Improved Algorithms for Vertex Cover with Hard
Capacities on Multigraphs and Hypergraphs
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Citation
Cheung, Wang Chi, Michel X. Goemans, and Sam Chiu-Wai
Wong. “Improved Algorithms for Vertex Cover with Hard
Capacities on Multigraphs and Hypergraphs.” Proceedings of the
Twenty-Fifth Annual ACM-SIAM Symposium on Discrete
Algorithms (December 18, 2013): 1714–1726. © 2014 the
Society for Industrial and Applied Mathematics
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http://dx.doi.org/10.1137/1.9781611973402.124
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Improved Algorithms for Vertex Cover with Hard Capacities on
Multigraphs and Hypergraphs
Wang Chi Cheung∗, Michel X. Goemans†, Sam Chiu-wai Wong‡
Abstract
In this paper, we consider the minimum unweighted Vertex
Cover problem with Hard Capacity constraints (VCHC) on
multigraphs and hypergraphs. Given a graph, the objective
of VCHC is to find a smallest multiset of vertices that cover
all edges, under the constraints that each vertex can only
cover a limited number of incident edges, and the number
of available copies of each vertex is bounded. This problem
generalizes the classical unweighted vertex cover problem.
Here we restrict our attention to unweighted instances, since
the weighted version of VCHC is as hard as the set cover
problem, as shown by Chuzhoy and Naor (FOCS 2002).
We obtain improved approximation algorithms for
VCHC on multigraphs and hypergraphs. This problem
has first been studied by Saha and Khuller (ICALP 2012).
They proposed a 38-approximation for multigraphs, and a
max {6f, 65}-approximation for hypergraphs, where f is the
size of the largest hyperedge. In this paper, √
we significantly
improve these approximation ratios to 1 + 2/ 3 < 2.155 and
2f respectively. In the case of multigraphs, our approximation ratio is very close to the longstanding bound of 2 for
the classical vertex cover problem. Our algorithms consist of
a two-step process, each based on rounding an appropriate
linear program. In particular, for multigraphs, the analysis
in the second step relies on identifying a matching structure
within any extreme point solution.
Furthermore, we consider the partial VCHC problem in
which one only needs to cover all but ` edges. We propose
a generic reduction from partial VCHC on f -hypergraphs
to VCHC on (f + 1)-hypergraphs, with a small loss in
the approximation factor. In particular, we present a
(2f + 2)(1 + )-approximation algorithm for partial VCHC
on f -hypergraphs.
1 Introduction
The minimum vertex cover problem is one of the most
well-studied combinatorial optimization problems. In
this classical problem, we are given a graph G = (V, E)
and a weight for every vertex v ∈ V , and the objective is to select a minimum weight subset U ⊂ V such
that for all e ∈ E, e is incident to at least one vertex
u ∈ U . It is well known that this problem has a 2approximation algorithm by linear programming techniques (Hochbaum[8] and Bar-Yehuda and Even [1]).
∗ MIT, wangchi@mit.edu. Supported by graduate fellowship
from A*STAR (Agency for Science, Technology And Research,
Singapore).
† MIT, goemans@math.mit.edu. Supported in part by NSF
contract CCF-1115849 and by ONR grant N00014-05-1-0148.
‡ MIT, samwong@mit.edu.
This also extends to an f -approximation algorithm for
f -hypergraphs, which are hypergraphs with largest edge
size ≤ f . On the other hand, assuming the Unique
Game Conjecture, [9] shows that, for any f ≥ 2, approximating the vertex cover problem on f -hypergraphs
better than f − is NP-hard, for any > 0.
In this paper, we offer improved approximation
algorithms for the unweighted minimum vertex cover
with hard capacity constraints (VCHC) on multigraphs
and hypergraphs. In these problems, we are given a
graph G = (V, E), which can have parallel edges. The
objective is to find a minimum vertex cover, but under
the constraints that each vertex v can cover at most kv
incident edges and that we can only select up to mv
copies of v for each vertex v. These problems generalize
the classical unweighted vertex cover problem.
Throughout this paper, we focus on unweighted instances, i.e. all vertices have weight 1. It is because
the weighted version of VCHC for simple graphs is already as hard as set cover (Chuzhoy and Naor [4]), while
the unweighted versions for multigraphs and hypergraphs admit constant approximation algorithm (Saha
and Khuller [10]). This property makes many standard
techniques such as primal-dual algorithms, LP rounding or iterative rounding methods not directly applicable for a constant approximation, since very often
those techniques provide the same approximation ratio
on weighted and unweighted instances.
1.1 Prior Work. The work on minimum capacitated
vertex cover problems falls in two categories: hard capacities, where there is an upper bound mv on the
number of available copies of v, and soft capacities,
where there is no upper bound, i.e. mv = ∞. The
weighted vertex cover problem with soft capacity constraints is first proposed by Guha et al. [7], which
give a 2-approximation primal-dual algorithm. Subsequently, another 2-approximation via dependent randomized rounding is provided in Gandhi et al. [6].
On the other hand, the vertex cover problem with
hard capacity constraints (VCHC) on simple graphs is
first studied by Chuzhoy and Naor [4]. They propose an
elegant 3-approximation algorithm for the unweighted
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problem, through randomized rounding followed by a
patching procedure. On the other hand, it is shown in
[4] that the weighted version of VCHC is as hard as the
set cover problem. Thus, subsequent work on VCHC
has focused on unweighted instances. In particular, a
2-approximation algorithm for VCHC on simple graphs
has been obtabied by Gandhi et al. [5], by refining the
approach in [4].
Nevertheless, as pointed out by Saha and Khuller
[10], the algorithms in [4, 5] have no approximation
guarantee for multigraphs. These are indeed randomized algorithms, and the presence of parallel edges induces positive correlation for some random variables in
their analyses. This hinders the use of certain concentration inequalities vital for obtaining a constant approximation guarantee. In fact, achieving a constant approximation guarantee for VCHC for multigraphs is an
open problem posed in [4]. Recently, this was settled by
Saha and Khuller [10], who derive a 34-approximation
algorithm in the case of unit multiplicities (mv = 1 for
all v ∈ V ), and a 38-approximation for general multiplicities. They also give a max{6f, 65}-approximation
algorithm for f -hypergraphs. These algorithms are
based on a rounding paradigm, followed by a randomized clustering procedure on an appropriate multisetmulticover instance.
The partial vertex cover problem is a generalization
of the vertex cover problem in which we are given an
integer ` and we only need to cover all but ` edges in
the graph. Capacitated versions are similarly generalized. The partial vertex cover problem without capacity
constraints was first studied by Bshouty and Burroughs
[3], which give a 2-approximation algorithm. For partial capacitated vertex cover with soft capacities, BarYehuda et al. [2] derive a 2-approximation algorithm
for simple graphs and a 3-approximation algorithm for
multigraphs. Both algorithms work for weighted instances and are based on local ratio techniques. For the
unweighted partial vertex cover with hard capacities,
Saha and Khuller [10] have announced without details
an O(f )-approximation algorithm for f -hypergraphs.
1.2 Our Contributions and Approach. Our main
contributions are the following. We obtain:
√
• A 1 + 2/ 3 < 2.155-approximation algorithm for
VCHC on multigraphs, which improves over the
previous approximation ratio of 38 in [10]. In particular, our ratio is very close to the longstanding
bound of 2 for classical vertex cover problem.
