6.5.1 Orthonormal representations
Definition 6.5.1. Anorthonormal representationofG, abbreviated asONR, con-sists of a set of unit vectorstu1, . . . , unu ĎRd(for somedě1) satisfying
uTiuj“0 @ti, ju PE.
IfSis a stable set inGand theui’s form an ONR ofGof dimensiond, then the vectorsui labeling the nodes ofS are pairwise orthogonal, which implies thatděαpGq. It turns out that the stronger lower bound: děϑpGqholds.
Lemma 6.5.2. The minimum dimensiondof an orthonormal representation of a graphGsatisfies:ϑpGq ďd.
Proof. Let u1,¨ ¨ ¨ , un P Rd be an ONR of G. Define the matrices U0 “ Id, Ui “uiuTi P Sd fori P rns. Now we define a symmetric matrixZ P Sn`1 by settingZij “ xUi, Ujyfori, jP t0u Y rns. One can verify thatZis feasible for the program (6.14) definingϑpGq(check it) withZ00“d. This givesϑpGq ďd.
6.5.2 The theta body THpGq
It is convenient to introduce the following set of matrices X P Sn`1, where columns and rows are indexed by the sett0u YV:
MG“ tY PSn`1:Y00“1, Y0i“Yii piPVq, Yij “0pti, ju PEq, Y ľ0u, (6.17) which is thus the feasible region of the semidefinite program (6.15). Now let THpGqdenote the convex set obtained by projecting the setMGonto the sub-spaceRV of the diagonal entries:
THpGq “ txPRV :DY PMG such thatxi“Yii@iPVu, (6.18) called thetheta bodyofG. It turns out that THpGqis nested between STpGqand QSTpGq.
Lemma 6.5.3. For any graphG, we have thatSTpGq ĎTHpGq ĎQSTpGq.
Proof. The inclusion STpGq ĎTHpGqfollows from the fact that the characteris-tic vector of any stable set lies in THpGq(check it). We now show the inclusion THpGq Ď QSTpGq. For this pick a vector xP THpGqand a clique C ofG; we show thatxpCq ď1. Sayxi “Yii for all iPV, whereY PMG. Consider the principal submatrixYCofY indexed byt0u YC, which is of the form
YC“
ˆ1 xTC xC DiagpxCq
˙ ,
where we setxC “ pxiqiPC. Now,YC ľ0 implies that DiagpxCq ´xCxTC ľ 0 (taking a Schur complement). This in turn implies:eTpDiagpxCq ´xCxTCqeě0, which can be rewritten asxpCq ´ pxpCqq2ě0, givingxpCq ď1.
In view of Lemma 6.4.3, maximizing the all-ones objective function over THpGqgives the theta number:
ϑpGq “max
xPRV
teTx:xPTHpGqu.
As maximizingeTxover QSTpGqgives the LP boundα˚pGq, Lemma 6.5.3 im-plies directly that the SDP boundϑpGqdominates the LP boundα˚pGq:
Corollary 6.5.4. For any graphG, we have thatαpGq ďϑpGq ďα˚pGq.
Combining the inclusion from Lemma 6.5.3 with Theorem 6.2.2, we deduce that THpGq “ STpGq “ QSTpGq for perfect graphs. It turns out that these equalities characterize perfect graphs.
Theorem 6.5.5. For any graphGthe following assertions are equivalent.
1. Gis perfect.
2. THpGq “STpGq 3. THpGq “QSTpGq.
4. THpGqis a polytope.
6.5.3 More on the theta body
There is a beautiful relationship between the theta bodies of a graphGand of its complementary graphG:
Theorem 6.5.6. For any graphG,
THpGq “ txPRVě0:xTzď1@zPTHpGqu.
In other words, we know an explicit linear inequality description of THpGq;
moreover, the normal vectors to the supporting hyperplanes of THpGqare pre-cisely the elements of THpGq. One inclusion is easy:
Lemma 6.5.7. IfxPTHpGqandzPTHpGqthenxTzď1.
Proof. LetY PMG andZ PMG such thatx“ pYiiqandz “ pZiiq. LetZ1 be obtained fromZ by changing signs in its first row and column (indexed by0).
ThenxY, Z1y ě 0 asY, Z1 ľ 0. Moreover,xY, Z1y “ 1´xTz (check it), thus givingxTzď1.
Next we observe how the elements of THpGqcan be expressed in terms of orthonormal representations ofG.
Lemma 6.5.8. ForxPRVě0,xPTHpGqif and only if there exist an orthonormal representationv1, . . . , vnofGand a unit vectordsuch thatx“ ppdTviq2qiPV. Proof. Letd, vibe unit vectors where thevi’s form an ONR ofG; we show that x“ ppdTviq2q PTHpGq. For this, letY PSn`1denote the the Gram matrix of the vectorsdandpviTdqviforiPV, so thatx“ pYiiq. One can verify thatY PMG, which impliesxPTHpGq.
For the reverse inclusion, pick Y P MG and a Gram representationw0, wi (i P V) of Y. Set d “ w0 andvi “ wi{}wi} fori P V. Then the conditions expressing membership of Y in MG imply that the vi’s form an ONR of G, }d} “1, andYii“ pdTviq2for alliPV.
To conclude the proof of Theorem 6.5.6 we use the following result, which characterizes which partially specified matrices can be completed to a positive semidefinite matrix – you will prove it in Exercise 6.1.
Proposition 6.5.9. Let H “ pW, Fq be a graph and let aij (i “ j P W or ti, ju P F) be given scalars, corresponding to a vector a P RWYF. Define the convex set
Ka“ tY PSW :Y ľ0, Yij “aij @i“j PW andti, ju PFu (6.19) (consisting of all possible positive semidefinite completions ofa) and the cone
CH “ tZ PSW :Zľ0, Zij “0@ti, ju PFu (6.20) (consisting of all positive semidefinite matrices supported by the graphH). Then, Ka ‰ Hif and only if
ÿ
iPW
aiiZii`2 ÿ
ti,juPF
aijZij ě0 @Z PCH. (6.21)
Proof. (of Theorem 6.5.6). Letx P RVě0 such thatxTz ď1 for allz P THpGq;
we show that x P THpGq. For this we need to find a matrixY P MG such thatx“ pYiiqiPV. In other words, the entries ofY are specified already at the following positions: Y00 “ 1, Y0i “ Yii “ xi fori P V, and Yti,ju “0 for all ti, ju PE, and we need to show that the remaining entries (at the positions of non-edges ofG) can be chosen in such a way thatY ľ0.
To show this we apply Proposition 6.5.9, where the graphH is Gwith an additional node 0 adjacent to all i P V. Hence it suffices now to show that xY, Zy ě0for allZ PSľ0t0uYV withZij “0ifti, ju PE. Pick suchZ, with Gram representation w0, w1,¨ ¨ ¨, wn. ThenwiTwj “ 0 if ti, ju P E. We can assume without loss of generality that all wi are non-zero (use continuity if some wi
is zero) and up to scaling thatw0 is a unit vector. Then the vectors wi{}wi} (iPV) form an ONR ofG. By Lemma 6.5.8 (applied toG), the vectorzPRV withzi “ pw0Twiq2{}wi}2 belongs to THpGqand thusxTz ď 1 by assumption.
Therefore,xY, Zyis equal to 1`2ÿ
iPV
xiw0Twi`ÿ
iPV
xi}wi}2ěÿ
iPV
xi
ˆpwT0wiq2
}wi}2 `2wT0wi` }wi}2
˙
“ÿ
iPV
xi ˆw0Twi
}wi} ` }wi}
˙2 ě0.