To get a sense of how LaTeX2WP works, and of what is still missing, here is the unedited output of the program given the latex source of my Max Cut paper (the most recent arxiv submission).

Note that the \ref commands have become clickable links.

You can see that footnotes and bibliography are not supported. Also, a \ref{} command cannot go in a section name. The rest of the LaTeX source of the paper, however, is handled gracefully. My cryptography notes are posted as unedited output of the converter.

\maketitle

We describe a new approximation algorithm for Max Cut. Our algorithm runs in time, where is the number of vertices, and achieves an approximation ratio of . On instances in which an optimal solution cuts a fraction of edges, our algorithm finds a solution that cuts a fraction of edges.

Our main result is a variant of spectral partitioning, which can be implemented in nearly linear time. Given a graph in which the Max Cut optimum is a fraction of edges, our spectral partitioning algorithm finds a set of vertices and a bipartition of such that at least a fraction of the edges incident on have one endpoint in and one endpoint in . (This can be seen as an analog of Cheeger’s inequality for the smallest eigenvalue of the adjacency matrix of a graph.) Iterating this procedure yields the approximation results stated above.

A different, more complicated, variant of spectral partitioning leads to an time algorithm that cuts fraction of edges in graphs in which the optimum is .

**1. Introduction **

In the Max CUT problem, we are given an undirected graph with non-negative weights on the edges and we wish to find a partition of the vertices (a *cut*) which maximizes the weight of edges whose endpoints are on different sides of the partition (such edges are said to be *cut* by the partition). We refer to the *cost* of a solution as the fraction of weighted edges of the graph that are cut by the solution.

It is easy, given any graph, to find a solution that cuts half of the edges, providing an approximation factor of for the problem. The algorithm of Goemans and Williamson \cite{GW94}, based on a Semidefinite Programming (SDP) relaxation, has a performance ratio of on general graphs, and it finds a cut of cost in graphs in which the optimum is . Assuming the unique games conjecture, both results are best possible for polynomial time algorithms \cite{K02:unique,KKMO04,MOO05} (see also \cite{ODW08}). Arora and Kale \cite{AK07} show that the Goemans-Williamson SDP relaxation can be near-optimally solved in nearly linear time in graphs of bounded degree (or more generally, in weighted graphs with bounded ratio between largest and smallest degree). We show in Appendix 9 that, using a reduction \cite{T01}, the Arora-Kale algorithm can be used to achieve the approximation performance of the Goemans-Williamson algorithm on all graphs in nearly-linear time.

A different, SDP-based, algorithm by Charikar and Wirth \cite{CW04} finds a solution that cuts at least a in graphs in which the optimum is . This result too is tight, assuming the unique games conjecture \cite{KOD06}.

No method other than SDP is known to yield an approximation better than for Max Cut, and such approximation has been ruled out for large classes of Linear Programming Relaxations \cite{VK07,SST07}.

** Our Results **

Our main result is a variant of the spectral partitioning algorithm with the following property: given a graph in which the Max CUT optimum cost is , it finds a set and a partition of into two disjoint sets of vertices such that the number of edges with one endpoint in and one endpoint in is at least a of the total number of edges incident\footnote{An edges is *incident* on a set of vertices if at least one of the endpoints belongs to .} on . More precisely, we show that the number of edges having both endpoints in or both endpoints in , plus half the number of edges having an endpoint in and an endpoint in is at most a fraction of the edges incident on . (See Theorem 1 and the subsequent discussion.) We will ignore the additive factors in the subsequent discussion in this section.

To derive an approximation algorithm for Max CUT, given a graph we apply the partitioning algorithm and find sets as above, we remove the vertices in from the graph, recursively find a partition of the residual graph, and then put back the vertices of on one side of the partition and vertices of on the other side. This means that we cut all the edges that are cut in the recursive step, plus all the edges with one endpoint in and one endpoint in , plus at least half of the edges between and . The recursion is stopped when less than half of the edges incident on are cut, in which in case we return a greedy partition of the residual graph.

