, at every vertex ${v \in V}$, we will create ${2^{|L|}}$ binary variables, one for every ${L}$-indexed binary string, and collect these into a function

$\displaystyle g_v : \{\pm1\}^L \rightarrow \{\pm1\}.$

Picture:

Next we generate the equations. Here’s the motivation: we want to do this in such a way that given a satisfying labelling for LABEL-COVER, nearly all the MAX-E3LIN equations can be satisfied. One idea is as follows: for every edge ${e}$, letting ${\pi = \pi_e}$,

• Take a ${K}$-indexed binary string ${x = (x_k)_k}$ at random. Take an ${L}$-indexed binary string ${y = (y_\ell)_\ell}$ at random.
• Define the ${L}$-indexed binary ${z = (z_\ell)_\ell}$ string by ${z = \left( x_{\pi(\ell)} y_\ell \right)}$.
• Write down the equation ${f_u(x) g_v(y) g_v(z) = +1}$ for the MAX-E3LIN instance.

Thus, assuming we had a valid coloring of the graph, we could let ${f_u}$ and ${g_v}$ be the dictator functions for the colorings. In that case, ${f_u(x) = x_{\pi(\ell)}}$, ${g_v(y) = y_\ell}$, and ${g_v(z) = x_{\pi(\ell)} y_\ell}$, so the product is always ${+1}$.

Unfortunately, this has two fatal flaws:

1. This means a ${1}$ instance of LABEL-COVER gives a ${1}$ instance of MAX-E3LIN, but we need ${1-\delta}$ to have a hope of working.
2. Right now we could also just set all variables to be ${+1}$.

We fix this as follows, by using the following equations.

Definition 8 (Equations of reduction)

For every edge ${e}$, with ${\pi = \pi_e}$, we alter the construction and say

• Let ${x = (x_k)_k}$ be and ${y = (y_\ell)_\ell}$ be random as before.
• Let ${n = (n_\ell)_\ell}$ be a random ${L}$-indexed binary string, drawn from a ${\delta}$-biased distribution (${-1}$ with probability ${\delta}$). And now define ${z = (z_\ell)_\ell}$ by

$\displaystyle z_\ell = x_{\pi(\ell)} y_\ell n_\ell .$

The ${n_\ell}$ represents “noise” bits, which resolve the first problem by corrupting a bit of ${z}$ with probability ${\delta}$.

• Write down one of the following two equations with ${\frac{1}{2}}$ probability each:

\displaystyle \begin{aligned} f_u(x) g_v(y) g_v(z) &= +1 \\ f_u(x) g_v(y) g_v(-z) &= -1. \end{aligned}

This resolves the second issue.

This gives a set of ${O(|E|)}$ equations.

I claim this reduction works. So we need to prove the “completeness” and “soundness” claims above.

4. Proof of Completeness

Given a labeling of ${G}$ with value ${1}$, as described we simply let ${f_u}$ and ${g_v}$ be dictator functions corresponding to this valid labelling. Then as we’ve seen, we will pass ${1 - \delta}$ of the equations.

5. A Fourier Computation

Before proving soundness, we will first need to explicitly compute the probability an equation above is satisfied. Remember we generated an equation for ${e}$ based on random strings ${x}$, ${y}$, ${\lambda}$.

For ${T \subseteq L}$, we define

$\displaystyle \pi^{\text{odd}}_e(T) = \left\{ k \in K \mid \left\lvert \pi_e^{-1}(k) \cap T \right\rvert \text{ is odd} \right\}.$

Thus ${T}$ maps subsets of ${L}$ to subsets of ${K}$.

Remark 9

Note that ${|\pi^{\text{odd}}(T)| \le |T|}$ and that ${\pi^{\text{odd}}(T) \neq \varnothing}$ if ${|T|}$ is odd.

Lemma 10 (Edge Probability)

The probability that an equation generated for ${e = \{u,v\}}$ is true is

$\displaystyle \frac{1}{2} + \frac{1}{2} \sum_{\substack{T \subseteq L \\ |T| \text{ odd}}} (1-2\delta)^{|T|} \widehat g_v(T)^2 \widehat f_u(\pi^{\text{odd}}_e(T)).$

Proof: Omitted for now\dots $\Box$

6. Proof of Soundness

We will go in the reverse direction and show (constructively) that if there is MAX-E3LIN instance has a solution with value ${\ge\frac{1}{2}+2\varepsilon}$, then we can reconstruct a solution to LABEL-COVER with value ${\ge \eta}$. (The use of ${2\varepsilon}$ here will be clear in a moment). This process is called “decoding”.

