Chapter 4 The Maximum Induced Matching Problem
4.3 Regular bipartite graphs
4.3.1 APX-completeness
In this section we show that the Maximum Induced Matching problem is APX- complete in the class ofr-regular bipartite graphs for anyr≥3. The proof follows from an
approximation preserving reduction. We use the L-reduction as dened in [Papadimitriou and Yannakakis, 1991]. LetP be a maximisation problem. For every instancexofP, and
every solution y of x, let cP(x, y) be the cost of the solution y. Let optP(x) be the cost
of an optimal solution. If c≥1 is a constant and there is a polynomial time algorithm
that computes a solutiony(x), such that∀x, cP(x, y(x))≥ 1coptP(x), then the algorithm
is said to approximate P to within a ratio of c. If this holds for a constant c >1, then
P is said to be constant-factor approximable and it belongs to the class APX. If, for any
positive ε, P has a polynomial time algorithm which approximates P to within a ratio
of≤1 +ε, then we say thatP has a polynomial-time approximation scheme (PTAS).
Denition 38. Let P and Qbe two maximisation problems. An L-reduction from P to Q is a four-tuple(t1, t2, α, β), where t1 andt2 are polynomial time computable functions andα andβ are positive constants with the following properties:
(a) t1maps instances ofP to instances ofQand for every instancexofP,optQ(t1(x))≤
αoptP(x).
(b) For every instancex ofP,t2 maps pairs(t1(x), y0)(wherey0 is a solution oft1(x)) to a solutionyofxso that|optP(x)−cP(x, t2(t1(x), y0))| ≤β|optQ(t1(x))−cQ(t1(x), y0)|. As shown in [Papadimitriou and Yannakakis, 1991], ifP andQare maximisation
problems and there is an L-reduction fromPtoQthen ifQhas a PTAS,P must also have
a PTAS. Conversely, the denition of APX-hardness implies that ifP is APX-complete,
thenQis APX-hard. If furthermore Qis in APX, then it is APX-complete.
For any nite set D of positive integers, we say a graph G is a D-graph if D is
the set of vertex degrees inG. For example a{k}-graph is a non-emptyk-regular graph.
Theorem 39. Let D be a nite set of positive integers such that maxd∈Dd ≥ 3, then
Maximum Induced Matching is APX-complete in the class of bipartiteD-graphs. In
particular, it is APX-complete in the class of k-regular bipartite graphs for anyk≥3.
Proof. The Maximum Induced Matching problem is known to be approximable to within a constant factor in k-regular graphs [Zito, 1999]; so it remains to show it is
APX-hard.
For any xed k ≥ 3, we dene the gadget Hk = (Vk, Ek) (see Figure 4.2) as
The set of vertices is dened byVk=L1∪L2∪L3∪L4∪L5∪L6 with L1={11, . . . ,1k}, L2 ={21, . . . ,2k}, L3 ={31, . . . ,3k(k−1)}, L4={41, . . . ,4k(k−1)}, L5 ={51, . . . ,5(k−1)2}, L6 ={61, . . . ,6(k−1)(k−2)}. Fori= 1, . . . , k−1, we denote S3i ={3(i−1)k+1, . . . ,3ik}, S4i ={4(i−1)k+1, . . . ,4ik}, S5i ={5(i−1)(k−1)+1, . . . ,5i(k−1)}, S6i ={6(i−1)(k−2)+1, . . . ,6i(k−2)}. So |S3i|=|S4i|=k,|S5i|=k−1and |Si6|=k−2.
The set of edges Ek is dened as follows:
(1) L1∪L2 induces a matching of size k: (1i,2i)∈Ek, i= 1, . . . , k.
(2) (2i,3(i−1)(k−1)+j)∈Ek, i= 1, . . . , k, j = 1, . . . , k−1.
(3) L3∪L4 induces a matching: (3i,4i)∈Ek, i= 1, . . . , k(k−1).
(4) For everyi= 1, . . . , k−1,Si
4 andS5i induce aKk,k−1. (5) For everyi= 1, . . . , k−1,S3i andS6i induce aKk,k−2.
