approximation of max-cut on graphs of bounded …945792/...approximation of max-cut on graphs of...

IN DEGREE PROJECT COMPUTER SCIENCE AND ENGINEERING, SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2016

Approximation of Max-Cut on Graphs of Bounded Degree

MIKAEL FLORÉN

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF COMPUTER SCIENCE AND COMMUNICATION

Approximation of Max-Cut on Graphs ofBounded Degree

Approximation av Max-Cut i grafer med begränsat gradtal

MIKAEL FLORÉN

Degree Project in Computer Science, Second Cycle - DA225XSupervisor: Per AustrinExaminer: Johan HåstadEmail: [email protected]

April 12, 2016

AbstractThe Max-Cut problem is a well-known NP-hard problem,for which numerous approximation algorithms have beendeveloped over the years. In this thesis, we examine thespecial case where the degree of vertices in the graph isbounded. With minor modifications to existing algorithms,we are able to obtain an improved approximation ratio forgeneral bounded-degree graphs. Furthermore we show ad-ditional improvements for graphs with at least a constantfraction of odd-degree vertices. We also identify some otherpossible areas for improvement in the general bounded-degree case.

Referat

Max-Cut-problemet är ett välkänt NP-svårt problem, förvilket ett flertal olika approximationsalgoritmer har utveck-lats över åren. I det här arbetet undersöks specialfallet dågrafens noder har begränsat gradtal. Med mindre föränd-ringar av existerande algoritmer lyckas vi uppnå en förbätt-rad approximationskvot för generella gradtals-begränsadegrafer. Vi visar också ytterligare förbättringar för grafermed minst en konstant andel noder med udda gradtal. Viidentifierar också några andra möjliga områden för förbätt-ring i det allmänna gradtals-begränsade fallet.

Contents

1 Introduction 11.1 NP-hard problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Max-Cut problem . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Thesis objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background 52.1 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Semidefinite programming . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Goemans-Williamson algorithm . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Algorithm description . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Algorithm analysis . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Specialization for graphs of bounded degree . . . . . . . . . . . . . . 112.4.1 Modified algorithm . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Algorithm analysis . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Results 133.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Bounded-degree improvement . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Candidate Vertices . . . . . . . . . . . . . . . . . . . . . . . . 143.2.3 Updated Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.4 Approximation Ratio . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Vertices of odd degree . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.1 Probability for odd candidate vertex . . . . . . . . . . . . . . 203.3.2 Ratio of odd-degree vertices . . . . . . . . . . . . . . . . . . . 213.3.3 Approximation ratio . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Assuming unbalance around all bad vertices . . . . . . . . . . . . . . 243.4.1 Utilizing the conjecture . . . . . . . . . . . . . . . . . . . . . 243.4.2 Motivation for the conjecture . . . . . . . . . . . . . . . . . . 263.4.3 An approach for proving the conjecture . . . . . . . . . . . . 283.4.4 Handling balanced pairs . . . . . . . . . . . . . . . . . . . . . 29

4 Discussion 33

4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.1 Probability of unbalanced neighbors . . . . . . . . . . . . . . 334.2.2 Utilizing balanced pairs . . . . . . . . . . . . . . . . . . . . . 344.2.3 Using Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . 34

Bibliography 37

Chapter 1

Introduction

1.1 NP-hard problems

Within computer science, there are problems of various difficulties. A commondistinction is whether a problem can be solved in polynomial time (often referred toas being efficiently solvable). Efficiently solvable decision problems (i.e. problemswith a yes-or-no answer) belong to the class P. Similarly, the class NP containsthose decision problems where a solution can be verified efficiently. We know thatP ⊆ NP, but it is not known whether the two are equal. In fact, it is widely believedthat P 6= NP, which is used as a working hypothesis.

Based on the above, NP-hard problems are those which are at least as hard asthe hardest problems in NP. Formally, we say that a problem X is NP-hard ifevery problem in NP can be reduced to an instance of X in polynomial time. Inother words, a solver for X can be used to solve any problem in NP with similarefficiency as the original solver. Furthermore, NP-complete problems are those thatare both in NP and NP-hard, making them the hardest problems in NP. It is worthnoting that NP-hard problems are not limited to decision problems, which meansthe definition extends to other natural problem formulations like optimization andsearch problems. The fact that there are no known efficient algorithms for NP-hardproblems, despite many years of research, provides some anecdotal evidence for theidea that P 6= NP.

Since NP-hard problems probably can not be solved efficiently, it is interesting to lookat polynomial-time approximation algorithms for such problems. This is motivatedby the possibility of finding a “reasonably good” solution in time that is hopefullymuch smaller than an exact solution would require. For maximization problems, analgorithm that guarantees a result that is at least a factor ρ of the optimal solutionis known as a ρ-approximation algorithm. In the case of randomized algorithms, asis the case in this thesis, the same term can be used to refer to algorithms finding

1

CHAPTER 1. INTRODUCTION

solutions with an expected value that is at least ρ times the optimal value.

1.2 The Max-Cut problem

The Max-Cut problem is one of Karp’s original NP-complete problems. Given anundirected graph with unweighted edges, the objective is to find a set S of verticesthat maximizes the number of edges between S and S (i.e. the cut edges). Anexample problem instance is shown in Figure 1.1.

Figure 1.1: Illustration of a Max-Cut problem instance (left) and itssolution (right). A filled circle corresponds to a vertex in S, while anempty circle corresponds to those in S. The dashed edges are those thatare cut by this assignment.

For Max-Cut, the best known polynomial-time approximation algorithm is due toGoemans and Williamson (1995), and achieves an expected approximation ratio ofαGW ≈ 0.878. This ratio is optimal assuming the so called unique games conjecture1,but improvements are possible for special instances of the problem. Feige, Karpinski,and Langberg (2002) show that for a graph of maximum degree ∆, the algorithmcan be improved to yield a αGW + ε∆ approximation ratio, where ε∆ = 1

233∆4 . Bothof these algorithms will be discussed in greater depth in the next section.

It is interesting to look at non-approximability results, to see what results we couldhope for at best. The randomized reduction described by Trevisan (2001) can beused to show that it is not possible to efficiently obtain an approximation ratio betterthan αGW +O( 1√

∆) (assuming the unique games conjecture). In other words, fairly

large improvements may be possible.

As this is an NP-hard problem, improvements here have some inherent theoreticalrelevance. Not only are improvements for the Max-Cut problem itself interesting, itmay also be possible to generalize those improvements for usage with other prob-lems. For instance, problems such as Max 3SAT and Max Bisection are in some

1First formulated by Khot (2002). Similarly to how the P = NP conjecture implies the diffi-culty of obtaining certain exact solutions, this conjecture implies the difficulty of obtaining certainapproximation ratios.

2

1.3. THESIS OBJECTIVE

ways similar to the Max-Cut problem, and may benefit from improvements here.There are also practical applications in circuit layout design and statistical physics(Goemans and Williamson, 1995).

1.3 Thesis objective

The goal of this thesis is to investigate whether further improvements can be madeto the approximation ratio for graphs with bounded degree, either by improving thealgorithm itself, or through a better analysis.