The idea is as follows: if ${S}$ is a small set such that ${\widehat f_u(S)}$ is large, then we can pick a key from ${S}$ at random for ${f_u}$; compare this with the dictator functions where ${\widehat f_u(S) = 1}$ and ${|S| = 1}$. We want to do something similar with ${T}$.

Here are the concrete details. Let ${\Lambda = \frac{\log(1/\varepsilon)}{2\delta}}$ and ${\eta = \frac{\varepsilon^3}{\Lambda^2}}$ be constants (the actual values arise later).

Definition 11

We say that a nonempty set ${S \subseteq K}$ of keys is heavy for ${u}$ if

$\displaystyle \left\lvert S \right\rvert \le \Lambda \qquad\text{and}\qquad \widehat{f_u}(S) \ge \varepsilon^2.$

Note that there are at most ${\varepsilon^{-2}}$ heavy sets by Parseval.

Definition 12

We say that a nonempty set ${T \subseteq L}$ of labels is ${e}$-excellent for ${v}$ if

$\displaystyle \left\lvert T \right\rvert \le \Lambda \qquad\text{and}\qquad S = \pi^{\text{odd}}_e(T) \text{ is heavy.}$

In particular ${S \neq \varnothing}$ so at least one compatible key-label pair is in ${S \times T}$.

Notice that, unlike the case with ${S}$, the criteria for “good” in ${T}$ actually depends on the edge ${e}$ in question! This makes it easier to keys than to select labels. In order to pick labels, we will have to choose from a ${\widehat g_v^2}$ distribution.

Lemma 13 (At least ${\varepsilon}$ of ${T}$ are excellent)

For any edge ${e = \{u,v\}}$, at least ${\varepsilon}$ of the possible ${T}$ according to the distribution ${\widehat g_v^2}$ are ${e}$-excellent.

Proof: Applying an averaging argument to the inequality

$\displaystyle \sum_{\substack{T \subseteq L \\ |T| \text{ odd}}} (1-2\delta)^{|T|} \widehat g_v(T)^2 \left\lvert \widehat f_u(\pi^{\text{odd}}(T)) \right\rvert \ge 2\varepsilon$

shows there is at least ${\varepsilon}$ chance that ${|T|}$ is odd and satisfies

$\displaystyle (1-2\delta)^{|T|} \left\lvert \widehat f_u(S) \right\rvert \ge \varepsilon$

where ${S = \pi^{\text{odd}}_e(T)}$. In particular, ${(1-2\delta)^{|T|} \ge \varepsilon \iff |T| \le \Lambda}$. Finally by \Cref{rem:po}, we see ${S}$ is heavy. $\Box$

Now, use the following algorithm.

• For every vertex ${u \in U}$, take the union of all heavy sets, say

$\displaystyle \mathcal H = \bigcup_{S \text{ heavy}} S.$

Pick a random key from ${\mathcal H}$. Note that ${|\mathcal H| \le \Lambda\varepsilon^{-2}}$, since there are at most ${\varepsilon^{-2}}$ heavy sets (by Parseval) and each has at most ${\Lambda}$ elements.

• For every vertex ${v \in V}$, select a random set ${T}$ according to the distribution ${\widehat g_v(T)^2}$, and select a random element from ${T}$.

I claim that this works.

Fix an edge ${e}$. There is at least an ${\varepsilon}$ chance that ${T}$ is ${e}$-excellent. If it is, then there is at least one compatible pair in ${\mathcal H \times T}$. Hence we conclude probability of success is at least

$\displaystyle \varepsilon \cdot \frac{1}{\Lambda \varepsilon^{-2}} \cdot \frac{1}{\Lambda} = \frac{\varepsilon^3}{\Lambda^2} = \eta.$

(Addendum: it’s pointed out to me this isn’t quite right; the overall probability of the equation given by an edge ${e}$ is ${\ge \frac{1}{2}+\varepsilon}$, but this doesn’t imply it for every edge. Thus one likely needs to do another averaging argument.)