• A 2f -approximation for VCHC on hypergraphs,
which improves over the previous approximation
ratio of max{65, 6f } in [10].
• A generic reduction from partial VCHC on f hypergraphs to VCHC on (f + 1)-hypergraphs.
In particular, we obtain a (2f + 2)(1 + )approximation for partial VCHC on f -hypergraphs.
Our algorithm for multigraphs is a two-step process,
each based on rounding an appropriate linear program.
In the first step, we solve a natural LP relaxation to
VCHC on multigraphs. Based on an optimal solution
to this LP, we select a multiset U of vertices such that
every edge has one of its ends in U ; viewed as a set, U
is a vertex cover. However, U might not have enough
capacity to cover all edges. Thus, in the second step,
for each uncovered edge, we associate its deficit with
its end in U . We then construct a covering LP whose
solution provides enough coverage to offset the deficits
of all vertices u ∈ U . By rounding up an extreme point
solution to this covering LP and adding it to U , we have
now enough caacity to cover all the edges.
We would like to contrast our algorithm with previous algorithms on VCHC. On one hand, the algorithms
in [4, 5, 10] are also two-step processes, similar to ours.
On the other hand, they perform randomized rounding
in their second steps, while we solve an optimization
problem. Therefore, unlike [4, 5], our approach allows us
to bypass the issue of positive correlation among parallel
edges. Furthermore, our optimization approach allows
us to carry out a tighter analysis than that in [10], which
involves a complicated analysis on its randomized clustering procedure. This leads to a significant improvement in the approximation guarantee. Our analysis in
the second step relies on identifying a matching structure within any extreme point solution of the covering
LP, which is vital for establishing the approximation
ratio.
Our algorithm for hypergraphs is also a two-step
process. Similar to the case of multigraphs, we first solve
a natural LP relaxation of VCHC on hypergraphs for an
optimal solution, and define a multiset of vertices U such
that every hyperedge has some vertex in U . However,
in the second step, the construction of the covering LP
requires more work. Indeed, an uncovered edge may
have more than one vertex in U , and it is unclear which
u ∈ e ∩ U to associate the deficit of the hyperedge e
with. We resolve this problem by splitting the deficit
of e among e ∩ U appropriately. We then set up the
covering LP in a similar way to the case of multigraphs,
and round up its extreme point solution and include the
correponding vertices into the solution.
Organization. In Section 2, we define the minimum Vertex Cover with Hard Capacities problem
(VCHC) formally. In Section 3, we present a family of approximation algorithms for VCHC on multigraphs based on a rounding threshold, and this high-
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lights the main ideas in
√ the analysis. Section 4 provides used extensively in the literature
an improved (1 + 2/ 3)-approximation algorithm for
X
min
xv
VCHC on multigraphs by randomizing the threshold.
v∈V
In Section 5, we obtain a 2f -approximation algorithm
to VCHC on f -hypergraphs. In Section 6, we demon- (2.1a) s.t. y(e, u) + y(e, v) = 1
strate a generic reduction from partial VCHC to VCHC. (2.1b)
y(e, u) ≤ xu
X
Concluding remarks are in Section 7.
(2.1c)
y(e, u) ≤ ku xu
2 Problem Statement
We start by specifying some notation for multigraphs.
A multigraph G = (V, E) may contain several edges
between two vertices u and v, and they will all be
denoted by uv even though we can uniquely identify
them as different members ei ’s in E. For an edge
e = uv between u and v, we often write that u ∈ e
(and similarly that v ∈ e).
The minimum Vertex Cover problem with Hard Capacity constraints (VCHC) is formally defined as follows. An instance of VCHC is specified by (V, E, k, m),
where
[4, 5, 10].
∀e = uv ∈ E
∀e = uv ∈ E, u ∈ e
∀u ∈ V
e∈δ(u)
(2.1d)
(2.1e)
xu ≤ mu
x, y ≥ 0
∀u ∈ V
The following lemma shows that, when constructing
a feasible solution to VCHC, we only need the integrality
of x, and not of y, as was established in Chuzhoy and
Naor [4]. This follows easily by the integrality of flows
in networks with integer capacities. We refer the reader
to [4] for a proof, or to the forthcoming Lemma 5.1 for
a more general statement.
Lemma 2.1. (Chuzhoy and Naor [4]) If (x, y) is
feasible for LP 2.1, and x is integral, there exists an
• G = (V, E) is the input multigraph,
integral y 0 such that (x, y 0 ) is feasible for LP 2.1, and
y 0 can be found efficiently by a maximum flow compu• For each v ∈ V , mv denotes the maximum number tation.
of copies of v one can select,
3 A
3-Approximation
to
VCHC
on
Multigraphs
• For each v ∈ V , kv is the number of incident edges
(a copy of) v can cover.
In this section, we present and analyze a (1 + 1/α)approximation algorithm ALGα for VCHC on multiA solution to VCHC consists of (x, y)
= graphs, where α ∈ [0, 1/2]. Choosing α = 1/2 will then
({xv }v∈V , {y(e, v)}e∈E,v∈e ). Here, xv is the num- yield a 3-approximation. On the other hand, the purber of copies of vertex v selected, and the assignment pose of presenting ALGα for α ∈ [0, 1/2] is to offer invariable y(e, v) ∈ {0, 1} represents whether edge e is sight into the two-step process of the algorithm, and to
covered by v, for each e ∈ E and v ∈ e. A solution highlight the key ideas in its analysis. This also serves
(x, y) is feasible for VCHC iff
as a warm-up for the improved algorithm in the next
section, where α is selected randomly.
1. For all v ∈ V : xv ∈ {0, 1, · · · , mv },
3.1 The (1 + α1 )-Approximation Algorithm. The
2. For all e = uv ∈ E: y(e, u) + y(e, v) = 1 (i.e. any high level idea of ALGα , where α ∈ [0, 1/2], is as
edge must be covered by one of its endpoints),
follows. We first solve LP 2.1 for an optimal solution
(x∗ , y ∗ ), and round up all vertices v with x∗v ≥ α.
3. For all v ∈ V : |{e : y(e, v) = 1}| ≤ kv xv (i.e. the Since those vertices might not have enough capacity
total number of edges assigned to v does not exceed to cover all edges1 , we select the remaining vertices by
its total capacity).
considering another LP, namely LP 3.2. Finally, we
assign the coverage of the edges to the selected vertices
The objective of VCHC is to find
P a feasible solution by Lemma 2.1
(x, y) for VCHC that minimizes v∈V xv , the size of
We now describe our approximation algorithm
the vertex cover. We note that VCHC generalizes ALGα . In anticipation of the next section, we introduce
the classical minimum vertex cover problem, which is another algorithm VC(x̂, ŷ, α), which is a subroutine in
already NP-hard. In light of this, we are providing ALGα and the approximation algorithm in Section 4.
efficient algorithms able to find approximate solutions to
1 as opposed to the situation in the classical vertex cover
VCHC. Our approach is based on rounding a fractional
solution to the following LP relaxation, which has been problem without capacities.
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δ 0 (u)
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E
W
uw ∈ E : u ∈ U, w ∈ W }. For this reason,
we design another linear program (LP 3.2) defined
on W whose solutions will provide us with enough
capacity to cover all the edges of the multigraph.
This statement is motivated in the construction
below and will be proved later in the section.
