Tight hardness of approximation for Max Cut

We will not show the full proof of Reference:thm-ug-maxcut (let alone Reference:thm-raghavendra), but we will illustrate some of the ideas behind it. A priori it seems very strange that a result like that could be proved. A \((c,s)\) integrality gap is some finite mathematical object with particular properties. How can the existence of such an object prevent the existence of an efficient algorithm?

The idea is that such an integrality gap can be used as a gadget in a reduction from the (conjectured to be) hard computational problem \(UG_{1-\epsilon,\epsilon}(\ell)\) into an instance of \(CSP(\cP)\). The Unique Games conjecture posits that for some particular values \(c^*=1-\epsilon\) and \(s^* =\epsilon\) it is computationally hard (specifically NP-hard) to distinguish, given an instance \(I\) of \(UG_{1-\epsilon,\epsilon}(\ell)\), between the case that \(val(I)\geq c^*\) and the case that \(\val(I)\leq s^*\). The reduction we are looking for is some efficient map \(\varphi\) mapping an instance \(I\) of \(UG_{1-\epsilon,\epsilon}(\ell)\) into an instance \(\varphi(I)\) of \(CSP(\cP)\) satisfying:

• Completeness: If \(\val(I)\geq c^*\) then \(\val(\varphi(I))\geq c-\epsilon\).
• Soundness: If \(\val(I)\leq s^*\) then \(\val(\varphi(I))\leq s+\epsilon\)

We will not give a full description of the reduction, but will only mention some of its key features. Recall that the alphabet of \(I\) in \(UG_{1-\epsilon,\epsilon}(\ell)\) is the set \([\ell]=\{1,\ldots,\ell\}\). The reduction will use as a gadget an error correcting code \(ECC\) mapping \([\ell]\) to \(\bits^L\) for some \(L\), and will map an instance \(I\) of \(\Pi\) that uses \(n\) variables into a max-cut instance \(I'\) on \(nL\) vertices that are divided into \(n\) \(L\)-sized blocks. If \(x\in [\ell]^n\) is an assignment that achieves value at least \(c^*\) for the instance \(I\), then we obtain a bipartition with cut value \(c-\epsilon\) for \(I'\) by cutting the \(L\) variables of the \(i^{th}\) block according to the \(L\)-length string \(ECC(x_i)\). The particular code we will use is known as the long code. It is the map \(ECC\from[\ell]\to\bits^L\) for \(L=2^\ell\), for every \(i\in [\ell]\) and \(w\in\bits^\ell\), we define \(ECC(i)_w=w_i\). It will be more convenient for us to think of (potentially corrupted) codewords of \(ECC\) as functions \(f\) mapping \(\bits^\ell\) to \(\bits\), where the actual (i.e., non corrupted) codewords correspond to the dictator functions of the form \(f_i(w)=w_i\).

The Max-Cut gadget desiderata

To show that our reduction is sound, we need to show that given any bipartition \(f\from\bits^{nL}\to\bits\) of the vertices of \(I'\) that cuts more than \(s+\epsilon\) fraction of the edges, we can decode it into an assignment \(x\in[\ell]^n\) that satisfies at least an \(s^*\) fraction of the original constraints of \(I\). It turns out that the key property is to show a way how to decode the restriction of the bipartition \(f\) to any particular block into a particular symbol in \(\Sigma\). Informally, the “gadget” we need for the reduction, is a graph \(H\) with vertex set \(V=\bits^\ell\) having the following properties:

• Completeness: For every \(i\in[\ell]\), the cut value of the bipartition corresponding to the dictator \(f_i\) is at least \(c-\epsilon\).
• Soundness: For every \(f\from\bits^\ell\to\bits\) that is “far” from a dictator function, the value of the cut corresponding to the bipartition \(f\) is at most \(s+\epsilon\).

Constructing a gadget from an integrality gap

Roughly speaking, the idea behind the construction is as follows. Recall that the \((x_{GW}-o(1),\alpha_{GW}x_{GW}+o(1))\) integrality gap was obtained by taking the graph \(G\) whose vertices are the vectors in the \(d\) dimensional unit sphere and we put an edge between two vertices \(u,v\in \R^d\) if \(\iprod{u,v} \leq \rho_{GW}+\epsilon\) for \(\rho_{GW} = 1-2x_{GW}\). Our gadget graph \(H\) over the Boolean cube \(\bits^\ell\) will be inspired by \(G\), in the sense that we connect \(w,z\in\bits^\ell\) if their correlation (i.e., the fraction of coordinates they agree on minus the fraction of coordinates they disagree on) is at most \(\rho_{GW}+\epsilon\). Let us now try to analyze this graph on an intuitive level.