3

Chapter 2

Background

2.1 Linear Programming

Linear programming is a technique that can be applied to various optimizationproblems. First, consider the linear equation systems that can be solved by Gaussianelimination. That is, we are given a matrix A and a vector b, and wish to find anothervector x such that Ax = b. In linear programming, this equation is replaced by theinequality Ax ≥ b (where the inequality is applied componentwise). The constraintx ≥ 0 (which is just a special case of Ax ≥ b) may be added as well.

These constraints define a region of possible solutions. Within this region, we wishto minimize a linear combination of the components of x. This linear combinationcan be written as cTx, referred to as the objective function, for some coefficientvector c. Both of these concepts have relatively simple geometrical interpretations,as shown in Figure 2.1. Using these definitions, Kleinberg and Tardos (2006) givethe following formulation for the standard form of a Linear Programming problem:

Given an m×n matrix A, and vectors b ∈ Rm and c ∈ Rn, find a vectorx ∈ Rn to solve the following optimization problem:

min(ctx such that x ≥ 0; Ax ≥ b).

Given a problem of this type, there are algorithms that give a solution in polynomialtime. This is useful not only because it gives a way to solve certain problemsefficiently, but it can also be used as a tool for approximation algorithms.

5

CHAPTER 2. BACKGROUND

X2

X1

00

1

1

2

2

3

3

4

4

5

5

6

6

−c

Figure 2.1: Illustration of a 2-dimensional linear programming problem.The grey area is the feasible region of solutions. The objective functionis illustrated using the vector −c, which points in the direction of theoptimal solution.

2.2 Semidefinite programming

The algorithms in the following sections are based on semidefinite programmingtechniques. A semidefinite program (SDP) is a kind of generalisation of a linearprogram, which works with matrices rather than vectors.

A core concept is that of a positive semidefinite matrix. A symmetric n-by-n matrixX is called positive semi-definite if vTXv ≥ 0 for all v ∈ Rn. This constraint ona semidefinite program essentially replaces the non-negativity constraint of a linearprogram, and is denoted X 0. There are other ways to state this requirement (seee.g. Lancaster and Tismenetsky (1985)):

Theorem 2.2.1. Given a symmetric n-by-n matrix X, the following are equivalent:

1. X is positive semidefinite

2. All eigenvalues of X are non-negative

3. There exists a m-by-n (where m ≤ n) matrix B such that BTB = X

Then, we introduce the following notation for a linear function C of a matrix X:

6

2.3. GOEMANS-WILLIAMSON ALGORITHM

C •X =

n∑i=1

n∑j=1

CijXij

Using this, we can write a semidefinite program on the following form:

MinimizeX

C •X

subject to: Ai •X ≥ bi, i = 1, . . . ,mX 0

Similar to a linear program, we have an objective function that we wish to minimize,defined by the matrix C, as well as a number of linear constraints set by the matricesAi.

Klerk (2002) describes some complexity results for SDPs. For one, it is not alwayspossible to find an optimal solution to semidefinite programs in polynomial time.However, there are algorithms that can find a solution within ε > 0 additive errorof the optimal solution in time that is polynomial in input size and log(1/ε). In thecontext of efficient approximation algorithms, this is good enough, as we can choosea value of ε that is small enough for our purposes while still only taking polynomialtime to obtain our solution.

2.3 Goemans-Williamson algorithm

The best known approximation algorithm for the general Max-Cut problem is the onepresented by Goemans and Williamson (1995). The algorithm utilises semidefiniteprogramming combined with a random hyperplane rounding technique to partitionthe graph, obtaining an expected approximation ratio of αGW ≈ 0.878.

2.3.1 Algorithm description

To understand this algorithm, it is useful to first look at the Max-Cut problemformulated as an integer quadratic program. Here, we have a graph with the thevertex set V = 1, ..., n. Given this, the following program gives the weight of themaximum cut:

7


Maximize1

2

∑(i,j)∈E

(1− yiyj)

subject to: yi ∈ −1, 1 ∀i ∈ V

Since this is exactly the Max Cut problem, solving it is still NP-hard, and we need touse some relaxed version of the problem to obtain a polynomial running time. Thesolution used here is to allow the values yi to be multidimensional vectors vi ∈ Sn(where Sn is the n-dimensional unit sphere). In doing this, we also need to modifythe objective function in a way that is equivalent to the above formulation for vectorsin a single dimension. A natural way to do this is to change the product to a dotproduct, which gives the following formulation:

Maximize1

2

∑(i,j)∈E

(1− vi · vj)

subject to: vi ∈ Sn ∀i ∈ V

We then need to show that this formulation is equivalent to a semidefinite program.Using the decomposition A = BTB mentioned in the previous section, we can simplylet each vi correspond to the ith column of B. This way, aij = vi · vj . Thus, thematrix A being symmetric corresponds to the fact that vi · vj = vj · vi. Additionally,we need the restriction that aii = 1 to ensure that each vector vi is a unit vector.With this restriction, the following SDP can be formulated:

Maximize1

2

∑(i,j)∈E

(1− aij)

subject to: aii = 1 ∀i ∈ VA 0

Solving this problem, we obtain a set of vectors vi corresponding to the set of verticesin the original graph. Now, we want to use these vectors to determine how to createa partition of the vertices into two sets. The solution is quite simple: we begin bychoosing a random vector r, uniformly distributed on Sn. Finally, we let one sideof the partition be S = i|vi · r ≥ 0 (and the other being S). This is equivalentto choosing a random hyperplane, and partitioning the vectors according to whichones lie “above” and “below” the plane.

8

2.3. GOEMANS-WILLIAMSON ALGORITHM

2.3.2 Algorithm analysis

We begin by presenting the following lemma regarding the probability of any givenedge being cut:

Lemma 2.3.1. The probability of an edge (i, j) being cut by the hyperplane roundingis proportional to the angle between the corresponding vectors, and can be explicitlywritten as:

Pr[sgn(vi · r) 6= sgn(vj · r)] =arccos(vi · vj)

π

vi

vjθij

Figure 2.2: Illustration of the vectors and projected hyperplane.

Proof. Picking any edge (i, j), we can consider the plane containing both the vec-tors vi and vj (illustrated in Figure 2.2). The edge will be cut precisely when theprojection of the hyperplane onto this plane is between the two vectors. Due to thespherical symmetry of the distribution of the vector r, we know that the angle of itsprojection onto any plane will be uniformly distributed across a circle. Thus, theprojection of the hyperplane it defines will also have an angle uniformly distributedover [0, 2π]. Due to symmetry, it suffices to examine cases where sgn(vi · r) = 1.This means that if the angle between the vectors is θij , the probability for the cutis θij/π. In terms of vi and vj , we have:

Pr[sgn(vi · r) 6= sgn(vj · r)] =arccos(vi · vj)

π

Theorem 2.3.2. There exists a semidefinite based algorithm that approximates theMax-Cut problem within an expected ratio of at least αGW = min0≤θ≤π

2π

θ1−cos θ .

9


Proof. The expected weight E[W ] of the cut obtained by the algorithm is simply thesum of the expected contributions of each edge. The contribution of an edge (i, j)is equal to the probability that it is cut, and we can write the sum as follows:

E[W ] =∑i<j

· Pr[sgn(vi · r) 6= sgn(vj · r)] (2.1)

where sgn(x) = 1 if x ≥ 0 and −1 otherwise. Combining this formulation of theexpected weight of the cut with Lemma 2.3.1, we obtain

E[W ] =∑i<j

arccos(vi · vj)π

.