0
u
U
For edge e = uw with u ∈ U and w ∈ W , our
partial solution x0 allows u to cover ŷ(e, u), but
there might be a deficit of ŷ(e, w) for this edge
which we can associate to u. The total deficit
associated to u ∈ U is
X
X
r(u) =
ŷ(e, w) =
ŷ(e, w),
Z
Figure 1: ALGα partitions the vertex set V into
W, U, Z. Note that there is no edge inside W ∪ Z.
e∈δ 0 (u)
w∈W :e=uw∈E
using the notation δ 0 (u) = δ(u) ∩ E 0 . On the other
hand, if we do select a vertex w ∈ W to be part of
our vertex cover, its value x̂w is scaled up to 1 and
therefore its coverage to e = uw can be increased
from ŷ(e, w) to ŷ(e, w)/x̂w . Vertex w, if selected,
can provide
ALGα is formally stated as follows:
1. Solve LP 2.1 for an optimal solution (x∗ , y ∗ ).
2. Run the algorithm VC(x∗ , y ∗ , α), which returns a
solution (x0 , y 0 ) feasible for VCHC.
X
M (u, w) =
Next, VC(x̂, ŷ, α) is an algorithm that takes as input
a solution (x̂, ŷ) feasible for LP 2.1 and a threshold
α ∈ [0, 1/2]. It outputs an (x0 , y 0 ) feasible for VCHC.
VC(x̂, ŷ, α) is formally stated as follows:
e=uw∈E 0
ŷ(e, w)
x̂w
units of coverage to u ∈ U , where we are summing
over all edges between u and w in our multigraph.
This motivates the following covering linear program:
X
min
zw
1. Given the input (x̂, ŷ, α), partition V into U ∪W ∪Z
where, as illustrated in Fig 1:
• U = {u ∈ V : x̂u ≥ α},
w∈W
• W = {w ∈ V : 0 < x̂w < α},
(3.2a)
• Z = {z ∈ V : x̂z = 0}.
s.t.
X
M (u, w)zw ≥ r(u) ∀u ∈ U
w∈W
(3.2b)
Note that there is no edge in E within W ∪ Z.
Indeed, such an edge e = uv with u, v ∈ W ∪ Z
would imply
0 ≤ zw ≤ 1
∀w ∈ W.
In this step, we compute an optimal extreme point
solution z ∗ to this linear program.
1 = ŷ(e, u) + ŷ(e, v) ≤ x̂u + x̂v < 2α ≤ 1,
∗
4. Define x0w = dzw
e for all w ∈ W . Note that x0 is
now integral.
which is a contradiction.
5. Compute an integral assignment y 0 via network
flows as in Lemma 2.1, and output (x0 , y 0 ).
2. Define the partial solution x0 , where
(
dx̂u e if u ∈ U
0
xu =
0
if u ∈ Z,
and note that we have not defined x0w for w ∈ W
yet.
3. The capacities provided by x0u for u ∈ U are
enough to cover the edges entirely within U and
also the edges between U and Z, but may not be
enough to fully cover the edges in E 0 := {e =
Before the analysis, we would like to compare
our algorithm with the previous algorithms on VCHC
[4, 5, 10]. The first step of both our algorithm and
the previous ones is based on LP rounding. The
difference is in the second step, where essentially all
previous algorithms obtain a feasible instance from x∗
(the optimal solution to 2.1) by randomized rounding
and patching (see e.g. [4, 5]). Our insight is that, instead
of using x∗ , writing down the actual requirements
explicitly as in LP 3.2 produces a sparse extreme
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• For u ∈ U , we have
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point solution which can be easily converted to a
feasible solution. Our approach is deterministic and the
structure of the extreme point solution enables a more
elegant analysis.
xu = dx̂u e ≤ mu ,
as x̂u ≤ mu and mu is integral.
• For w ∈ W , by the definition of W , we have x̂w > 0.
3.2 Feasibility of ALGα . Before establishing the
It implies that mw ≥ 1, thus xw = zw ≤ 1 ≤ mw .
approximation guarantee, we first prove that the output
0 0
• For z ∈ Z, we have xz = 0 ≤ mz .
(x , y ) of ALGα is indeed a feasible VCHC in the
following theorem:
Finally, it remains to check (2.1c). For w ∈ W , we have
X
X ŷ(e, w)
Theorem 3.1. Let (x̂, ŷ) be feasible for LP 2.1, and
kw x̂w
zw ≤
zw
y(e, w) =
0 ≤ α ≤ 12 . Consider any feasible solution z to the
x̂w
x̂w
e∈δ(w)
e∈δ(w)
corresponding LP 3.2 in VC(x̂, ŷ, α). Then for
= kw zw = kw xw .


dx̂v e if v ∈ U
For u ∈ U , we have
xv = zv
if v ∈ W
X


y(e, u)
0
if v ∈ Z,
e∈δ(u)
there exists y such that (x, y) is feasible for LP 2.1.
In the following proof, we explicitly define a y such that
(x, y) is feasible. While we need to check for all the
constraints (2.1a)-(2.1e), the main part of the proof is
to show that x has enough capacity to cover all the
edges in E 0 . This justifies the construction of LP 3.2 in
VC(x̂, ŷ, α).
X
=
1−
e=uw∈δ 0 (u)
ŷ(e, u) + |δ 0 (u)| −
e∈δ(u)\δ 0 (u)
X
≤
X
ŷ(e, w)
zw
x̂w
M (u, w)zw
w∈W
ŷ(e, u) + |δ 0 (u)| − r(u)
e∈δ(u)\δ 0 (u)
X
=
ŷ(e, u) + |δ 0 (u)| −
e∈δ(u)\δ 0 (u)
X
=
• For e = uw ∈ E 0 , where u ∈ U , w ∈ W , define
X
ŷ(e, w)
e∈δ 0 (u)
X
ŷ(e, u) +
e∈δ(u)\δ 0 (u)
ŷ(e, w)
zw , y(e, u) = 1 − y(e, w).
x̂w
X
ŷ(e, u) +
e∈δ(u)\δ 0 (u)
Proof. In this proof, we define a y such that (x, y) is
feasible:
y(e, w) =
X
=
ŷ(e, u) ≤ ku x̂u ≤ ku xu .
e∈δ 0 (u)
P
Finally, for z ∈ Z, e∈δ(z) y(e, z) = 0 = xz . Altogether,
(x, y) is feasible to LP 2.1.
∗
∗
for w ∈ W , Theorem 3.1
e ≥ zw
As we set x0w = dzw
then implies that the integral solution x0 is a feasible
capacitated vertex cover. We emphasize that we do not
y(e, u) = ŷ(e, u), y(e, v) = ŷ(e, v).
need the optimality of (x̂, ŷ) for Theorem 3.1 to hold,
We claim that (x, y) is feasible for LP 2.1. First, for all and that LP 3.2 depends on the solution (x̂, ŷ) as well
e ∈ E, v ∈ e, we have 0 ≤ y(e, v) ≤ 1, and this implies as α; this will be important in the next section. Finally,
that (2.1a), (2.1e) are satisfied. We show that (2.1b) is Theorem 5.1 will generalize Theorem 3.1.
also satisfied for each vertex:
3.3 Approximation Guarantee for ALGα . In
• When u ∈ U and e ∈ δ(u), we have
this section, we establish the approximation guarantee
of our proposed algorithm. First, the cost C of the outy(e, u) ≤ 1 ≤ dx̂u e = xu ,
put solution (x0 , y 0 ) is bounded as follows:
• For e = uv ∈ E \ E 0 , define
• When w ∈ W and e ∈ δ(w), we have
y(e, w) =
C=
ŷ(e, w)
zw ≤ zw = xw ,
x̂w
X
x0v
v∈V
=
X
dx∗u e +
u∈U
• When z ∈ Z and e ∈ δ(z), we have y(e, z) = 0 = xz .