The completeness property is fairly straightforward. Indeed, if \(f:\bits^\ell\to\bits\) is a “dictator” function of the form \(f(w)=w_i\) then for a random edge \((w,z)\), the probability that \(w_i\neq z_i\) is at least \(1/2 + (\rho_{GW}+\epsilon)/2 = x_{GW}+\epsilon/2\).

The soundness property is much subtler, not least because we did not even define what being “far from a dictator” means. Let us take one particular example. Suppose that \(f\from\bits^\ell\to\bits\) is a linear threshold function of the form \(f(w)=1\) if \(\sum \alpha_i w_i > \tau\) and \(f(w)=0\) otherwise, for some coefficients \(\alpha_1,\ldots,\alpha_\ell,\tau\). If all the \(\alpha\) coefficients but one are zero then \(f\) is a dictator, and so one way of saying that \(f\) is “far” from being a dictator is that all the coefficients are of roughly equal magnitude (say the same up to some constant).In Fourier analysis parlance, this is known as the property of \(f\) having small maximum influence. In this case we can use the central limit theorem to argue that \(\sum \alpha_i w_i\) is roughly the same as a normal variable with the same mean and variance. So, the probability that for a random edge \((w,z)\), the bipartition \(f\) will cut \((w,z)\) in the sense \(f(w)\neq f(z)\) is essentially the same as the probability that we can distinguish between two \(\rho\)-correlated normal variables, but by the same calculations that we’ve done before, this will be at most \(\arccos(\rho)/\pi\) which in our case would be at most \(\alpha_{GW}x_{GW}\).

More generally, to analyze the soundness of this gadget, Khot et al. (2004) used a powerful generalization of the central limit theorem known as the invariance principle (Mossel, O’Donnell, and Oleszkiewicz 2005).In fact the papers were in the reverse order. The motivation behind (Mossel, O’Donnell, and Oleszkiewicz 2005) was precisely to complete the soundness analysis of (Khot et al. 2004). Roughly speaking, the invariance principle means that if \(f\) is “far” from a dictator in some technical sense, then it cannot distinguish between the case that its input comes from the \(0/1\) Bernoulli distribution or the Gaussian distribution with the same moments. But then if \(f\) would have too good of a cut value, that would refute the isoperimteric result that underlies the integrality gap. The invariance principle can be thought of as an inverse theorem, saying that only “nice” functions (i.e., dictators or functions close to them) can and do distinguish between the sphere (or Gaussian space) and the cube. This completes our (admittedly quite partial and sketchy) outline of the proof.

What’s unique about unique games?

We have not seen the full reduction, and so can not at this point truly appreciate why it needs to rely on the unique games conjecture. Indeed, the label cover problem, which is superficially quite similar to unique games, is in fact NP-hard (see Reference:def-label-cover below). A priori one could perhaps hope that there is a “minor modification” of this reduction so it can handle the case where the original constraint satisfaction problem instance \(I\) is “non unique” and hence be based merely on \(P\neq NP\).

On a technical level, the current proof relies on the uniqueness property to reduce the task of verifying that the original constraints were satisfied to the “code checking” task of verifying that any assignment that has good value for the gadget is related to a true codeword. One could hope to get a more sophisticated gadget that would enable such “consistency checking” as well. Indeed, the current best candidate approaches to proving the unique games conjecture boil down to coming up with such gadgets. There are some obstacles to such an approach, showing that it would require more significant modifications. First, there is a sub-exponential time algorithm for unique games (Arora, Barak, and Steurer 2010) which implies that any such reduction based on the label cover problem (which is believed to be exponentially hard) would need to use some kind of “powering” step with a polynomial blow up the instance size in addition to any gadget. It also shows that there is a sense in which the Unique Games problem is qualitatively easier than Label Cover.

Also, the relation between gadgets and integrality gaps is not yet fully understood. For example, while the current gadgets are based on degree two integrality gaps, it turns out that they are inherently not integrality gaps for higher degree, as the invariance principle itself (or, more accurately, close variants of it) has a constant degree sum of squares proof (Barak et al. 2012). We do not know whether it means that a gadget whose soundness proof is based on the invariance principle cannot be used to obtain such NP hardness reductions.

We will return to the Unique Games Conjecture later in this course. Given current research, it seems that understanding its truth is closely coupled with the question of understanding the extent of the power of the sum of squares algorithm.