−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

vi · vj

E[W ] = arccos (vi · vj)/πSDP(G) = (1− vi · vj)/2αGW · SDP(G)

Figure 2.3: Plot of the functions being compared. The dashed lineαGW ·SDP(G) lies tangent to E[W ] at vi · vj = ρ∗ = −0.688 (i.e.θ = 2.331)

To determine the approximation ratio, we need to compare this value to the optimalcut. Since the SDP used above is a relaxation of the original problem, we knowthat SDP(G) ≥ OPT(G). In other words, a ρ such that E[W ] ≥ ρSDP(G) impliesthat E[W ] ≥ ρOPT(G). Thus, the value for αGW can be obtained by analyzing theratio of the contributions of individual edges in E[W ] and SDP(G) and finding itsminimum value:

10

2.4. SPECIALIZATION FOR GRAPHS OF BOUNDED DEGREE

αij =arccos(vi · vj)

π·

2

(1− vi · vj)=θijπ

·2

1− cos θ

αGW = min0≤θ≤π

2

π

θ

1− cos θ≈ 0.878567

This value is obtained for θ = θ0 ≈ 2.331 (shown in Figure 2.3). We now knowthat the contribution of each edge is at least a fraction αGW of the optimal value inexpectation. Using linearity of expectation 1, we can conclude that the sum is alsoat least a fraction αGW of the optimal value.

2.4 Specialization for graphs of bounded degree

Feige, Karpinski, and Langberg (2002) build upon this method by adding a localimprovement step. This addition is shown to improve the approximation ratio forgraphs of bounded degree ∆, obtaining a ratio of at least α∆ = αGW + 1

233∆4 .

2.4.1 Modified algorithm

First, additional constraints of the form (2.2) and (2.3) (known as triangle inequali-ties) are added to the SDP for all i, j, k ∈ [n]. This is still a relaxation of the originalMax-Cut problem, which will be shown in the next subsection.

vi · vj + vi · vk + vj · vk ≥ −1 (2.2)

vi · vj − vi · vk − vj · vk ≥ −1 (2.3)

Once this SDP is solved, the hyperplane rounding technique is used as before. Then,a local improvement step is performed, where misplaced vertices are moved to theother side of the partition. The algorithm thus comes down to the following steps:

Algorithm ACut:

1. Solve the modified SDP, and round the resulting vectors using the randomhyperplane technique.

1The expected value of the sum is equal to the sum of the individual expected values, regardlessof whether they are independent or not (see e.g. Blom et al. (2005))

11


2. Each vertex that is on the same side of the partition as more than half itsneighbors is moved to the other side of the partition. This is repeated untilno such vertices can be found.

When performing the local improvement step, the order in which the vertices aremoved may affect the end result. However, to obtain the approximation ratio men-tioned earlier in this section, these improvements can be made in an arbitrary order.Essentially, this is because the degree is bounded, meaning each vertex moved mayonly affect a limited number of other vertices. Thus, given that some number of im-provements are initially possible, this means that at least some constant (dependingon ∆) fraction of these will yield an improvement regardless of order.

Validity of triangle inequalities

As mentioned by Feige and Goemans (1995) these triangle inequalities can be shownto be valid based on the fact that they hold for boolean variables ∈ −1, 1. Thereare some symmetries in these equations that simplify this proof. In both (2.2)and (2.3), any assignment of vi, vj , vk will give the same result as its inverse (i.e.v′i = −vi, etc). In (2.2), we also have the case that swapping the values of any twovariables will also give the same result. Thus, (2.2) effectively only has two cases:one where all values are equal, and one where they are not.

vi vj vk vi · vj + vi · vk + vj · vk1 1 1 31 1 -1 -1

Table 2.1: Possible values for the left-hand side of (2.2).

As for (2.3), the only difference is that vk is replaced by −vk. As vk ∈ −1, 1, thismeans that the left-hand side of the equation can take on exactly the same valuesas in (2.2).

Thus, the smallest possible value in the boolean case is −1. The upper limit of 3 isalready implied by the variables being unit vectors.

2.4.2 Algorithm analysis

As the analysis of this algorithm is very similar to the one used for our own analysis(which can be found in Section 3.2), we will not cover this analysis here. The inter-ested reader can find the original proof in Feige, Karpinski, and Langberg (2002).

12

Chapter 3

Results

3.1 Summary of results

In this chapter, we will present and prove several results. The first is an improvementof the additive term of the approximation ratio such that the denominator is cubicin ∆:

Theorem 3.2.5. There exists a semidefinite based algorithm that for every ∆ >0 approximates the Max-Cut problem on graphs with bounded degree ∆ within anexpected ratio of αGW + ε∆ where ε∆ = 1

234∆3 .

The second theorem deals with graphs that both have a bounded degree and atleast some constant fraction of odd-degree vertices. In this case, the additive termis improved further, with a denominator that is quadratic in ∆:

Theorem 3.3.4. Assume the graph G has bounded degree ∆, and some fractionω of its vertices are of odd degree. Then there is an SDP-based algorithm thatapproximates the Max-Cut problem on G with an expected ratio of at least αGW +ε∆,where ε∆ = ω

224∆2 .

Furthermore, in Section 3.4 we examine what happens if it is likely that the partitionsaround an all bad vertex are of unequal size, even if that vertex has an even degree.To do this, we assume the following conjecture:

Conjecture 3.4.1. Assume some set of vectors vi, i ∈ [d], where for each i, j ∈[d] we have vi

|vi| ·vj|vj | ≥ −0.2, is partitioned using the random hyperplane rounding

technique. Then there is at least a probability c that the two sides of the partitionare not of equal size, for some positive constant c, which does not depend on d.

13

CHAPTER 3. RESULTS

We present some arguments for why this conjecture is reasonable, but do not proveit. Utilizing this conjecture, we go on to prove the following theorem, which wouldallow the additive term to be proportional to 1

∆2 :

Theorem 3.4.3. Assuming Conjecture 3.4.1, the following holds.

Given a graph G with bounded degree ∆, there is an SDP-based algorithm that ap-proximates the Max-Cut problem on G with an expected ratio of at least αGW + ε∆,where ε∆ = c

223∆2 .

3.2 Bounded-degree improvement

In this section, we will show how some slight alterations to the algorithm yield animproved approximation ratio of αGW + 1

234∆3 .

3.2.1 Notation

First, we explain some of the notation used in the rest of the section.

A bad edge (i, j) is one for which the corresponding inner product vi · vj ∈ [−0.689,−0.687].This interval roughly corresponds to the worst-case scenario of the Goemans-Williamsonalgorithm, in which ρ∗ = −0.688 . . ., which occurs when the angle between thosevectors is θ0.

An all bad vertex i is one for which every edge (i, j) is bad.

When discussing the neighborhood of an all bad vertex i of degree d, we can limitourselves to a (d + 1)-dimensional space in which the corresponding vectors lie.We represent these vectors as (α, β, γ) ∈ Rd+1, where α ∈ R, β ∈ R, and γ ∈Rd−1. Specifically, the vertex in question is represented by vi = (1, 0, 0). Oneof its neighbors is u1 = (α1, β1, 0) (chosen such that β1 > 0), and the rest areuj = (αj , βj , γj). Similarly, we have the hyperplane normal r = (αr, βr, γr). Notably,for all j ∈ [d], αj ∈ [−0.689,−0.687] (since i is all bad).