(3.3)
Moreover, (2.1d) is satisfied for all vertices:
≤
X
u∈U
1718
X
∗
dzw
e
w∈W
dx∗u e
+
X
∗
zw
+ |Wf |,
w∈W
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by the Society for Industrial and Applied Mathematics.
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∗
where Wf := {w ∈ W : 0 < zw
< 1} is the set of
∗
vertices in W with fractional value
z
. Now we proceed
P
∗
to bounding the second term w∈W zw
and third term
|Wf |.
First,
the following bound on the second
P we show
∗
term w∈W zw
, by our choice of LP 3.2.
To exhibit the matching, notice that det(A) 6= 0 as
the constraint matrix is of full rank. This implies that
there
Qt exists a permutation σ on {1, 2, · · · , t} such that
i=1 M (ui , wσ(i) ) 6= 0. Furthermore, if M (u, w) 6= 0,
then uw must be an edge. Therefore, there is a matching
between W 0 and U that fully matches W 0 , and by
restricting to the edges incident to Wf , we get the
Lemma 3.1. In VC(x̂, ŷ, α), where (x̂, ŷ) is a feasi- desired matching between W and U .
f
ble solution to LP 2.1 and 0 ≤ α ≤ 1/2, the soFinally, to bound |Wf |, we have the following:
lution {x̂w }w∈W
P
P is feasible for LP 3.2. Therefore,
∗
t
z
≤
X
w∈W w
w∈W x̂w .
|Wf | ≤ |W 0 | ≤
(ŷ(ui wσ(i) , ui ) + ŷ(ui wσ(i) , wσ(i) ))
Proof. By definition of W , we know that 0 ≤ x̂w < α <
i=1
t
1 for all w ∈ W , so (3.2b) is satisfied. For (3.2a), we
X
X
≤
(x̂ui + x̂wi ) ≤
x̂v . have for all u ∈ U
i=1
X X ŷ(e, w)
x̂w
M (u, w)x̂w =
x̂w
w∈W
w∈W e=uw
X
=
ŷ(e, w) = r(u).
X
v∈V
Combining all the pieces, we establish the approximation guarantee of ALGα .
Theorem 3.3. ALGα is a 1 + α1 -approximation algorithm for the VCHC problem on multigraphs.
e∈δ 0 (u)
So {x̂w }w∈W is feasible for LP 3.2.
Proof. For Algα , we have that (x̂, ŷ) = (x∗ , y ∗ ). By
(3.3), we have that the cost of the solution returned
Next, we bound |Wf | by exploiting the sparsity of an
satisfies
extreme point of LP 3.2. In fact, we establish the X
X
X
X
existence of a matching between Wf and U that covers
x0v ≤
dx∗u e +
x∗w +
x∗v
all vertices in Wf . This structural result allows us to v∈V
u∈U
w∈W
v∈V
X
X
∗
partly to the vertices
charge the cost of rounding up zw
X
1
1
∗
∗
+1
+1
≤
xu + 2
xw ≤
x∗v ,
in U . The charging here crucially uses the fact that the
α
α
u∈U
v∈V
w∈W
objective function is unweighted.
Theorem 3.2. For any extreme point solution z ∗ to where the first inequality follows by Lemma 3.1∗ and
∗
3.2, let Wf = {w ∈ W : 0 < zw
< 1} as above. Then Theorem 3.2, the second one from the fact that xu ≥ α
there exists a matching between Wf and UPthat fully for all u1 ∈ U , and the third one by the fact that
0 ≤ α ≤ 2.
matches W . Furthermore, we have |W | ≤
x̂ .
f
f
v∈V
v
Finally, choosing α = 0.5 gives the desired 3Proof. Since z ∗ is an extreme point solution of LP
approximation algorithm.
3.2, there exists a set of |W | linearly independent
√
inequalities in LP 3.2 that are satisfied by z ∗ with
4 A (1 + 2/ 3)-approximation to VCHC on
equalities. By renaming the vertices in U and W , we
Multigraphs via Random Threshold
have a system of |W | linearly independent equalities in
In
the
previous section, we establish that the approx∗
|W |
|W | variables (recall z ∈ [0, 1] ) of the following form:
imation algorithm ALGα has a worst-case approxima



tion ratio of 1 + 1/α for α ∈ [0, 1/2]. In particular, the
r(u1 )
M (u1 , w1 ) · · · M (u1 , wt )




..
.
.
.
smallest worst-case ratio is 3 by choosing α = 1/2. How..
..
· · ·  ∗  .. 

.
 . ever, a given instance need not attain the worst-case
z = 


 r(ut ) 
 M (ut , w1 ) · · · M (ut , wt )
ratio for every value of α. This suggests trying every
0(|W |−t)×t
I|W |−t
0 or 1s
possible value α and outputting the best solution proHere, the first t equalities are from the constraints (3.2a) duced. However, for the sake of the analysis, it is easier
and correspond to {u1 , · · · , ut } ⊆ U , and the last |W |−t to consider a randomized algorithm where the rounding
equalities are from the constraints (3.2b). Let’s call the threshold α is chosen according to a probability distriupper left matrix A, and let W 0 = {w1 , · · · , wt } ⊂ W bution A. In this section, we show that for a suitable
be the set of w ∈ W that corresponds to the first t such distribution and a slight modification of the algocolumns in the above set of equalities. Observe that rithm ALGα , the corresponding randomized approxi∗
Wf is a subset of W 0 since zw
∈ {0, 1} for w ∈
/ W 0 . mation algorithm ALG0A has a performance guarantee
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√
of 1 + 2/ 3 < 2.155, which improves upon the factor 3
from the previous section. Furthermore, the algorithm
can be easily derandomized by trying all relevant values
of α.
We first describe informally our modification of the
algorithm for a given value of α. Instead of applying
VC(x∗ , y ∗ , α) as in the previous section, we apply VC
on a modified LP solution (x̂, ŷ) obtained by first
rounding up the x∗u for u ∈ U and also assigning a
larger fraction of the edges in E 0 to the vertices in U .
Since we maintain feasibility of (x̂, ŷ) in LP 2.1, the
properties of VC still hold, but this allows us to derive
a stronger bound when considering a random choice for
the threshold.
4.1 An Improved Approximation Algorithm.
Our improved algorithm is parametrized by a probability distribution A for α, where A satifies PA (α ∈
[0, 1/2]) = 1. We now present the algorithm ALG0A
formally:
1. Sample a random threshold α according to the
distribution A.
Lemma 4.1. The modified solution (x̂, ŷ) in ALG0A is
feasible for LP 2.1.
Lemma 4.1 asserts that by rounding up x∗u to x̂u for
all u ∈ U , it gives enough room to scale up y ∗ (e, u)
to ŷ(e, u) for e ∈ δ 0 (u) for all u ∈ U , while satisfying
the constraints in LP 2.1. The proof of Lemma 4.1 is
elementary, and is deferred to Appendix A.
Next, we show that randomizing the threshold gives
an approximation ratio better than 3 by choosing the
following probability distribution A:
(
1−β
x
if 0 ≤ x ≤ β
β
1−x
.