We present an analysis of the recursive procedure due to Moses Charikar, which improves an analysis of ours which appeared in a previous version of this paper. The basic idea is that if we look at a generic step of the execution of the algorithm, if the optimal solution in the original graph is , and the current residual graph holds a fraction of the original edges, then we know that the optimum in the current residual graph is at least , and the spectral algorithm cuts at least a fraction of the edges incident on . When the recursion ends, it is because the spectral algorithm cuts less than half of the edges incident on , and so the optimum of the residual graph at the end of the recursion must be less than , meaning that the residual graph at the end of the recursion contains at most a fraction of the edges of the original graph. Putting together this information, a calculation shows that the algorithm cuts at least a fraction of edges of the graph. The ratio is always at least .

When applied to graphs in which the optimum is close to (in fact, to any graph in which the optimum is smaller than ), our algorithm may simply return a random partition. Thus, it fails to provide any non-trivial approximation to the Max CutGain problem, which is the same as the Max Cut problem, except that we count the number of cut edges *minus *. (Equivalently, we count the number of cut edges minus the number of uncut edges.) For Max CutGain we develop a more sophisticated spectral partitioning algorithm with the following property: given a graph in which the Max Cut optimum is , our algorithm finds sets such that the number of edges incident on cut by the partition exceeds the number of uncut edges by at least a fraction of the edges incident on . Iterating this algorithm allows us to find a cut for the entire graph of cost at least .

Our algorithm applies also to the case in which edges have negative weights, and it approximates a general class of quadratic programs. Given a symmetric real-valued matrix with zeroes on the diagonal, if there exists a vector such that , our algorithm finds a vector such that . (The algorithm of Charikar and Wirth finds a vector such that .)

** Relation to Cheeger’s Inequality **

In the case of regular graphs, our main result, Theorem 1, may be seen as an analog of Cheeger’s inequality \cite{A86} for the smallest (rather than second largest) eigenvalue of the adjacency matrix of the graph. We discuss this analogy in Section 5

** Relation to the Goemans-Williamson Relaxation **

Our algorithm may also be seen as a primal-dual algorithm that produces, along with a cut, a feasible solution to the semidefinite dual of the Goemans-Williamson relaxation such that the cost of the cut is at least times the cost of the dual solution. We describe this view in Section 6.

We know of only one other primal-dual approximation algorithm based on Semidefinite Programming: one of the algorithms for uniform sparsest cut of Arora, Rao, and Vazirani \cite{ARV04}.

** Other Relations to Previous Work **

It has been known that one can use spectral methods to certify an upper bound to the Max CUT optimum of a given graph. In particular, if is a -regular graph of adjacency matrix , and has eigenvalues , then one can easily show\footnote{The earliest reference that we are aware of for (1) is \cite[Lemma 3.1]{Alon96}.} that

(Our Lemma 2 is essentially a restatement of this fact.)

What is new is that we are able to prove a *converse*, in Lemma 3, and show that a non-trivial consequence follows whenever is close to .

As mentioned above, it was known that if and only if has a bipartite connected component. In particular, if is connected and not bipartite then . Alon and Sudakov \cite{AS00} consider the question of how small, in such case, can the gap be. They show that, if is connected and not bipartite, it has maximum degree and diameter , and is the smallest eigenvalue of the adjacency matrix , then . Our result implies the weaker bound in a -regular graph.

The “converse expander mixing lemma” of Bilu and Linial \cite{BL06} has some similarity with our approach to Max CutGain. Bilu and Linial show that if is a -regular graph, is the adjacency matrix, and are the eigenvalues of , then if it follows that there are sets such that the number of edges between and differs from what one would expect in a random -regular graph by a multiplicative error factor . In our main result for Max CutGain (Theorem 7) we have a stronger assumption, that , but we need to derive a much stronger conclusion, namely that the number of edges between and not only exceeds the number of edges that one would expect in a random -regular graph (a fact that can be probably proved with the same quantitative result of Bilu-Linial), but in fact exceeds the number of edges which are entirely contained in or entirely contained in .