3.2.2 Candidate Vertices

The idea of the local improvement step in Feige, Karpinski, and Langberg (2002) isthat there are some misplaced vertices, where moving them will increase the valueof the cut, as shown in Figure 3.1.

14

3.2. BOUNDED-DEGREE IMPROVEMENT

In order to estimate how likely it is that a vertex i is misplaced, they examinethe case where the vector vi lies above, but very close to, the random hyperplane.Assume this happens with some probability δ1. Since αr is small, the position of theneighbors uj relative to the hyperplane will largely be decided by the hyperplanecorresponding to (βr, γr). This means that if the degree of i is odd, it will end upon the same side of the hyperplane as more than half its neighbors with probabilityabout 1/2. Otherwise, continue with the case that a neighbor u1 is also close to andabove the random hyperplane, which we assume happens with some probability δ2.In that case, we also have the probability 1/2 for more than half of the remainingvertices to end up on the same side as i. In the latter case, we conclude that theprobability of a misplaced vertex is at least roughly δ1δ2/2.

Based on this idea, we now define a set of candidate vertices. These are essentially thesame vertices that were examined by Feige, Karpinski, and Langberg (2002), exceptsome of the constants involved are different. Later in this section we will show thatthese are indeed misplaced vertices, and make use of their other properties to proveour approximation ratio.

i i

Figure 3.1: A vertex i that is initially misplaced (left) that is then movedto the other side of the partition (right). This increases the cut by 1.

Definition 3.2.1. Let i be an all bad vertex of degree d, with corresponding vectorvi. Then i is a δ-candidate vertex with respect to r if the following is true:

1. vi · r = αr ∈ [δ/8, δ]

2. βr ∈ [4δ, 32δ]

3. |γj · r| > 14∆ · ||γj || for all j ∈ [d].

4. γj · r ≥ 0 for more than half j ∈ [d].

In short, the first requirement says that vi most lie close to and above the hyperplane.The second essentially states that one neighbor should also be close to and above thehyperplane. The third and fourth requirements make sure that the final componentsof more than half the remaining vectors are non-negative and “large enough”, whichis important for the proofs later in this section.

15

CHAPTER 3. RESULTS

Lemma 3.2.1. Given δ ≤ 128∆

, a δ-candidate vertex with d neighbors will have atleast bd/2c+ 1 neighbors that lie on the same side of the partition and that are notδ-candidate vertices themselves.

Proof. First, we consider the vertices j for which γj · r ≥ 0. We want to make surethat the inner products uj · r > δ, since that means that all of them lie above thehyperplane, and that none of them are candidate vertices (as they do not fulfill therequirement of uj · r ∈ [δ/8, δ]). The rest of this proof follows some of the later stepsused for Lemma 4.6 in Feige, Karpinski, and Langberg (2002). We first consider thecase where βj ≥ 0.7:

uj · r = αjαr + βjβr + γj · γr ≥ (4βj − 0.689)δ +1

4∆||γj || > 2δ

For the case where βj < 0.7, we first show a lower limit on βj , using the triangleinequality (2.2) for the vectors vi, u1, uj :

vi ·u1 + vi ·uj + u1 ·uj ≥ −1

=⇒ u1 ·uj − 2 · 0.689 ≥ −1

=⇒ α1αj + β1βj + 0γj ≥ 1 + 1.378

=⇒ β1βj ≥ 0.378− 0.6892 ≥ −0.097

=⇒ βj ≥ −0.097/(1− 0.6892) ≥ −0.14

This means that βj ∈ [−0.14, 0.7], which we can use to establish a lower limit on||γj ||. Using the fact that ||uj || = 1, we get ||γj ||2 = 1−α2

j−β2j ≥ 1−0.6892−0.72 ≥

0.038 > 164 . This means that ||γj || > 1/8, which use in the following way:

uj · r = αjαr + βjβr + γj · γr

≥ (32 ·βj − 0.689)δ +1

4∆||γj ||

≥ (−4.48− 0.689)δ +1

25∆> −6δ + 8δ = 2δ

where the last step is obtained by using δ ≤ 128∆

. Thus, we can conclude that thevertices for which γj · r ≥ 0 all lie above the hyperplane, and none of them arecandidate vertices. By the fourth requirement on candidate vertices, we know thatthis is true for more than half the neighbor vertices, which completes the proof.

16

3.2. BOUNDED-DEGREE IMPROVEMENT

As the above shows that each candidate vertex improves the size of our cut, we nowwant to examine the probability of an all bad vertex being a candidate vertex. Todo this, we first need the following lemma regarding the projection of a vector v onr, found in Feige, Karpinski, and Langberg (2002) (which follows from the fact thatthe components ri of the vector defining the hyperplane follow a normal distributionN(0, 1) ):

Lemma 3.2.2. Let v be some vector in Rn of norm ||v||, and let r = (r1, . . . , rn)∈ Rn be a n-dimensional vector where each ri has normal distribution N(0, 1). Thesize of the projection of v on r can be bounded as follows:

Pr[|v · r| ≥ δ||v||] = Pr[|r1| ≥ δ] ≥ 1− δPr[v · r ∈ [0, δ]||v||] = Pr[r1 ∈ [0, δ]] ∈ [δ/8, δ/2]

Where δ ∈ [0, 1].

We now move on to the actual probability for candidate vertices:


, an all bad vertex is a d-candidate vertex withprobability at least 1

221∆2 over the choice of r.

Proof. We can consider each of the requirements for a candidate vertex separately.This works since the first two requirements are concerned with variables not involvedin any others (i.e. αr and βr). The last two requirements both depend on γr, butthey are only concerned with its magnitude and sign respectively, and are thusindependent.

1. Using Lemma 3.2.2 we know that αr ∈ [δ/8, δ] occurs with probability ≥ δ24.

2. Using the same lemma, βr ∈ [4δ, 32δ] occurs with probability ≥ 2δ.

3. By Lemma 3.2.2, we have the probability 1 − 14∆ for each j that |γj · r| >

14∆ · ||γj ||. By Boole’s inequality1, the probability of at least one inner productbeing too small is at most

∑dj=1 Pr[|γj · γr| < 1

4∆ ||γj ||] ≤ d · 14∆ ≤ 1/4. In

other words, the absolute values of the inner products are large enough withprobability at least 3/4.

4. Since γ1 = 0, we know that γ1 · r ≥ 0. For the other neighbors, we can simplyuse the fact that r and −r are equally likely. For any r where at least half theneighbors are below the hyperplane, −r would instead place at least half theneighbors above the hyperplane. Taking into account u1, this means that morethan half the neighbors are above the hyperplane. Thus, we have a probabilitythat is at least 1/2 that γj · r ≥ 0 for more than half of the neighbors.

1Also known as the union bound. This states that for any finite set of events, the probabilityof at least one event happening is at most equal to the sum of the individual probabilities (see e.g.Blom et al. (2005)).

17

CHAPTER 3. RESULTS

With δ = 128∆

, we have a final probability of at least δ24

· 2δ · 34 · 1

2 = 3δ2

26≥ 1

221∆2 .