(4.5)
PA (α ≤ x) =
1
if x > β
√
Here β = 2 3 − 3 ≈ 0.464. The rationale behind the
choice of A is explained in the proof of the following
theorem:
Theorem 4.1. Let C be the cost of the output of
ALG0A ,√where
P A is as defined in (4.5). Then E[C] ≤
(1 + 2/ 3) v∈V x∗v . Hence, ALG0A returns a (1 +
√
2/ 3)-approximation to VCHC on multigraphs.
2. Solve LP 2.1; let (x∗ , y ∗ ) be an optimal solution.
The proof of Theorem 4.1 can be found in Appendix
B. The algorithm ALG0A can be easily derandomized
3. Similar to Step 1 in algorithm VC, define a partiby trying for the threshold α all values in the set
tion of V into U ∪ W ∪ Z, and E 0 as follows:
Γ = {x∗v |x∗v ≤ β}, and by outputting the best solution.
∗
Indeed the thresholding for some α is unaffected if we
• U = {u ∈ V : xu ≥ α},
increase
α to the next value within Γ.
∗
• W = {w ∈ V : 0 < x < α},
w
• Z = {z ∈ V :
x∗z
= 0},
5
A
2f -approximation
to
VCHC
on
f -hypergraphs
In this section, we propose a 2/α-approximation al4. Modify (x∗ , y ∗ ) to form (x̂, ŷ) as follows:
gorithm ALGhα for VCHC on f -hypergraphs, where
α ∈ [0, 1/f ], by extending the framework in Section
dx∗v e if v ∈ U
x̂v =
∗
3. Recall that an f -hypergraph is one with the size of
xv
otherwise,
largest hyperedge at most f . Although setting α =
and
1/f gives the smallest approximation ratio, we keep
the parametrization with α to maintain full generality.
• for e = uw ∈ E 0 , where u ∈ U and w ∈ W ,
Now, the following is an LP relaxation of VCHC on an
set
f -hypergraph:
∗
dx
e
X
,
(4.4)
ŷ(e, u) = min 1, y ∗ (e, u) u∗
min
xv
xu
• E 0 = {e = uw ∈ E : u ∈ U, w ∈ W }.
v∈V
ŷ(e, w) = 1 − ŷ(e, u),
(5.6a)
• for e ∈ E \ E 0 and v ∈ e, set ŷ(e, v) = y ∗ (e, v).
5. Run the subroutine VC(x̂, ŷ, α). Let (x0 , y 0 ) be the
returned solution.
s.t.
X
y(e, u) = 1
∀e ∈ E
u:u∈e
(5.6b)
(5.6c)
First, to show that ALG0A indeed returns a VCHC,
(5.6d)
it suffices to show that it is legitimate to input (x̂, ŷ) to
(5.6e)
VC in Step 5. It is justified by the following Lemma:
1720
y(e, u) ≤ xu
X
y(e, u) ≤ ku xu
∀e ∈ E, u ∈ e
∀u ∈ V
e∈δ(u)
xu ≤ mu
x, y ≥ 0
∀u ∈ V
Copyright © 2014.
by the Society for Industrial and Applied Mathematics.
P
e
Note that 0 ≤ γve ≤ 1, and
v∈e∩U γv = 1.
0
Therefore, for each e ∈ E , its deficit is fully
distributed among the vertices in e ∩ U . The total
amount of deficit of u ∈ U is
X
X
r(u) =
γue
y ∗ (e, w).
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LP 5.6 has a similar interpretation as LP 2.1. Before
presenting ALGhα , we note that when constructing a
feasible solution (x, y) to VCHC on hypergraphs, we
only need the integrality of x, but not y.
Lemma 5.1. Suppose (x, y) is a feasible solution for LP
5.6 with x integral. Then we can compute efficiently
an integral y 0 such that (x, y 0 ) is a feasible VCHC by a
maximum flow computation.
e∈δ 0 (u)
Next, we consider the amount of coverage w ∈ W
can provide. Suppose we include w into our cover.
Then for e ∈ δ(w) and u ∈ e ∩ U , w can offer
a coverage of γue y ∗ (e, w)/x∗w for u, since we can
scale up the coverage of e from w accordingly when
we scale up x∗w to 1. However, if we include
a subset {w1 , · · · , ws } ⊂ e ∩ W in our vertex
cover, P
this subset can only offer a coverage of
s
min{1, i=1 γue y ∗ (e, wi )/x∗wi } for u on edge e, since
each edge needs at most 1 unit of coverage.
Lemma 5.1 generalizes Lemma 2.1, and the proof is offered in Appendix C. Next, we state our approximation
algorithm ALGhα to VCHC on f -hypergraphs below:
1. Solve LP 5.6 for an optimal solution (x∗ , y ∗ ). Partition V into U ∪ W ∪ Z, where:
• U = {u ∈ V : x∗u ≥ α},
• W = {w ∈ V : 0 < x∗w < α},
• Z = {z ∈ V : x∗z = 0}.
To overcome this non-linearity in our formulation
of the covering LP, we scale down the coverage
γue y ∗ (e, w)/x∗w to γue y ∗ (e, w)/(f − 1)x∗w . This resolves the issue of non-linearlity, since the total
amount of coverage of e ∩ W on e for any u ∈ e ∩ U
is always less than or equal to 1. Now, vertex w, if
selected, can provide
Note that no hyperedge is completely contained in
W ∪ Z. It is because e ⊂ W ∪ Z implies that
X
X
1=
y(e, v) ≤
xv ≤ f max xv < 1,
v:v∈e
v:v∈e
w∈e∩W
v∈e
which is a contradiction.
M (u, w) =
2. Define the partial solution
(
dx∗u e if u ∈ U
0
.
xu =
0
if u ∈ Z
X γ e y ∗ (e, w)
u
∗ (f − 1)
x
e:u,w∈e w
units of coverage to u ∈ U . Altogether, the
discussion motivates the following linear program:
Note that we have not defined x0w for w ∈ W yet.
min
3. Similar to the case in multigraphs, the capacities
provided by x0u for u ∈ U are enough to cover the
edges lying entirely within U ∪ Z. However, these
capacities may not be enough to fully cover the
edges in E 0 := {e ∈ E : e ∩ W 6= ∅}, where e ∩ W
denotes the set of vertices in W that are incident to
e. Thus we design a covering LP to select a subset
of W into the vertex cover, in order to provide full
coverage for E 0 .
0
0
For an edge
P e ∈ E ∗, our partial solution x covers a
fraction u∈e∩U
y (e, u) of e, but there might be
P
a deficit of w∈e∩W y ∗ (e, w) for the coverage of e.
Unlike the case of multigraphs, this deficit can be
associated to more than one vertex in e ∩ U when
|e ∩ U | > 1. We distribute the deficit of e among
e ∩ U as follows. We associate with u a partial
deficit of
X
y ∗ (e, u)
.
γue
y ∗ (e, w), where γue = P
∗
v∈e∩U y (e, v)
w∈e∩W
X
zw
w∈W
(5.7a)
s.t.
X
M (u, w)zw ≥ r(u) ∀u ∈ U
w∈W
(5.7b)
0 ≤ zw ≤ 1
∀w ∈ W.
When f = 2, it is in fact LP 3.2. Now, solve LP
5.7 for an extreme point solution z ∗ .
∗
4. Define x0w = dzw
e for all w ∈ W . Note that x0 is
now integral.