The main difference between our proof and the proof of Bilu and Linial is that the combinatorial quantity that they relate to is the normalized multilinear form , for a certain matrix , while the combinatorial quantity that we wish to relate to is the normalized homogeneous quadratic form , for a different matrix . Generally, it is considerably harder to round continuous relaxations of quadratic forms of the latter type compared to multilinear forms of the first kind. (See e.g. the introduction of \cite{CW04} and their discussion of their results versus the results of Alon and Naor \cite{AN06}.)

The idea of iteratively removing parts of an instance in which one has a good solution appears in various works on the sparsest cut problem (for example in the way Spielman and Teng \cite{ST04} find a balanced separator using their “nibble” procedure), and it was used to approximate the Max CUT problem (in the version in which one wants to minimize the number of uncut vertices) by Agarwal et al. \cite{ACMM05}. In the algorithm of Agarwal et al., as in our algorithm, the basic procedure that is being iterated finds a set of vertices and a bipartition of such that most of the edges incident on have one endpoint in and one endpoint in .

**2. Sparsification **

It follows from the Chernoff Bound that if we are given a graph and we sample edges with replacement\footnote{If the graph is unweighted, we sample from the uniform distribution over the edges; otherwise we sample from the distribution in which each edge has a probability proportional to its weight.} then, with high probability, every cut has the same cost in the original graph as in the new graph, up to an additive error .\footnote{Note that the sparsified graph is an unweighted multigraph, and that the sparsification process is considerably simpler than the one used for algorithms for sparsest cut and other graph minimization problems.}

For this reason, all the dependency on in the running time of our algorithm can be changed to a dependency on with an arbitrarily small loss in the approximation factor.

**3. The Spectral Algorithm **

In this section we prove our main result.

Theorem 1 (Main)There is an algorithm that, given a graph for which the optimum of the Max CUT problem is at least , and a parameter , finds a vector such thatwhere is the weight of edge and is the (weighted) degree of vertex .

The algorithm can be implemented in nearly-linear randomized time .

To understand the statement of Theorem 1, let be the vector returned by the algorithm, and call the set of vertices with negative coordinates in , and the set of vertices with positive coordinates. Then, up to constant factors, the numerator counts the number of edges incident on which fail to have one endpoint in and one endpoint in , the denominator counts the number of incident incident on . More specifically, the numerator counts four times the edges that are entirely contained in or entirely contained in , and twice the edges that have one endpoint in and one endpoint in . The denominator every edge incident on once or twice, depending on whether one or both the endpoints of the edge are in .

The following form of the conclusion of Theorem 1 will be convenient in our analysis: given the vector , call the number of edges incident on , the number of “uncut” edges that have both endpoints in or both endpoints in , and the number of “cross” edges that have exactly one endpoint in ; then

Let be the adjacency matrix of our input graph (hence is the weight of the edge between and ), and be the diagonal matrix such that is the weighted degree of vertex and for .

Theorem 1 follows by combining the following two results, and noting that, for , .

Lemma 2If the optimum Max CUT in has cost at least , there is a vector such thatFurthermore, for every , we can find in time a vector such that

Lemma 3Given a vector such that , we can find in time a vector such that

Lemma 2 has a simple proof, and it can be seen as a statement about the semidefinite dual of the Goemans-Williamson relaxation, as discussed in Section 6. Lemma 3 is the main result of this paper.

** 3.1. Proof of Lemma \protect\ref **

}

Consider the optimization problem

Let be an optimum cut for , and define the vector such that if and otherwise. Then equals twice the difference between the number of edges not cut by and the number of edges that are cut, which is at most . As for , we have

Thus is a feasible solution to (3) of cost at most , and if is the optimal solution to (3), then we must have

To prove the “furthermore” part of the lemma, we observe that the optimization problem in (3) is equivalent to

where is the matrix that such that if , and otherwise. In turn, the optimization problem in (4) is the problem of computing the smallest eigenvalue of , which is the same as computing the largest eigenvalue of the positive semidefinite matrix .