3.2.3 Updated Analysis

Lemma 3.2.4. Let (X,Y ) be a partition obtained from the hyperplane roundingtechnique with c candidate vertices (with respect to the vector r defining the hyper-plane). Moving each of these candidate vertices to the opposite side of the partitionwill increase the size of the cut by at least c.

Proof. By Lemma 3.2.1, more than half the neighbors of a candidate vertex i willbe vertices on the same side as i that are not themselves candidate vertices. Thismeans that moving any single candidate vertex to the other side of the partitionincreases the cut by at least 1 (as we are now cutting more than half the edges,while previously we were cutting less than half the neighbors).

Now, we consider what happens when one or more of the neighbors are candidatevertices themselves. In Feige, Karpinski, and Langberg (2002) this is the sourceof interference between moved nodes; moving one vertex might make it so that itis no longer profitable to move another. With these candidate vertices, that is nolonger a risk as the analyzed benefit of moving a candidate vertex only includesnon-candidate vertices2.

Thus, each of the c candidate vertices will contribute 1 edge to the cut, for a totalincrease of c edges.

3.2.4 Approximation Ratio

Theorem 3.2.5. There exists a semidefinite based algorithm that for every ∆ >0 approximates the Max-Cut problem on graphs with bounded degree ∆ within anexpected ratio of αGW + ε∆ where ε∆ = 1

234∆3 .

Proof. First, we solve the SDP and choose a random vector r, where each ri hashas normal distribution N(0, 1), to define our hyperplane for rounding as in the firststep of Algorithm 2.4.1. By Goemans and Williamson (1995), if this SDP solutionhas value Z, we have an expected cut of αGWZ after hyperplane rounding.

2In fact, due to the pessimism in this analysis, we assume that edges to adjacent candidatevertices were cut before the move, and no longer will be afterwards, resulting in a loss of one edge.Since both candidates will be moved, each edge between candidates will remain the same as it waspreviously, meaning that any adjacent candidate vertices will give a cut 1 higher than this analysisimplies.

18

3.3. VERTICES OF ODD DEGREE

We first assume that at least a fraction (1 − 12∆) of the edges in E are bad. This

means that at most |E|/∆ vertices are not all bad, and at least n − |E|/∆ ≥ n/2vertices are all bad. By Lemma 3.2.3, an all bad vertex has probability at least

1221∆2 to be a candidate vertex. Thus, the expected number of candidate vertices isn

222∆2 ≥ |E|221∆3 . By Lemma 3.2.4, the cut is increased by at least this value in the local

improvement step, giving us an approximation ratio that is at least (αGW + 1221∆3 ).

Now, we examine the case where we have a large number of good (i.e. not bad) edges.By the analysis of Goemans and Williamson (1995), each such edge will in expecta-tion give an approximation ratio that is at least αGW + 1

212. If the contribution of

these edges to the SDP value is at least εZ, the expected cut is

E[W ] ≥ (αGW +ε

212)Z

Otherwise, we can ignore the good edges and perform the local improvement on theremaining subgraph. This gives an expected cut of

E[W ] ≥ (αGW +1

221∆3)(1− ε)Z

≥ (αGW +1

221∆3− ε)Z

Choosing ε = 1222∆3 gives us an expected approximation ratio of at least αGW+ 1

234∆3 .

3.3 Vertices of odd degree

In the original analysis of Feige, Karpinski, and Langberg (2002), the existence ofvertices with even degree is somewhat problematic. The whole argument surround-ing the vector u1 is based on this, which directly adds a factor proportional to 1

∆ toε∆ (see the proof of Lemma 3.2.3). This means that a graph with only vertices of odddegree will definitely show some improvement in the approximation ratio. Buildingon the previous improvements, we will show that we can obtain an additive termεodd∆ = Ω( 1

∆2 ) for such graphs.

19

CHAPTER 3. RESULTS

3.3.1 Probability for odd candidate vertex

For this section, we will use odd candidate vertices, which are similar to the candidatevertices of the previous section, but with a slightly different definition:

Definition 3.3.1. Let i be an all bad vertex of odd degree d, with correspondingvector vi. Then i is an odd candidate vertex with respect to r if the following istrue:

1. vi · r = αr ∈ [δ/8, δ]

2. |γj · r| > 14∆ · ||γj || for all j ∈ [d].

3. γj · r ≥ 0 for more than half j ∈ [d].

Since we are guaranteed to never have the same number of neighbors on both sidesof the hyperplane, the condition on the value of βr has been removed. This alsolets us use the notation uj = (αj , γj) for the neighbors of i. Other differences arethat we require that the vertex has an odd degree, and that the value of δ has beenaltered (with the particular value being motivated in the proof of Lemma 3.3.2.


, an odd-degree all bad vertex is an odd δ-candidatevertex with probability at least 1

210∆over the choice of r.

Proof. The analysis uses the same overall ideas as before. We begin by consideringan all bad vertex i, and in the same way (using Lemma 3.2.2) as before we have theprobability δ

24that vi · r ∈ [δ/8, δ].

As before, we examine the case where |γj · γr| ≥ 14∆ ||γj || for all vectors uj . Despite γj

being one dimension larger, the probability bound given before remains unchanged,meaning this happens with probability at least 3

4 . By the same argument as before,we also have that more than half the products γj · γr are positive with probabilityat least 1

2 . Taken together, we have a probability δ26≥ 1

210∆of this occurring.


, an odd δ-candidate vertex with d neighbors willhave at least d+1

2 neighbors on the same side of the partition and that are not oddcandidate vertices themselves.

Proof. As in the proof of Lemma 3.2.1, we consider the vertices j for which γj · r ≥ 0.Again, the goal is to make sure that uj · r > δ. Since we are no longer concernedwith the βj component, the equation is slightly simpler:

|uj · r| ≥ −0.689δ +1

4∆·√

1− (−0.689)2 ≥ 1

4∆· 0.724− 0.689 · δ

20


By using δ ≤ 124∆

, this product is greater than δ. As before, this means that thevertices for which γj · r ≥ 0 all lie above the hyperplane, and none of them areodd candidate vertices. We also know that these are more than half the neighborvertices, which completes the proof.

3.3.2 Ratio of odd-degree vertices

In order to use a proof similar to that of Theorem 3.2.5 for the approximation ratioin a graph with a given ratio ω of odd-degree vertices, we will need some additionalresults. Specifically, in the case where good edges are ignored, we want to make surethat we can retain a similar ratio (i.e. a ratio Θ(ω)) of odd-degree vertices.

Lemma 3.3.3. Assume G is a graph with a fraction ω vertices that are of odd degree,and some number of “good” edges. Then it is possible to remove “good” edges in away that retains a fraction of odd-degree vertices that is at least ω, with an averageof less than 2 “good” edges adjacent to each odd-degree vertex.

Proof. Consider the following algorithm:

Algorithm Aremove:

1. While there exists a cycle C of good edges, remove all edges in C.

2. Iteratively remove each good edge that is adjacent to at least one even-degreevertex, until no such edges are left.

We first look at the number of odd-degree vertices after Aremove is performed. Step 1of the algorithm doesn’t change the parity of any vertex (since each vertex involvedin a cycle is adjacent to two edges of that cycle), so the number of odd-degreevertices is unchanged. In step 2, removing an edge can never decrease the numberof odd-degree vertices, since only two vertices are affected and at least one of themis turned from an even degree to an odd degree. Thus, we can conclude that thefraction of odd-degree vertices is still at least ω.