5. Compute an integral assignment y 0 via network
flows as in Lemma 5.1, and output (x0 , y 0 ).
5.1 The Analysis of ALGh
α . First, we prove the
following which shows the feasibility of the algorithm.
Theorem 5.1. Let (x̂, ŷ) be feasible for LP 5.6, and
0 ≤ α ≤ 1/f . Consider any feasible solution z to the
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corresponding LP 5.7. Then for


dx̂v e if v ∈ U
xv = zv
if v ∈ W


0
if v ∈ Z,
The first inequality is by x∗u ≥ α for every u ∈
U , and the second is by Lemma 5.2. To bound
r, by the argument in Theorem 3.2, we know that
there
Qr exists distinct vertices u1 , · · · , ur ∈ U such that
j=1 M (uj , wj ) 6= 0. That means
there exists y such that (x, y) is feasible for LP 5.6
r ≤ |U | ≤
1 X ∗
xu .
α
u∈U
The proof of Theorem 5.1 can be found in Appendix
D. By putting (x̂, ŷ) = (x∗ , y ∗ ) and z = dz ∗ e, Theorem Overall the bound for the cost is
5.1 shows that ALGhα returns a feasible VCHC. Note
X
X
2 X ∗
2 X ∗
that Theorem 5.1 generalizes Theorem 3.1.
xu + (f − 1)
x∗w ≤
xv .
x0v ≤
α
α
Next, we establish the approximation guarantee of
u∈U
w∈W
v∈V
v∈V
h
0 0
ALGα . Now, the total cost of the solution (x , y ) is
When f = 2, Theorem 5.2 gives a guarantee of 4, while
X
X
X
∗
x0v =
dx∗u e +
dzw
e.
Theorem 3.3 gives a guarantee of 3. It is because in the
v∈V
u∈U
w∈W
argument for general hypergraphs, there is no matching
To bound
cost, we first have the following structure associated with an extreme point solution of
P the total
∗
LP 5.7. Hence, this leads to a slightly coarser analysis.
bound on w∈W zw :
Lemma 5.2. The solution zw = (f − 1)x∗w , for all 6 A Reduction from Partial VCHC to VCHC
w ∈ W , is a feasibleP
solution to LP 5.7. In particular, In this section, we consider the Partial Vertex Cover
P
∗
∗
≤
(f
−
1)
z
w∈W w
w∈W xw .
with Hard Capacities constraints (Partial-VCHC),
∗
Proof. First, we have 0 ≤ (f − 1)xw ≤ (f − 1)/f ≤ 1 which is a generalization of VCHC. An instance
of Partial-VCHC is specified by the tuple I =
for every w ∈ W . For every u ∈ U , we have
(V, E, k, m, `), and is defined as follows.
We are
X
∗
provided
with
a
VCHC
instance
with
parameters
M (u, w)(f − 1)xw
(V, E, k, m), but now we are only required to cover all
w∈W
but ` edges, instead of covering all. In particular, when
X X γ e y ∗ (e, w)
u
∗
x
(f
−
1)
=
` = 0, it is a VCHC problem. We provide the following
w
x∗ (f − 1)
w∈W e:u,w∈e w
reduction from Partial-VCHC to VCHC:
X X
e ∗
=
γu y (e, w) = r(u).
Theorem 6.1. For a given integer f ≥ 2, suppose
eδ 0 (u) w∈e∩W
we have an ηf +1 -approximation algorithm to VCHC on
Note that Lemma 5.2 generalizes Lemma 3.1. Finally, (f + 1)-hypergraphs. Then for any > 0, there is
the following Theorem establishes the approximation an ηf +1 (1 + )-approximation algorithm to the PartialVCHC on f -hypergraphs, which runs in poly(|V |1/ |E|)
ratio of ALGhα :
time.
Theorem 5.2. ALGhα returns a 2/α-approximation
for the VCHC on hypergraphs. In particular, when We prove Theorem 6.1 by reducing a Partial-VCHC
α = 1/f , it returns a 2f -approximation.
instance on an f -hypergraph to a VCHC instance on
an (f + 1)-hypergraph.
∗
Proof. Let Wf := {w ∈ W : zw
∈ (0, 1)} =
∗
{w1 , · · · , wr } be the set of w ∈ W with fractional zw
Proof. Given a f -hypergraph Partial-VCHC instance
values. In particular, we have |Wf | = r. The total cost I = (V, E, k, m, `), define a new (f + 1)-hypergraph
can be bounded as follows:
VCHC instance I 0 = (V 0 , E 0 , k 0 , m0 ), where
X
X
X
∗
• V 0 = V ∪ {s}, where s is a new vertex.
x0v =
dx∗u e +
dzw
e
v∈V
u∈U
w∈W
• E 0 = {e ∪ {s} : e ∈ E}. For each e ∈ E, denote
e0 = e ∪ {s}.
X
1 X ∗
∗
≤
xu +
zw
+r
α
u∈U
w∈W
X
1 X ∗
≤
xu + (f − 1)
x∗w + r.
α
u∈U
• kv0 = kv if v ∈ V , and ks0 = `.
• m0v = mv if v ∈ V , and m0s = 1.
w∈W
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We compute a ηf +1 -approximation (x0 , y 0 ) to I 0 , and References
output the solution (x, y), where x is obtained by
restricting x0 to V , and y(e, v) = y 0 (e0 , v) for all e ∈
[1] R. Bar-Yehuda and S. Even. A linear-time approximaE, v ∈ e. Note that (x, y) is feasible to I, since the total
tion algorithm for the weighted vertex cover problem.
number of edges assigned is
J. Algorithms, 2(2):198–203, 1981.
X X
X X
0 0
[2]
R.
Bar-Yehuda, G. Flysher, J. Mestre, and D. Rawitz.
y (e , v)
y(e, v) =
e∈E v∈V :v∈e
e0 ∈E 0 v∈V :v∈e0
0
= |E | −
X
0
y (e, s)
[3]
e0 ∈E 0
≥ |E| − `,
so that (x, y) covers all but at most ` edges. Next, we
bound the cost as follows:
X
X
xv ≤
x0v ≤ ηf +1 opt(I 0 )
v∈V
v∈V 0
[5]
≤ ηf +1 (opt(I) + 1) = ηf +1
[4]
1
1+
opt(I)
opt(I).
Approximation of partial capacitated vertex cover.
SIAM J. Discrete Math., 24(4):1441–1469, 2010.
N. H. Bshouty and L. Burroughs. Massaging a linear
programming solution to give a 2-approximation for a
generalization of the vertex cover problem. In Proceedings of the 15th Annual Symposium on Theoretical
Aspects of Computer Science, pages 298–308. Springer,
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J. Chuzhoy and J. Naor. Covering problems with hard
capacities. SIAM J. Comput., 36(2):498–515, 2006.
R. Gandhi, E. Halperin, S. Khuller, G. Kortsarz, and
A. Srinivasan. An improved approximation algorithm
for vertex cover with hard capacities. J. Comput. Syst.
Sci., 72(1):16–33, 2006.
R. Gandhi, S. Khuller, and A. Srinivasan. Approximation algorithms for partial covering problems. J.
Algorithms, 53(1):55–84, 2004.
S. Guha, R. Hassin, S. Khuller, and E. Or. Capacitated
vertex covering with applications. In SODA, pages
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D. Hochbaum. Approximation algorithms for the set
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S. Khot and O. Regev. Vertex cover might be hard
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[6]
Here, opt(I) denotes the optimal value for instance I.
