The Capacitated Vertex Cover Problem with Relaxed Constraints
Theorem 13. Algorithm DUAL-HCVC computes a (δ + 2)-approximation for the unweighted capacitated domination problem with hard capacities in polynomial time
≤ 2 · d∗(v)· ye.
Let hOPT denote an optimal demand assignment for the input graph and xhOPT denote the corresponding multiplicity function. By charging the cost incurred by vertices in V3to the cost incurred by hOPT, we have the following lemma.
Lemma 12. We have∑
v∈V3x(h)v ≤ δ · w(hOPT).
Proof. For any v ∈ V3. We have x(h)v = 1 since d∗(v) < cv. For any e ∈ E[v] such that he,v > 0, we know that hOPT e,v′ > 0 for some v′ ∈ e. In other words, for such a vertex v there exist an edge and a vertex e, v′, where e ∈ E[v] and v′ ∈ e, such that he,v > 0, hOPT e,v′ > 0, and x(hv′OPT) ≥ 1. We charge the cost incurred by v in our solution to the cost incurred by v′ in the optimal solution.
Consider any v′ ∈ V such that x(hv′OPT) > 0 and any e ∈ E[v′] such that hOPT e,v′ > 0.
According to the design of our algorithm, we know that the demand of e is assigned in one particular iteration, and hence it can only be assigned to at most one vertex in V3 since at most one vertex could be classified into V3 in any iteration. Therefore v′ can be charged by at most δ′v times. Since δv′ ≤ δ, it follows that∑
v∈V3x(h)v ≤ δ · w(hOPT).
By combining the upper-bounds we obtained for the three sets V1, V2, and V3, and the dis-cussion in subsection §4.2.2, we have the following theorem.
Theorem 13. Algorithm DUAL-HCVC computes a (δ + 2)-approximation for the unweighted capacitated domination problem with hard capacities in polynomial time.
4.4 (k, (1 +
k−11)(f − 1))-augmented-cover for Augmented Cover
In this section, we establish the following theorem:
Theorem 14. For any integer k ≥ 2, we can compute a (
k, (1 +k−11 )(f − 1))
-augmented-cover for HCVC in polynomial time.
Let m′v = k· mvdenote the augmented multiplicity function for each v ∈ V . We invoke al-gorithm Dual-HCVC on the instance Π′ = (V, E, d, w, c, m′). Let h be the demand assignment and Ψ = (y, z, g, η) be the dual solution output by the algorithm for Π′.
The following observation is crucial in establishing the bi-approximation ratio: The dual solution Ψ, which was computed for instance Π′, is also feasible for input instance Π.
Lemma 15. Ψ is feasible for LP (4.2) with respect to Π. In other words, we have
∑
e∈E
de· ye−∑
v∈V
mv · ηv ≤ OP T (Π).
Proof. The statements follow directly since LP (4.2) has the same feasible region for Π and Π′.
It is also worth mentioning that, the assignment h computed by Dual-HCVC already gives an augmented (
k, (1 +k−11 )f)
-augmented-cover. To obtain our claimed ratio, however, we further modify some of the demand assignments in h to achieve better utilization on the residue capacity of the vertices.
Let VS denote the set of vertices that have been included in S. For each v ∈ V such that Dh(v) < cv, let ℓv denote the function given by Lemma 10 with respect to v.
We use h∗to denote the resulting assignment to obtain, where h∗ is initialized to be h. For each e∈ E, we repeat the following operation until no such vertex pair can be found:
• Find a vertex pair u∈ e \ VSand v ∈ e such that
Intuitively, in assignment h∗if some demand is currently assigned to a vertex in V \ VSthat requires multiple multiplicities, then we try to reassign it to vertices that have surplus residue capacity (according to the function ℓv) to balance the load. Note that, the reassignments are performed only between vertices not belonging to VS.
We use h∗ to denote the resulting assignment to obtain, where h∗ is initialized to be h.
In assignment h∗, if some demand is currently assigned to a vertex in V \ VS that requires multiple multiplicities, we try to reassign it to vertices that have surplus residue capacity (ac-cording to the function ℓv in Lemma 10) to balance the load.
The following lemma shows that, the cost incurred by vertices in V \ VS can be distributed to the dual variables of the edges.
Lemma 16. We have consider the following three exclusive cases separately:
1. If Dh∗(v) > cv, then by Proposition 8 we have
In this case we charge the cost incurred by v to the demand it serves, where each unit of demand, say, from edge e, gets a charge of 2· ye.
2. If Dh(v) > cv ≥ Dh∗(v), then we know that x(hv ∗) = 1. By Proposition 8, we have
wv · x(hv ∗) = wv = cv· zv < ∑
e∈E[v]
he,v· ye,
where the last inequality follows from the assumption that Dh(v) > cv. In this case we charge the cost of v to the demand that was assigned to it in the original assignment h, where each unit demand gets a charge of ye.
3. If cv ≥ Dh(v), then we know that h∗e,v ≤ ℓv(e) for all e ∈ E[v] by Lemma 10 and the way how h∗is modified. Therefore, we have x(hv ∗) = 1 and Lemma 10 states that
wv· x(hv ∗)= ∑
e∈E[v]
ℓv(e)· ye.
In this case, we charge the cost incurred by v to the demand that is located in ℓv, each of which gets a charge of ye.
Consider any unit of demand from an edge e∈ E and the number of charges it gets in the above three cases. Depending on the assignment h∗, we have the following three cases.
(a) If the unit demand is assigned to a vertex in VS, then it is charged at most (f − 1) times, i.e., at most once in case (3) above by its remaining incident vertices.
(b) If the unit demand is assigned to a vertex v ∈ V \ VSwith Dh∗(v) > cv, then it is charged twice in case (1) above by v.
(c) If the unit demand is assigned to a vertex v ∈ V \ VSwith Dh∗(v)≤ cv, then it is charged at most f times, i.e., at most once by all of its incident vertices in case (2) and (3) above.
Since f ≥ 2, summing up the discussion we obtain the statement as claimed.
The following lemma provides a lower bound for OPT(Π) in terms of the net sum of the that h∗e,v > 0. Therefore, it follows that
mv · ηv ≤ mv· cv· ye ≤ 1 k · ∑
e∈E[v]
h∗e,v· ye. (4.3)
By Inequality (4.3) and Lemma 15, it follows that
∑
where the last inequality follows from Inequality (4.4).
In the following we establish the bi-criteria approximation factor and prove Theorem 14.
Lemma 18. We have
Proof. By Lemma 9, we have Dh(v) = m′v · cv = k· mv · cv for any v ∈ VS. Therefore,
The former item is upper-bounded by k · OPT(Π) by Lemma 15. Combing the above with Lemma 17, we obtain
Lemma 19. The running time of procedure Dual-HCVC with demand reassignment is O(|V |2· Tmaxf low(V ) + f · |E|).
Proof. The running time of reassignment is O(f · |E|) because we check all incident vertices for each edge. By Lemma 7, we have that the running time of our augmented cover algorithm is O(|V |2· Tmaxf low(V ) + f · |E|).