Since all cycles of good edges are removed, the graph spanned by the remaining goodedges is a forest (i.e. some number of disjoint trees). As we also remove all goodedges adjacent to even-degree vertices, we know that every vertex in this forest isof odd degree. Since a tree has an average degree of < 2, we can conclude that theaverage number of good edges adjacent to each odd-degree vertex is less than 2.

21

CHAPTER 3. RESULTS

3.3.3 Approximation ratio

Theorem 3.3.4. Assume the graph G has bounded degree ∆, and some fractionω of its vertices are of odd degree. Then there is an SDP-based algorithm thatapproximates the Max-Cut problem on G with an expected ratio of at least αGW +ε∆,where ε∆ = ω

224∆2 .

Proof. As in the proof of Theorem 3.2.5, we solve the SDP and choose a ran-dom vector r. Given an optimal vector configuration vi with weight Z, weknow by Goemans and Williamson (1995) that the expected value of the cut af-ter the hyperplane rounding is at least αGWZ. As before, a bad edge is one wherevi · vj ∈ [−0.689,−0.687].

We first assume that at least a (1− ω2∆) fraction of the edges are bad. This means

that at most ω|E|∆ ≤ ωn

2 vertices are not all bad. Thus at least ωn− ωn2 = nω

2 verticesare all bad and of odd degree. Combining Lemmas 3.3.1 and 3.3.2, each of thesehave an expected contribution of δodd∆ ≥ 1

210∆edges in the local improvement step.

Thus, the approximation ratio in this case is at least

αGW +nω

2·

1

210∆·

1

|E|≥ αGW +

ω

210∆2(3.1)

where the inequality follows from n2|E| ≥

1∆ .

Otherwise, we have at least ω2∆ |E| edges that are not bad. Again, we use the

analysis by Feige, Karpinski, and Langberg (2002), stating that these edges eachhave an approximation ratio that is at least αGW + 1

212. We also assume that the

contribution of these edges towards Z is at least εZ for some small, positive ε. Thenwe have that

E[W ] ≥ (αGW +ε

212)Z

The final case is when the good edges contribute less than εZ. In this case, we canignore some of those edges when analyzing our improvement step. Note that theseedges do not have to be ignored when actually running the algorithm, as doing socan only make the cut worse - this is merely a way of analyzing a worst-case scenario.Doing so will allow us to reuse the proof for the case where many edges are bad.However, we also want to make sure that we don’t change too many odd-degreevertices into ones of even degree when doing so.

This is possible by choosing to first ignore those edges (i, j) where the product vi · vjis less than that of a bad edge (i.e. vi · vj < −0.689). Doing this, we run the riskof decreasing the ratio of odd degree vertices. Here we will show that the impact

22


of this is guaranteed to be relatively small. For each edge ignored, we know thatvi · vj < −0.689. This means that the contribution from each edge towards Z isZij =

1−vi · vj2 ≥ 0.844. However, we also know that the good edges (of which the

ignored edges are a subset) contribute less than εZ. The number of ignored edgescan thus be at most εZ

0.844 . Since each edge affects at most 2 vertices, the number of“lost” odd-degree vertices is no more than

2εZ

0.844≤ 3ε|E| ≤ 3n∆ε

2≤ 2n∆ε

which guarantees that we have at least a fraction (ω − 2∆ε) odd-degree verticesremaining.

The edges where the product is greater than −0.687 do not impact the expectedcontribution negatively, and do not all have to be ignored. This can be seen bysimply considering that a larger inner product implies that the vectors are closertogether, which in turn means that they are more likely to be on the same side ofthe random hyperplane. Thus, whenever such an edge is involved, the probabilityof more than half the neighbors being on the same side increases.

On the other hand, these edges are not guaranteed to not affect other odd candidatevertices (consider e.g. vi · vj = 1, where vi · r = δ ⇔ vj · r = δ). Here, we can useLemma 3.3.3, letting us ignore most of the remaining good edges. Specifically, wewill still have at least a fraction (ω − 2∆ε) odd-degree vertices, with an average ofless than 2 good edges adjacent to each of them. Since vertices connected by thesegood edges may affect each other, each of these “nearly all bad” vertices may affectup to 2 other such vertices, giving a factor 1

2+1 in the improvement term. As before,each odd-degree vertex has an expected contribution of δodd∆ ≥ 1

210∆edges in the

local improvement step, for a total ratio of at least:

(1− ε)αGW +1

|E|·n(ω − 2∆ε) ·

1

210∆·1

3

≥ (1− ε)αGW + (ω − 2∆ε) ·1

211∆2

≥ αGW +ω

211∆2− ε(αGW +

1

210∆)

≥ αGW +ω

211∆2− ε

Choosing ε = ω212∆2 completes the proof.

23

CHAPTER 3. RESULTS

3.4 Assuming unbalance around all bad vertices

As mentioned in Section 3.3, vertices of even degree are somewhat problematic.Specifically, the problem is that their neighbors may be partitioned in a balancedmanner (i.e. where the two partitions are of equal size). In this section, we ex-plore what it would mean if the partitions are unbalanced with a reasonably largeprobability. Specifically, we assume the following conjecture:

Conjecture 3.4.1. Assume some set of vectors vi, i ∈ [d], where for each i, j ∈[d] we have vi

|vi| ·vj|vj | ≥ −0.2, is partitioned using the random hyperplane rounding

technique. Then there is at least a probability c that the two sides of the partitionare not of equal size, for some positive constant c, which does not depend on d.

There are at least some intuitive reasons to believe that this may be the case. Inessence, these are based on the fact that any negative correlation must be relativelysmall, and that in the completely uncorrelated case we know that an unbalancedpartition is likely to occur. In Section 3.4.2 we present both some more detailedarguments based on this intuition, as well as some possible approaches for provingthe conjecture.

3.4.1 Utilizing the conjecture

Probability for local improvement

Lemma 3.4.2. Assuming Conjecture 3.4.1, the following holds.

Let a be an all bad vertex of degree d. Then there is a probability of at leastP[Unbalance] ≥ c

210∆that more than half of its neighbor vertices lie on the same

side of the random hyperplane used for rounding.

Proof. Similar to the proof for Lemma 3.2.3, we use Lemma 3.2.2 to determine thatthere is a δ

24probability that va · r ∈ [δ/8, δ]. Again, we denote the neighboring

vectors as uj = (αj , γj).

