2.6 Power Schemes
2.6.1 Ordering Power Schemes
Let us consider the following partial order ’≺’ on the set of power schemes.
Definition 2.17. For any two power schemes P and Q, we write P ≺Q if and only if P grows slower than Q when considered as a function of link-length, i.e. for any two links i, j such thatli ≥lj it holds thatP(i)/P(j)≤Q(i)/Q(j).
In particular, it is easy to see thatL ≺ M ≺ U. More generally, we have Lt≺ Lt0 for t≤t0.
Definition 2.18. We say that a power schemePbelongs to the class ofsub-linear power schemes if P ≺ L. We say that P belongs to the class of super-linear power schemes if L ≺P.
Note that we includeLin both of these classes. This is because (as we will see afterwards) it shares common properties with other power schemes from both of these classes.
The following is a simple but useful fact that follows from the definition of the partial order ≺. For a set of linksS and a linki, we make the following notation:
S[i+] ={j ∈S :lj ≥li}and S[i−] ={j∈S :lj ≤li},
i.e. S[i+] (S[i−]) is the subset of S that consists of the links that are longer (shorter) than the link i.
Lemma 2.19. Suppose that power schemesP andQare such that P ≺Q. For any set of links S and a linki it holds that
ISRQ(S[i−], i)≤ISRP(S[i−], i) andISRP(S[i+], i)≤ISRQ(S[i+], i).
Proof. Consider any link j∈S[i−]\ {i}. SinceP ≺Q andli ≥lj, we haveP(i)/P(j)≤ Q(i)/Q(j); hence,
ISRQ(j, i) = min (
1, Q(j)lαi Q(i)dαji
)
≤min (
1, P(j)lαi P(i)dαji
)
=ISRP(j, i), which proves the first part. The second part is obtained similarly.
Chapter 3
The Complexity of Scheduling
In this chapter we show that the scheduling problem is hard for a large family of power schemes, including the uniform, linear and mean power schemes. We use a reduction from the N P-complete problem Partiton, similarly to [GOW07]. For power schemes Lt with t < 1, i.e. sublinear power schemes Lt, we show that the scheduling problem defined on simple instances as the ones with nodes placed on a straight line isN P-hard and it is N P-hard to approximate the scheduling problem for these instances within an approximation factor of ρ < 3/2. For the linear power scheme, we show that it is N P-hard to approximate the scheduling problem defined in general metric spaces within a factor ofρ <3/2.
3.1 Complexity of Scheduling on the Line with Sub-Linear Power Schemes
In this section we show that a special family of scheduling problems defined on straight line is N P-complete, and prove an inapproximability result. The decision problem of scheduling with respect to a power schemeP is stated as follows.
LineScheduling(P). Given a set of links Γ on the Euclidean plane, the SINR param-eters α > 1, β ≥ 1, N ≥ 0, and a natural number k > 0, the question is if there is a partition of that set into not more than k SINR-feasible subsets with respect to the power scheme P.
It is easy to see, that LineScheduling(P) is in N P: given a solution to the problem, one needs only to check if the subsets are SINR-feasible, which can be done in polynomial time. We consider power schemesLt with t6= 1, where for each link iLt(i) =clαti , and show that for these power schemes the problem is N P-complete. In order to prove
30
theN P-completeness, we reduce theN P-complete problemPartition (defined below;
see also [GJ79]) to (LineScheduling(Lt)) for all t 6= 1, by modifying the construction of [GOW07]. Let us denote by ΣA the sum of elements of a set of integers A: ΣA = P
a∈Aa.
Partition. Given a finite set A =a1, a2, . . . , an of positive integers, the question is if there is a subsetA0 ⊆A such that ΣA0 = ΣA\A0 holds.
When it is not ambiguous, we will identify an instance of Partition with the corre-sponding set of integersA.