To justify the third inequality above, we argue that
opt(I) + 1 ≥ opt(I 0 ). It is because given an optimal
[7]
solution (x∗ , y ∗ ) to I, including s in the cover and
assigning the uncovered edges to s will give a feasible
solution to I 0 .
[8]
Lastly, to obtain the stated approximation ratio,
we perform the following. For each 1 ≤ C ≤ 1/, and
for each multisets of vertices of total size C, we check
[9]
if it is feasible to I. This can be done by checking
if xv ≤ mv for all v and checking if the network
flow instance defined in Lemma 5.1 has optimal value [10]
≥ |E| − `. If it is the case, output the multi-set along
with the assignment defined by the optimal flow. This
is the optimal Partial-VCHC. Otherwise, we know that
opt(I) > 1/, and the approximation algorithm above A Proof of Lemma 4.1
will have an approximation ratio ≤ ηf +1 (1 + ).
We will prove that (x̂, ŷ) satisfies the constraints (2.1a)(2.1e). First, (2.1a) is satisfied by the definition of ŷ.
Combining with ALGh1/f , we have the following:
Also, (2.1d) is satisfied by the definition of x̂.
For x̂, clearly constraint (2.1e) is satisfied. For ŷ, we
Corollary 6.1. For f ≥ 2, there is a (2f + 2)(1 + )know
that 0 ≤ ŷ(e, u) ≤ 1 for all e ∈ δ(u), u ∈ U . Then
approximation algorithm to the partial-VCHC problem
1/
by
the
fact that ŷ(e, u) + ŷ(e, w) = 1 for all e = uw ∈ E,
on f -hypergraphs which runs in poly(|V | |E|) time.
we also have 0 ≤ ŷ(e, w) ≤ 1 for all e ∈ δ(w), w ∈ W ∪Z.
Therefore, (2.1e) is satisfied for all ŷ.
7 Conclusion
To prove for the remaining constraints (2.1b),
In summary, we have proposed new approximation al(2.1c), we first claim that for all w ∈ W and e ∈ δ(w),
gorithms to the minimum Vertex Cover with Hard Cawe have
pacities problem (VCHC), which√improve the approximation ratio from 38 to 1 + 2/ 3 < 2.155 for multi(A.1)
ŷ(e, w) ≤ y ∗ (e, w).
graphs, and from max{65, 6f } to 2f for f -hypergraphs.
Furthermore, we have presented a (2f + 2)(1 + ) ap- Indeed, for e = uw ∈ E 0 , where u ∈ U and w ∈ W ,
proximation for partial VCHC on f -hypergraphs. To
y ∗ (e,u)dx∗
ue
we have ŷ(e, u) ≥
≥ y ∗ (e, u). Thus, we have
x∗
u
conclude, we leave the open problem of whether there
∗
ŷ(e, w) = 1 − ŷ(e, u) ≤ 1 − y (e, u) = y ∗ (e, w).
is a 2-approximation to the VCHC on multigraphs.
Now, to prove for the constraint (2.1b), for w ∈ W ,
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we know from (A.1) that
ŷ(e, w) ≤ y ∗ (e, w) ≤ x∗w = x̂w .
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For u ∈ U , we know that
Lemma B.2. For all v ∈ V , we have the following
bound:
x∗
(B.2) dx∗v eP(x∗v ≥ α) + x∗v P(x∗v < α) + f (x∗v ) ≤ v .
β
Lemma B.2 immediately implies that ALG0A is a 1/β <
2.155-approximation algorithm. Our choice of the
For z ∈ Z, we have ŷ(e, z) = 0 = x̂z .
probability distribution A was engineered so as to
Finally, to prove for the constraint (2.1c), for w ∈ minimize the factor in the above lemma.
W , we have from (A.1) that
Proof. [Proof of Lemma B.2] When x∗v = 0 or x∗v ≥ 2,
X
X
∗
∗
the bound (B.2) is clearly true. The main cases are
ŷ(e, w) ≤
y (e, w) ≤ kw xw = kw x̂w .
when 0 < x∗v ≤ 1 and 1 < x∗v < 2, as analyzed below.
e∈δ(w)
e∈δ(w)
When 0 < x∗v ≤ 1, the bound (B.2) is equal to the
For u ∈ U , we have
following:
∗
∗
X
X
dx e
dx e
dx∗v eP(x∗v ≥ α) + x∗v P(x∗v < α) + f (x∗v )
y ∗ (e, u) ≤ u∗ ku x∗u = ku x̂u .
ŷ(e, u) ≤ u∗

xu
xu
∗
∗
e∈δ(u)
e∈δ(u)
 1−β xv ∗ + x∗v 1 − 1−β xv ∗ + 0 if 0 ≤ x∗v ≤ β
β 1−xv
β 1−xv
P
.
=
∗
Finally, for z ∈ Z, we have e∈δ(z) ŷ(e, z) = 0 = kz x̂z .
v
1 + 0 + 1 − 1−β 1−x
if β < x∗v ≤ 1
β
x∗
v
Altogether, (x̂, ŷ) is feasible to LP 2.1.
In each of the two cases, the bound is less than or equal
to x∗v /β, and in fact for the case 0 ≤ x∗v ≤ β it is an
B Proof of Theorem 4.1
For a realization of the random threshold α ∈ [0, β], equality. Also, it is easy to check that our choice of A
(3.3) and Lemma 3.1 imply that the cost C of ALG0A minimizes the factor in the above analysis.
When 1 < x∗v < 2, the bound (B.2) is equal to the
has the following upper bound:
following:
X
C≤
{dx∗v e1(x∗v ≥ α) + x∗v 1(x∗v < α)
dx∗v eP(x∗v ≥ α) + x∗v P(x∗v < α) + f (x∗v )
v∈V
=2 + 0 + P(α ≥ 1 − x∗v /2)
+ 1(v ∈ U, v is in the matching M )} .
1 − β 2 − x∗v
x∗
=2
+
1
−
< v,
Here, the matching M refers to the matching between
∗
β
xv
β
W 0 and U identified in Theorem 3.2. Taking expectation
∗
on α over the probability distribution A defined in (4.5), where the last inequality is true for all xv > 1.
we have
Finally we return to the proof of Lemma B.1:
X
∗
∗
∗
∗
E[C] ≤
{dxv eP(xv ≥ α) + xv P(xv < α)
Proof. [Proof of Lemma B.1] We first show that the
v∈V
bound
+ P(v ∈ U, v is in the matching M )} .
P(v ∈ U, v is in M ) ≤ P(α > 1 − x∗v )
First, we provide the following bound for the third term:
holds for all v ∈ V . Indeed, if v ∈ U is covered by an
Lemma B.1. The probability P(v ∈ U, v is in M ) has edge vw ∈ M , then we know that w ∈ W , which implies
that
the following upper bound:
(
1 = y ∗ (e, v) + y ∗ (e, w) ≤ x∗v + x∗w < x∗v + α.
P(α
> 1 − x∗v ) if 0 ≤ x∗v ≤ 1
∗
.