Given a positive semidefinite matrix with non-zero entries and of largest eigenvalue , and a parameter , it is possible to find a vector such that in randomized time \cite{KW92}. Applying the algorithm to , which, as proved above, has a largest eigenvalue which is at least , and which has non-zero entries, we find in randomized time a vector such that

and so

and, if we define , then

which we can rewrite

** 3.2. Proof of Lemma \protect\ref **

}

We now come to our main result.

The condition is equivalent to

Before starting the formal proof, we describe a heuristic argument that gives some intuition for the actual proof.

Our algorithm, which we call the 2-Thresholds Spectral Cut algorithm and abbreviate 2TSC, is as follows:

- Algorithm 2TSC
- For every vertex
- Define the vector as follows:

- Output the vector for which the ratio
is smallest

The algorithm can be implemented to run in time. We first sort the vertices according to the value of , and so we assume we have when we run 2TSC. At each step , we need to modify the vector only in positions and , and the cost of recomputing the ration is only , so that all the steps together take time .

We need to argue that, under the assumption of the Lemma, the algorithm outputs a vector such that the ratio in (2) is at most

In order to analyze 2TSC, we study the following randomized process:

- Pick a value uniformly in ;
- Define as follows:

Every that is generated by the probabilistic process with positive probability is considered by algorithm 2TSC at some stage; this implies that if algorithm 2TSC outputs a vector such that , then in the randomized process we must have with probability 1 and, in particular, .

We shall prove that and so we shall conclude that the output of algorithm 2TSC satisfies the Claim.

Since Equation (5) and the distribution are invariant under multiplying by a scalar, we may assume that , so that is chosen uniformly in .

A case analysis shows that, for every edge ,

To verify Equation (6) we need to distinguish the case in which and have different signs from the case in which they have the same sign. We assume without loss of generality that .

- If they have different signs, and, say, , then when , and zero otherwise. Indeed, if then .
So equals , which is equal to the right-hand side of Equation (6).

- If they have the same sign, then when , when .
Overall, equals . The right-hand-sise of Equation (6) is , which is only larger.

Note also that .

To complete our argument it remains to apply Cauchy-Schwarz and standard manipulations.

By our assumption,

and it is a standard calculation that

and so

This completes the proof that Algorithm 2TSC performs as required by the Lemma.

**4. Approximation for Max Cut **

In this section we analyze the following algorithm

- Algorithm: {\sc Recursive-Spectral-Cut}
- Input: graph , accuracy parameter
- Run the algorithm of Theorem 1 with accuracy parameter , and let be the solution found by the algorithm; call the weighted number of edges such that least one of or is non-zero, the weighted number of
*cut*edges such that are both non-zero and have opposite signs, and the weighted number of*cross*edges such that exactly one of is zero; - If , then find a partition of that cuts edges, and return it.
- If , then let , , , let be the graph induced by , recursively call {\sc Recursive-Spectral-Cut} on , and let be the partion found by the algorithm; return or , whichever is better.

Note that the algorithm runs in randomized time because each iteration takes time and there are at most iterations.

In a preliminary version of this paper we presented a simple argument showing that if , then the algorithm cuts at least fraction of edges. The following tighter argument is due to Moses Charikar (personal communication, July 2008).

Theorem 4If Algorithm {\sc Recursive-Spectral-Cut} receives in input a graph whose optimum is , with then it finds a solution that cuts at least a fraction of edges.

*Proof:* Consider the -th iteration of the algorithm, and let be the residual graph at that iteration, and let be the number of edges of . Then we observe that the Max Cut optimum in is at least .

Let be the set of vertices and the partition found by the algorithm of Theorem 1. Let be the residual graph at the following step, and the number of edges of . (If the algorithm stops at the -th iteration, we shall take to be the empty graph; if the algorithm discards and chooses a greedy cut, we shall take to be empty and to be the partition given by the greedy cut.)

We know by Theorem 1 that the algorithm will cut at least a fraction of the edges incident on .

Indeed, we know that at least a fraction of those edges are cut (for small value of , it is possible that , but the algorithm is always guaranteed to cut at least half of the edges incident on ). This means that any convex combination of and is still a lower bound on the fraction of edges incident on cut by the algorithm.