Remark. Whenever it is necessary, we can assume that the integers in question are distinct (note that this is assumed in [GOW07], but no justification is presented). This can be justified based on the proof of N P-completeness of Partition by a reduction from an N P-complete problem 3-Dimensional Matching [GJ79]. For each instance of 3-Dimensional Matching, an instance of Partition is constructed which is as follows: A=C∪ {b1, b2}withb1= ΣC+t, b2 = 2ΣC−t, where, by definition,C is a set of distinct integers (these integers are given by their binary representation and each two of them differ by the value of at least one bit), andt≤ΣC. It follows thatb1 ≥ΣC and b2 ≥ΣC; hence, in any valid partition, b1and b2 must be in different subsets. Ifb1 6=b2, we have a Partitioninstance with distinct integers, otherwiseb1 =b2 and the problem reduces to an instance with distinct integers by just removing b1 and b2. This means that the Partition problem restricted to sets of distinct integers is also N P-complete.
We refer the reader to [GJ79] for the details of the reduction.
Theorem 3.1. For each t < 1, LineScheduling(Lt) is N P-complete. Moreover, the corresponding optimization problem cannot be approximated within a factor less than 32 unless P =N P.
Proof. Let us fix a real number t < 1 and show that LineScheduling(Lt) is N P-complete. We will reduce thePartition problem to the case of the scheduling problem with k = 2 and N = 0, i.e. it is asked if the given set of links can be scheduled into two feasible subsets without environmental noise. Let an instance A={a1, a2, . . . , an} of Partition be given, and assume that the numbers ai are distinct (by virtue of the remark above). We construct the corresponding instance of LineScheduling(Lt) as follows. The set of links Γ consists of n+ 2 links: Γ ={1,2, . . . , n+ 2}. The links are arranged on a straight line with a given coordinate system (Fig.3.1). The coordinate of sender node si is (ΣA/ai)1/αfor i≤n. We place the receiver noderi at the coordinate (ΣA/ai)1/α+`, the value of ` to be defined afterwards. Note that li =`for i≤n. Let δ = mini6=j|(ΣA/ai)1/α−(ΣA/aj)1/α|. We have that δ > 0 because we assumed that the numbers ai are distinct. We choose `to be strictly less than δ. For all i, j≤n, we
Figure 3.1: The network example for Theorem 3.1.
have dij ≥δ−`. The receiver nodes rn+1 and rn+2 are placed at the same point with coordinate 0. The sender nodes sn+1 and sn+2 are placed at the point with coordinate
−`0, the value of `0 to be defined afterwards. It follows from the construction that ln+1 =ln+2 =`0, and the links n+ 1 and n+ 2 cannot be in the same feasible subset.
First let us show that ISRLt(Γ, i)<1/β for alli≤n, if the value of ` is chosen to be small enough. Consider any link i with i ≤ n and let us estimate the contribution of other links in ISRLt(Γ, i). First consider the links on the right side of i. Since the links are arranged on the line anddji≥δ−`, this sum can be bounded by the following:
where at the last step we used a well known bound for the series P∞ i=1 1
iα which can be proven using Cauchy method (see Lemma A.1 in the appendix). We take`to be small enough for the sum to be bounded by 4β1 (this can be done because (3.1) holds). Note that the contribution of the links on the left side of linki, except linksn+ 1 andn+ 2, inISRLt(Γ, i), can be bounded by the same value; hence,
We choose`0 =
Thus, we finish the construction of theLineScheduling(Lt)instance corresponding to A.
It is easy to check that Γ can be scheduled into two SINR-feasible subsets if and only if A has a partition. Indeed, suppose that A admits a valid partition (A0, A\A0), and means that the set of links corresponding to indices IA0 ∪ {n+ 1} is feasible, as well as the set of links corresponding to IA\A0 ∪ {n+ 2}, which gives a schedule of Γ into two index sets of a valid partition of A. This completes the proof ofN P-completeness.
In order to prove the inapproximability result, let us assume the contrary, i.e. sup-pose that there is a polynomial algorithm that approximates the scheduling problem w.r.t. Lt within a factor of ρ <3/2. Consider any instance A of the Partition prob-lem with distinct positive integers. Let us construct the corresponding instance Γ of LineScheduling(Lt). If we apply the hypothetical algorithm to Γ, we will find a schedule with a number of subsets differing from the optimum by a factor at most ρ.
If the optimum schedule length is at most two then the algorithm will schedule Γ into at most 2ρ < 3 subsets, i.e. at most 2 subsets. Thus, Γ can be scheduled into two subsets if and only if applying our approximation algorithm to Γ returns at most two subsets. This means that we can solve the instanceA in polynomial time, which might be possible only if P =N P.