In order to apply Conjecture 3.4.1, we need to make sure the inner products γi · γjare large enough (doing this for the full products ui ·uj is not sufficient, as theyare more likely to be on a different side of the hyperplane than a). Since a isall bad (again, an edge is bad when vi · vj ∈ [−0.689,−0.687]), we also have that||γj || ≥

√1− 0.6872 ≈ 0.724. We then combine this with the triangle inequality:

24

3.4. ASSUMING UNBALANCE AROUND ALL BAD VERTICES

va ·ui + va ·uj + ui ·uj ≥ −1

=⇒ 2(−0.687) + ui ·uj ≥ −1

=⇒ ui ·uj = αiαj + γi · γj ≥ −1− 2(−0.687) = 0.374

=⇒ γi · γj ≥ 0.374− (−0.689)2 ≥ 0.374− 0.475

=⇒ γi|γi|

·γj|γj |≥ −0.101

0.7242≥ −0.193

Using the union bound, we see that |γj · γr| ≥ c2∆ ||γj || for all vectors uj with proba-

bility at least 1− c2 (since the union bound gives the probability

∑dj=1 Pr[|γj · γr| <

c2∆ ||γj ||] ≤ d · c

2∆ ≤ c/2 for the opposite). As Conjecture 3.4.1 gives a c probabil-ity for an unbalanced partition, we have a probability that is at least c · (1 − c

2) ≥c · (1 − 1

2) = c2 for both these events to occur simultaneously. Due to symmetry,

we also have 12 probability that the larger partition is above the hyperplane. Taken

together with the condition on r1 (as in the proof of Lemma 3.2.3, with a probabilityof δ

24), we have a total probability of at least c

26δ.

Now, we want to choose a δ such that |uj · r| > δ. As in Lemma 3.3.2, this meansthere is no interference between the vertices considered by our analysis. As before,we have

|uj · r| ≥ −0.689δ +1

4∆||γj || ≥

1

4∆· 0.724− 0.689 · δ

and choosing δ ≤ 124∆

the product is greater than δ.

This means that the probability for an all bad vertex to be on the same side of thehyperplane as more than half of its neighbors is P[Unbalance] ≥ c

210∆.

New approximation ratio

Theorem 3.4.3. Assuming Conjecture 3.4.1, the following holds.

Given a graph G with bounded degree ∆, there is an SDP-based algorithm that ap-proximates the Max-Cut problem on G with an expected ratio of at least αGW + ε∆,where ε∆ = c

223∆2 .

Proof. This proof closely follows that of Theorem 3.2.5, with only minor differences.As such, this description is a bit more terse; refer to the former for more details onsome of the steps.

25

CHAPTER 3. RESULTS

First, assume that at most |E|2∆ edges are not bad, implying that at least n2 vertices areall bad. By Lemma 3.4.2, each such vertex is expected to contribute at least δ ≥ c

210∆edges in the local improvement step. Since the vertices covered by the analysis donot affect each other, the total expected contribution is n

2 · c210∆

≥ c210∆2Opt, for a

total approximation ratio of

αGW +c

210∆2

If we have more edges that are not bad, we first consider the case where theycontribute more than εZ to the objective value Z of the SDP. Each good edgehas an expected approximation ratio that is at least αGW + 1

212, giving us a total

approximation ratio of at least

αGW +ε

212

If the good edges contributed less than εZ, we can simply ignore them when runningthe algorithm, for an expected cut of at least

(αGW +c

210∆2)(1− ε)Z

Choosing ε = c211∆2 completes the proof.

3.4.2 Motivation for the conjecture

Intuitively, the conjecture seems reasonable. For a pair of vectors, the bound onvi · vj guarantees there is at least a 1 − arccos(−0.2)

π ≥ 0.435 probability that thetwo end up on the same side of the plane. Iteratively adding more vector pairsseems like it should only increase the probability of an unbalanced partition; morepairs increase the chance that at least one pair is unbalanced, and if more pairs areunbalanced it seems unlikely that they’d counteract each other with a very highprobability (similar to how individual pairs act).

One can also consider the worst-case scenario if the inner products were not lim-ited. In that case, the vectors could be pairwise antipodal (the relationship betweendifferent pairs would be irrelevant), meaning there is a 100% probability that thepartitions are of equal size. There is no obvious way of translating this scenario toone where the inner products are limited, as the products between different pairs

26


20 40 60 80 100 120 140 1605 · 10−2

0.1

0.15

0.2

0.25

0.3

d

Pr[Balan

ced]

SimplexNear-antipodal pairs

Figure 3.2: Measured probability of d vectors being distributed evenlyacross the two partitions by a random hyperplane.

can now be relevant. One approach that is easy to analyze is where the differentpairs are all uncorrelated:

Assume that all but the last pair have already been partitioned by the randomhyperplane. As the vectors are uncorrelated to everything outside the pair, thefinal pair still has the same constant probabilities for each way of partitioning it.Regardless of how the previous pairs have been partitioned, there is at most one wayof achieving partitions of equal size (balanced if the previous vectors were balanced,unbalanced in the opposite direction if they were a single pair off of being balanced).Even if the previous vectors were balanced or near-balanced with probability 1, therewould still be at least a constant probability for the final partitions to be unbalanced.

Another candidate for a worst-case scenario is where the vectors form a regularsimplex. In other words, the inner product between any two vectors is the same:−1d−1 . This essentially means the vectors are as far away as possible from as manyothers as possible. On the other hand, as d increases, this configuration should moreand more behave like a completely uncorrelated one, which would imply a probabilityof 1 − O(1/

√(d) (close to 1 for large values of d) for an unbalanced configuration.

Similar to the previous argument, this would imply at least a constant probability

27

CHAPTER 3. RESULTS

for unbalanced partitions.

To support this notion, we performed random hyperplane rounding both on regularsimplexes and the near-antipodal pairs discussed before (with no correlation betweendifferent pairs), the results of which are shown in Figure 3.2.

3.4.3 An approach for proving the conjecture

In this section, we discuss an approach we attempted, but did not manage to cometo any definite conclusion using. The idea is to investigate the variance (and possiblyhigher-degree moments) of the sum

X =∑

xi , xi ∈ −1, 1

where xi being positive or negative corresponds to which side of the partition i ison. Since E[X] = 0 (due to the symmetry of the random hyperplane), the varianceof this sum can be written as

E[X2] =∑i

∑j

E[xixj ]

= d+∑i 6=j

E[xixj ] = d+ 2∑i<j

E[xixj ]

= d+ 2

(d

2

)avgi 6=j(E[xixj ]) = d+ d(d− 1) avgi 6=j(E[xixj ])

= d · (1 + (d− 1) · avgi 6=j(E[f(vi · vj)])) (3.2)

where d is the number of vertices considered and avgi 6=j(E) is the average value ofE for all i, j ∈ [d] such that i 6= j. f(vi · vj) can be obtained from Lemma 2.3.1:

E[xixj ] = f(vi · vj)= −1 · Pr[Cut] + 1 · (1− Pr[Cut])= 1− 2 ·E[W]

= 1− 2

πarccos(vi · vj)

Using the first-degree Taylor expansion of f(x) and the fact that E[vi · vj ] ≥ −1d−1

3,we can estimate a lower bound for avgi 6=j(E[f(vi · vj)]):

3This can be shown by using f(x) = x in Equation (3.2), combined with E[X2] ≥ 0.

28


E[f(vi · vj)] ≈ E[2vi · vjπ

] =2

π·E[vi · vj ]

≥ 2

π·−1

d− 1(3.3)

Inserting this into (3.2) gives us:

E[X2] ≈ Θ(d)

However, it turns out that this is not necessarily good enough; for instance, wecould have X ≈ ±d with probability about 1

d , and roughly 1 − 1d probability that

the partitions are balanced.

The next step is to examine the fourth degree moment, E[X4]. In the case from theprevious paragraph, we would have E[X4] ≈ d3. Essentially, it should be enoughto show that E[X4] is “significantly” smaller than this. What exactly this meansdepends on how accurate the approximation (3.3) is, but E[X4] ≈ d2 is definitelygood enough (as that would imply a probability of at most ≈ 1

d2that X ≈ ±d).