If both and are at least , we are going to use the lower bound

If , then we use the lower bound

Finally, if both and are smaller than , we use the lower bound

Summing those bounds, we have that the number of edges cut by the algorithm is at least

Corollary 5Algorithm {\sc Recursive-Spectral-Cut} is a approximate algorithm for Max Cut.

*Proof:* Write . If then the algorithm finds a solution of cost and the approximation ratio is .

If , then the algorithm finds a solution of cost at least , and the approximation ratio is at least

Some calculus shows that, for , is minimized at (the smallest root of ) and is always at least .

**5. Relation to Cheeger’s Inequality **

In this section we compare our main result, Theorem 1, with Cheeger’s inequality \cite{A86}. We restrict our discussion to the case of regular graph.

If is a -regular graph, is its adjacency matrix, and , then has eigenvalues, counting multiplicities, which we shall call . It is always the case that , and that for every . The extremal cases are captured by the following well-known facts:

- if and only if is disconnected, that is, if and only if there is a set , , such that no edge of leaves .
- if and only if contains a bipartite connected component, that is, if and only if there is a set and partition of into disjoint sets , such that all edges incident on have one endpoint in and one endpoint in .

Cheeger’s inequality characterizes the cases in which is close to as those in which there is a set , such that the number of edges between and is small compared to .

If we define to be the *edge expansion* of ,

then we have Cheeger’s inequality

Similarly, Lemmas 2 and 3 characterizes the cases in which is close to as those in which there is a set and a partition of such that the number of edges incident on which fail to be cut by the partition is small compared to .

Define the *bipartiteness ratio* number of a graph to be

which is equivalent to

then we have

There are examples in which both inequalities in (8) are tight within constant factors.

If we take an odd cycle with vertices, then , because for every subset of vertices and for every bipartition of there is at least one failed edge, and the number of edges incident on is at most . In an odd cycle, however, and , and so is as large as .

To see the tightness of the other inequality, start from a -regular expander such that, say, . (Such graphs exist for constant .) Then construct by taking the disjoint union of the edges of and the edges of a -regular bipartite graph, so that the resulting graph is -regular with . There is a cut that cuts all the edges of the bipartite graph, so , but the smallest eigenvalue of is at least , meaning that is .

Our results, as stated in (8), are not just syntactically similar to Cheeger’s inequality: There are also similarities between the proof of Cheeger’s inequality and of Theorem 1. The analysis in Cheeger’s inequality relies on the study of the quadratic form

and it is based on the intuition that if (9) is small compared to then for most edges we have .

Our analysis was based on the study of the quadratic form

and the intuition that if (10) is small compared to then for most edges we have .

**6. Relation to the Goemans-Williamson Relaxation **

The dual of the Goemans-Williamson relaxation is

We can see Lemma 2 as stating a special case of the weak duality fact that the cost of every feasible solution to (11) is an upper bound to the optimal cut in the graph.

Indeed, if the optimal cut is of size , then no solution of cost can be feasible for (11). In particular, the solution has cost and cannot be feasible, meaning that cannot be feasible, and there is a vector such that .

In turn, Lemma 3 has the following primal dual interpretation: given a graph , there is an such that algorithm 2TSC finds such that , and the solution is feasible for (11), thus showing that the Max Cut optimum is at most .

Given this premise, we can now view algorithm {\sc Recursive-Spectral-Cut} as a primal-dual algorithm.

At step of the recursion, let be the number of edges in the residual graph , and and be the number of cut and cross edges in the solution found by the algorithm. Define so that is the upper bound on the Max Cut of given by the dual solution associated to the algorithm as above, and the algorithm satisfies . Then the dual solution at time also proves an upper bound to the Max Cut optimum of . Let ; then we have (i) a dual solution proving that the Max Cut of is , and we know that (ii) at every step we have . From fact (ii) and the analysis done in the proof of Theorem 1 we see the algorithm outputs a solution that cuts at least a fraction of edges, and it is able to output a feasible dual solution to the GW relaxation proving a upper bound to the optimum.

In particular, the ratio between the cost of the solution found by the algorithm and the upper bound provided by the dual solution is always at least .