≤ f (xv ) :=
x∗
if x∗v > 1
P α > 1 − 2v
Next, we show the stronger bound
x∗v
We note that the stronger bound derived in the
P(v
∈
U,
v
is
in
M
)
≤
P
α
>
1
−
2
range of x∗v > 1 crucially uses the modification step
(i.e. Step 4) in ALG0A . Also, note that the bound only when x∗ > 1. Since it is clearly true when x∗ ≥ 2, we
v
v
depends on the value of x∗v . The proof of Lemma B.1 is focus on the range 1 < x∗ < 2. Now, if v ∈ U is in the
v
given at the end of this appendix section.
matching, then there must exist w such that
Given the bound in Lemma B.1, we prove the
∗
following bound for the expected cost charged to each M (v, w) = X max 0, 1 − y ∗ (e, v) dxv e · 1 > 0.
x∗v
x∗w
vertex:
e:e=vw
ŷ(e, u) ≤ 1 ≤ dx∗u e = x̂u .
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Taken an edge e between
v and w such that We claim that (x, y) is feasible to LP 5.6.
o
dx∗
1
∗
ve
First, (5.6a) is satisfied. For e ∈ E − E 0 , it clearly
max 0, 1 − y (e, v) x∗ · x∗ > 0. Since 1 < x∗v < 2, it
v
w
holds. For e ∈ E 0 , we have
implies that
!
X
X
X
∗
ŷ(e,
w)
2
dx
e
y(e, v) =
γue 1 −
zw
1 > y ∗ (e, v) ∗v = y ∗ (e, v) ∗ .
(f − 1)x̂w
xv
xv
v∈e
u∈e∩U
Finally, we have
+
X
w∈e∩W
x∗v > 2y ∗ (e, v) = 2(1 − y ∗ (e, w)) ≥ 2(1 − x∗w ) ≥ 2(1 − α)
w∈e∩W
ŷ(e, w)
zw = 1,
(f − 1)x̂w
since v∈e∩U γue = 1.
Secondly, (5.6d) is clearly satisfied, since for u ∈ U ,
x̂u ≤ mu implies that xu = dx̂u e ≤ mu , and for w ∈ W ,
xw ≤ 1 ≤ mw , where the latter inequality is due to
x̂w > 0. For z ∈ Z, we have xz = 0 ≤ mz .
Thirdly, (5.6e) is satisfied. For the x variables, we
know that they are non-negative. For the y variables,
we have the following cases. For e ∈ E − E 0 , it is clear.
For y(e, w) where e ∈ E 0 , w ∈ W , it is also clear. For
y(e, u) where e ∈ E 0 , u ∈ U , we have
!
X
ŷ(e,
w)
zw
y(e, u) = γue 1 −
(f − 1)x̂w
w∈e∩W
!
X
1
e
(D.3)
≥ 0,
≥ γu 1 −
f −1
P
as desired.
C Proof of Lemma 5.1
Given an instance I = (V, E, k, m), and a solution (x, y)
to LP 5.6, where x is integral, construct the following
˜
network flow instance I:
• The set of nodes is s ∪ V ∪ E ∪ t, where s is the
source, t is the sink.
• The set of arcs and their capacities are defined as
follows:
– For all v ∈ V , there is an arc from s to v with
capacity kv xv .
– For all v ∈ V and e 3 v, there is an arc from
v to e with capacity xv .
w∈e∩W
(D.3),
– For all e ∈ E, there is an arc from e to t with since ŷ(e, w)/x̂w ≤ 1. We note that in showingŷ(e,w)
is
we
crucially
use
the
fact
that
the
contribution
capacity 1.
x̂w
ŷ(e,w)
scaled down to (f −1)x̂w .
Now, the feasibility of (x, y) for LP 5.6 implies that
Fourthly, (5.6b) is satisfied. For w ∈ W , we have
there is a flow of value |E| from s to t in the instance
y(e,
w) ≤ zw since
˜
˜
I. But since all capacities in I are integral, there also
exist an integral flow of value |E| from s to t. Now, for
ŷ(e, w)
zw
y(e, w) =
≤ zw .
zw ≤
all e ∈ E, v ∈ e, define y 0 (e, v) = 1 if there is one unit
(f
−
1)x̂
f
−1
w
0
flow through the arc (v, e), and y (e, v) = 0 otherwise.
Then (x, y 0 ) is a feasible solution to VCHC on instance For u ∈ U , we first define the notation δ 0 (u) =
I.
{e ∈ δ(u) : e ∩ W 6= ∅}. Going back to the checking of
(5.6b) for u, if e ∈ δ(u) \ δ 0 (u), it is clearly true. If
D Proof of Theorem 5.1
e ∈ E 0 , y(e, u) ≤ xu , since
In the proof, we define a y such that (x, y) is feasible as
X
ŷ(e, w)
follows:
y(e, u) ≤ 1 −
z w ≤ 1 ≤ xu .
(f − 1)x̂w
w∈e∩W
0
• For e ∈ E and v ∈ e, we define:
For v ∈ Z, we have y(e, v) = xv = 0.
– If v ∈ e ∩ W , y(e, v) = (fŷ(e,v)
−1)x̂v zv .
Finally, it remains to check (5.6c). For w ∈ W , we
have
– If v ∈ e ∩ U ,
!
X
X
ŷ(e, w)
X
ŷ(e, w)
zw
y(e, w) =
e
y(e, v) = γv 1 −
zw .
(f − 1)x̂w
e∈δ(w)
e∈δ(w)
(f − 1)x̂w
w∈e∩W
kw x̂w
≤
zw
– If v ∈ e ∩ Z, y(e, v) = 0.
(f − 1)x̂w
≤ kw zw = kw xw .
• For e ∈ E−E 0 , for all u ∈ e, define y(e, u) = ŷ(e, u).
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For u ∈ U , to check that its corresponding constraint in
(5.6c) is satisied, we first claim that
X
X
y(e, u) ≤
ŷ(e, u).
e∈δ 0 (u)
e∈δ 0 (u)
Given the claim, we can conclude that
X
X
X
y(e, u) =
y(e, u) +
e∈δ 0 (u)
e∈δ(u)
=
X
X
X
y(e, u) +
e∈δ 0 (u)
≤
y(e, u)
e∈δ(u)\δ 0 (u)
ŷ(e, u)
e∈δ(u)\δ 0 (u)
X
ŷ(e, u) +
e∈δ 0 (u)
ŷ(e, u)
e∈δ(u)\δ 0 (u)
≤ ku x̂u ≤ ku xu .
Finally, to verify the claim, we have:
X
y(e, u) =
e∈δ 0 (u)
X
γue
1−
e∈δ 0 (u)
=
X
w∈e∩W
γue −
=
γue −
e∈δ 0 (u)
≤
X
X
ŷ(e, w)
zw
(f − 1)x̂w
X
w∈W e∈E 0 :u,w∈e
e∈δ 0 (u)
X
X
X
!
γue ŷ(e, w)
zw
(f − 1)x̂w
M (u, w)zw
w∈W
γue − r(u)
e∈δ 0 (u)
=
X
γue −
e∈δ 0 (u)
(D.4)
=
X
e∈δ 0 (u)
X
e∈δ 0 (u)
γue
X
v∈e∩U
γue
X
ŷ(e, w)
w∈e∩W
ŷ(e, v) =
X
ŷ(e, u).
e∈δ 0 (u)
We crucially use the definition of γue in (D.4) to establish
(5.6c)
for u ∈ U . Finally, for z ∈ Z, we have
P
e∈δ(z) y(e, z) = 0 = kz xz . Thus, constraint (5.6c)
is satisfied for all v ∈ V . Altogether, (x, y) is feasible to
LP 5.6.
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