Expanding E[X4] and using x2i = 1, we get:

E[X4] =∑i

E[x4i ] +

∑i,j

(E[x2ix

2j ] + E[x3

ixj ]) +∑i,j,k

E[x2ixjxk] +

∑i,j,k,l

E[xixjxkxl]

≈ d+ d2 +∑i,j

E[xixj ] + d ·∑i,j

E[xixj ] +∑i,j,k,l

E[xixjxkxl]

For the next step, we would like to show that:

E[xixj ] ≈ Θ(1

d)

E[xixjxkxl] ≈ Θ(1

d2)

However, we were not able to prove that this is the case, leaving this as an openquestion.

3.4.4 Handling balanced pairs

Another possible approach is to show that even if Conjecture 3.4.1 does not hold,we can obtain an improved approximation ratio. In this section, we will describe an

29

CHAPTER 3. RESULTS

additional local improvement step. Where the previous improvement step workedon vertices with an uneven distribution of neighbors, this step deals with some ofthe vertices that are balanced (i.e. there is an equal amount of neighbors on eachside of the partition). Do note that the results presented here are not enough tocover every case where the conjecture is false, it may be a useful starting point.

More specifically, the case we are interested in is the one where two adjacent vertices iand j are both balanced, and i and j are on different sides of the partition. Referringto such (i, j) as a balanced pair, we can perform the following local improvement step:

• For each balanced pair (i, j), move both i and j to the opposite sides of thepartition

This could be incorporated in the ACut algorithm in a number of different ways,though combining it with the other local improvement step seems good as the twomay provide opportunities for each other.

i j

+ -

B(i)

+

-

B(j)

+

-

Figure 3.3: If i and j form a balanced pair, i is initially “positive”,while j would be “negative”. The sets B(i) and B(j) are each balanced,meaning they (individually) have an equal number of vertices on eachside of the hyperplane.

Lemma 3.4.4. If there are ωn balanced pairs in a graph with bounded degree ∆,performing the above local improvement step will increase the approximation ratioby at least ω

∆2 .

Proof. First, we look at the improvement to the cut from altering one pair (asillustrated by Figure 3.3). Disregarding the edge between i and j, both verticeshave d/2 (for their individual degrees d - which are not necessarily equal) neighborson the same side, and d/2−1 on the other side. If we swap the sides of i and j, thesevalues are reversed. In other words, we gain d/2− (d/2− 1) = 1 cut edges on each

30


side. Since the edge between i and j will still be cut, we thus have an improvementof 2 edges for each balanced pair.

Altering one such pair may affect up to 2(∆− 1) other pairs. Thus, performing theimprovement will yield at least 2ωn · 1

2(∆−1) ≥Optω∆2 additional cut edges.

31

Chapter 4

Discussion

4.1 Conclusions

We were able to show that relatively small alterations to existing algorithms allowus to obtain an improved approximation ratio. Specifically, we have an expectedapproximation ratio of at least αGW + 1

234∆3 for graphs of bounded degree ∆. Com-pared to the original approximation ratio by Feige, Karpinski, and Langberg (2002)(where the additive term was 1

233∆4 ), we have a better approximation ratio for any∆ > 2. This improvement stems from our algorithm avoiding the interference be-tween candidate vertices, which contributed a factor 1

∆ in the previous result.

We were also able to show an improved approximation ratio for graphs of boundeddegree ∆ with at least a constant ω fraction of vertices with odd degree. Specifically,the additive term ε∆ = 1

233∆4 from the general case was replaced with a term εodd∆ =ω

224∆2 . This is an improvement whenever ω > 129∆2 , which is true for large groups

of graphs (e.g. random graphs where the expected number of odd-degree vertices isreasonably large, and probably many graphs involved in real applications).

Assuming that Conjecture 3.4.1 holds, a similar result to the odd-degree one canbe applied to the general bounded-degree case as well, where the additive term isΩ( 1

∆2 ). Some reasons to believe this may be the case were presented, though noconclusive proof was found.

4.2 Future work

4.2.1 Probability of unbalanced neighbors

As mentioned before, proving Conjecture 3.4.1 could substantially improve the ap-proximation ratio for bounded-degree graphs. Of course, it is not known whether

33

CHAPTER 4. DISCUSSION

this is actually possible. The experimental evidence in Section 3.4.2 suggests thatthere is at least a chance the conjecture is true.

The intuitive notion that adding more pairs of vectors should decrease the chance ofthem being partitioned equally also suggests an approach for proving the conjecturethrough induction.

The moment-based approach described in Section 3.4.3 may also be useful in provingthe conjecture, though we were unable to do so during the work on this thesis.

4.2.2 Utilizing balanced pairs

There are a few obstacles remaining in order to be able to use the balanced pairhandling described in Section 3.4.4.

One approach is to assume that there is a large number of edges belonging to bal-anced vertices (or vertices with a high probability of being balanced). If so, weshould also have a relatively large number of balanced pairs. Otherwise, we havea large number of unbalanced vertices (or vertices with a high probability of beingunbalanced). This sounds good - unbalanced vertices is what we wanted in theprevious analysis. However, we also want the corresponding vector to lie close tothe hyperplane. Otherwise, its neighbors will most likely be on the other side ofthe plane - and this could be the reason the vertex had a high probability of beingunbalanced in the first place. Thus, further research is needed to determine whetherthis approach can yield any results.

Another way of looking at it is that the problematic cases are those where:

1. The neighbors ui are unbalanced with high probability

2. Their projections orthogonal to v are balanced with high probability

The converse of the first would imply that balanced pairs are common, while theconverse of the second would allow simple local improvements (essentially what isimplied by Conjecture 3.4.1). Thus, finding a way to handle those cases could beanother way to improve the approximation ratio.

4.2.3 Using Hierarchies

Semidefinite programs can be strengthened by adding additional constraints. Hier-archies give a structured approach for adding such local constraints. Chlamtac and

34

4.2. FUTURE WORK

Tulsiani (2012) describe several of these, but based on the comparison by Laurent(2003) the Lasserre hierarchy seems the most promising.

The Lasserre hierarchy works in levels, with increasing levels giving tighter relax-ations. Each level adds additional variables and corresponding constraints for in-creasingly large subsets of the original problem’s variables. Solving the relaxationat a given level t takes time nO(t) (i.e. a constant t gives a solution in polynomialtime).

It may be possible to obtain an improved approximation ratio by using these hi-erarchies. For instance, the constraints added at t = 2 are similar the triangleinequalities (2.2) and (2.3), which played an important role in the proof of the maintheorem by Feige, Karpinski, and Langberg (2002).

35

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Klerk, Etienne de (2002). Aspects of Semidefinite Programming: Interior Point Al-gorithms and Selected Applications (Applied Optimization). Springer. isbn: 978-0-306-47819-2.

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Trevisan, Luca (2001). “Non-approximability results for optimization problems onbounded degree instances”. In: Proceedings on 33rd Annual ACM Symposium onTheory of Computing, July 6-8, 2001, Heraklion, Crete, Greece. ACM, pp. 453–461. isbn: 1-58113-349-9.

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