**7. Quadratic Programming and the Max CutGain Problem **

Let be the adjacency matrix of a weighted graph with no self-loops, possibly with negative weights, let be the weighted degree of node , and . *Max-Cut Gain* is the optimization problem

In words, Max Cut Gain is the maximum, over all cuts, of the difference between the number of cut edges and the number of edges that are not cut, divided by the total number of edges. Equivalently, the optimum of Max Cut Gain is if and only if the optimum of Max Cut is . (The name of the problem comes from the fact that one is measuring how much one *gains* by using an optimum cut compared to a random cut, which only cuts a fraction of edges.)

Note that, up to the scaling that we do by dividing by , we are considering the problem

where is an arbitrary symmetric matrix with zeroes on the diagonal. Apart from the restriction to symmetric matrices, this is the same family of quadratic programs studied by Charikar and Wirth \cite{CW04}. It helps intuition, however, to continue to think about as the adjacency matrix of a weighted undirected graph.

We define the *gain ratio* of a graph the quantity

In the *gain ratio*, we consider all subsets of vertices, and all partitions of the set ; the objective function is the ratio between twice the difference of cut edges minus uncut edges among the edges induced by , divided by the volume of . If one imposed the additional constraint , then one would recover the Max Cut Gain problem.

Let be the smallest eigenvalue of the matrix ; then we see that

because

we conjecture that

but we are only able to prove the considerably weaker result that .

We use the following approach. Let be a real vector, and be a distribution over discrete vectors . We say that is a -good (randomized) rounding of if

We have the following simple fact:

Claim 1If is a vector such that , and is a a -good rounding of , then the support of contains a vector such that

*Proof:* We have

and so

and in particular there must exist a vector such that

Lemma 6 (Main)For every and every there is a -good rounding of such that .

*Proof:* Given , we assume without loss of generality that for every , and we consider the following distribution :

- Pick a threshold so that is uniformly distributed in ;
- For every vertex , pairwise independently:
- If or , then set ;
- If , then set with probability , and with probability .

We begin with the calculation of the expectations .

*Proof:* [Of Claim 2] The threshold is chosen according to a distribution whose density function is for ; conditioned on a specific choice of , the expectation of is 0 if or , and it is otherwise. Hence, we have

Claim 2 tells us that we can take . The following two claims give us that we can take , so that as required.

*Proof:* [Of Claim 3] Just note that, under the assumption of the claim, , and .

*Proof:* [Of Claim 4] Consider the expectation of , , conditioned on a fixed choice of .

whenever or . If is such that , then the conditional expectation of is . Overall, we have

So we have

where the last inequality follows from the fact that for every .

The lemma now follows.

In order to make the proof constructive, we need to show that we can find a vector in the sample space of as in the conclusion of the lemma. Suppose that the distribution of described above is such that .

A first observation is that there must be a threshold such that, conditioned on that particular choice of , we still have . Once we find such a threshold, we can search in the sample space of , which is of polynomial size.

It remains to describe how to find a threshold as above. Let us say that two thresholds are *combinatorially indistinguishable* if the sets of vertices and are equal, and call the set of vertices.

Then we have

and, similarly

so that it is always preferable to choose the smaller threshold. This means that for every equivalence class of combinatorially indistinguishable thresholds we only need to look at one of them, in order to find , and so we only need to consider at most thresholds. In particular, can be found in time. A nearly pairwise independent sample space of size can be used instead of a perfectly pairwise independent one so that the whole algorithm takes time , at the price of a additive loss in the approximation.

The following theorem summarizes our progress so far.

Theorem 7There is a nearly quadratic time algorithm that in input a graph such that finds a set and a partition of whose gain is at least .

*Proof:* We call the algorithm *Four-Threshold Spectral Cut*, or 4TSC.

- Algorithm 4TSC
- Input: Graph
- Let be the adjacency matrix of , be the matrix of degrees, . Find a vector such that , where is the smallest eigenvalue of . Set
- For every threshold in the set
- Let be a distribution of sample space that is -close to pairwise independence, and such that if or ; and such that with probability otherwise.

- Output the vector in the union of that maximizes

Using the construction of almost pairwise independent random variables of Alon et al. \cite{AGHP92}, each sample space has size , and can be computed in time. For each vector , the ratio can be computed in linear time.

By iterating the algorithm we derive our main result of this section.

Theorem 8There is a nearly cubic time algorithm that in input a graph such that finds a cut of of gain

**8. Conclusions **

The motivating question for this work was to find a combinatorial interpretation of the quantity in a -regular graph, akin to the interpretation of provided by the theory of edge expansion.

In establishing such an interpretation (in terms of the quantity that we call “bipartiteness ratio” in Section 5) we proved that a natural and easy-to-implement spectral algorithm performs non-trivially well with respect to the Max Cut problem. We expect the algorithm to perform well “in practice” and we are beginning an experimental study on which we will report in a follow-up paper.

It remains an interesting open question to find a “purely combinatorial” algorithm (namely, one not involving matrix computations) for Max Cut achieving an approximation factor better than 1/2.

** Acknowledgements **

I would like to thank an anonymous commenter for asking the question of the connection between spectral techniques and Max Cut, and Satyen Kale, James Lee and Salil Vadhan for providing helpful comments and references to the related literature.

I am grateful to Moses Charikar for communicating the proof of Theorem 4, which substantially improved my previous analysis, and for allowing me to present his improved analysis in this paper.

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\newpage

\appendix

**9. Appendix **

** 9.1. Efficiency of the Arora-Kale Algorithm **

Arora and Kale \cite{AK07} describe an algorithm for the Goemans-Williamson SDP relaxation of Max Cut which achieves an approximation ratio and runs in time given in input an unweighted multi-graph of maximum degree .\footnote{The Arora-Kale result is more general, but this statement is sufficient for our purpose} In particular, it is possible to find -approximate solutions to Max Cut in time , where is the approximation ratio of the Goemans-Williamson algorithm.

In this section we show that, using the Arora-Kale algorithm and a reduction from \cite{T01}, it is possible to approximate Max Cut within in time regardless of the degree distribution.\footnote{The running time can be reduced to if the representation of the graph is such that a random edge can be sampled in time, and the degree of a given vertex can be found in time.}

Given the sparsification result discussed in Section 2, it is sufficient to prove the following theorem, which is implicit in \cite{T01}.

Theorem 9There is a randomized algorithm and a deterministic algorithm with the following properties.

Given a graph , algorithm constructs in time a graph of maximum degree with vertices, such that the following happens with high probability: (i) , and (ii) given an arbitrary solution of cost in , algorithm constructs in time a solution of cost for .

*Proof:* We sketch how the argument in \cite{T01} applies to Max Cut.

Define the weighted graph as follows. (This graph will only be used in the analysis, and not explicitely constructed in the reduction.) For every vertex of degree , contains copies of ; for every edge in , we have edges in , one for every copy of and for every copy of , each such edge having weight .

We claim that approximating Max Cut in is equivalent to approximating Max Cut in . First, it should be clear that if is a cut in of cost , then if we define to be the set of all copies of vertices in , then is a cut of cost in . On the other hand, if is a cut of cost , then consider the distribution over cuts in in which a vertex is picked to be in with probability proportional to the fraction of copies of which are in ; the expected fraction of cut edges in is exactly , and using the method of conditional expectations we can find a cut of cost at least in linear time.

The graph is obtained by sampling with replacement edges from , using the distribution in which an edge is sampled with probability proportional to its weight. As discussed in Section 2, it follows from Chernoff bounds that a solution of cost in has cost in .

It remains to discuss the complexity of sampling : to sample one edge, we first pick a random edge of , and then we pick at random one of the copies of and one of the copies of ; this distribution is equivalent to randomly sampling one of the edges of with probabiltiy proportional to its weight. After time preprocessing, each edge of can be sampled in constant time.\footnote{The point of this discussion is that may have edges even if , for example if there are two vertices of degree . This means that it is not possible to explicitly construct in time, and so one must sample edges from without explicitly constructing .}

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