Note that every vertex of L1 is of degree 1 inHk while the other vertices are of
degree k. Note also that∀i∈ {1, . . . , k−1}, N(S3i)∩L2={2i,2i+1}.
For any graph G = (V, E) and any set of k vertices S = {v1, . . . , vk} ⊂ V, we
dene the graph G∪SHk obtained by adding an Hk to G and identifying L1 and S. More formally its set of vertices isV ∪L2∪L3∪L4∪L5∪L6 and(G∪SHk)[V] =Gand (G∪SHk)[S∪L2∪L3∪L4∪L5∪L6] =Hk. For any two graphsG= (V, E), G0 = (V0, E0)
we denoteG∪G0 = (V ∪V0, E∪E0).
Lemma 40. For anyk≥3,{(3i,4i), i= 1, . . . , k(k−1)}is a maximum induced matching of Hk.
Proof. Note rst that, since vertices 1i, i = 1, . . . , k are of degree 1 in Hk, for any
induced matching M of Hk containing an edge (2i,3(i−1)(k−1)+j), with i ∈ {1, . . . , k}
andj ∈ {1, . . . , k−1},M\ {(2i,3(i−1)(k−1)+j)} ∪ {(1i,2i)}is also an induced matching.
Consequently, without loss of generality we can restrict ourselves to the case where M
does not contain any edge (u, v), u ∈ L2, v ∈ L3. For every i = 1, . . . , k−1, we let
Mi = M ∩[{(1i,2i),(1i+1,2i+1)} ∪ {(u, v), u ∈ S3i, v ∈ S6i} ∪ {(u, v), u ∈ S4i, v ∈ S5i}]. Note that|Mi| ≤3. Since edges(u, v), u∈S3i, v∈S4i constitute an induced matching and
… … S61 S62 S51 S 52 S41 S31 11 21 k-2 k-1 31 41 51 61 S6k-2 S6k-1 S5k-2 S 5k-1 S4k-1 S3k-1 1k 2k 3k(k-1) 4k(k-1) 5(k-1)(k-1) 6(k-1)(k-2) L6 L1 L2 L3 L4 L5 61 62 11 12 13 21 22 23 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54
are only linked, inM, to edges belonging toMi,M0 =M\Mi∪ {(u, v), u∈S3i, v ∈S4i}is also an induced matching. Moreover, ifM contains an edge(u, v), u∈S3i, v ∈S4i, then it
does not contain any edge (u0, v0), u0∈S3i, v0 ∈S6i or any edge(u00, v00), u00∈S4i, v00∈S5i.
So|M0| ≥ |M|and consequently, there is a maximum induced matching ofHkcontaining
edges(u, v), u∈L3, v ∈L4. Since this matching is maximal, Lemma 40 follows. 2 A direct consequence of Lemma 40 is that there is a maximum induced matching inG∪SHk containing{(3i,4i), i= 1, . . . , k(k−1)}and consequently
iµ(G∪SHk) =iµ(G) +k(k−1) (4.1)
Moreover, if Gis bipartite and S is monochromatic for a xed 2-colouring of G,
thenG∪SHk is also bipartite.
We can now describe the reduction. Let G = (V, E) be any bipartite D-graph.
Let us rst note that for any positive integerd,G∪Kd,d is a bipartite(D∪ {d})-graph
and
iµ(G∪Kd,d) =iµ(G) + 1 (4.2)
On the other hand, for any d ∈ D, let u1d, . . . , upd, p ≥ 1 denote the vertices of
degreed. Letk≥3. We considerkcopies ofGdenoted byG1, . . . , Gkand for any vertex
v ∈ V(G) we let S(v) denote the set of copies of v in G1, . . . , Gk (so |S(v)| =k). We
then dene:
Tdk(G) = (G1∪. . .∪Gk)∪S(u1
d)Hk. . .∪S(u p d)Hk
Using relation (4.1) we immediately obtain:
iµ(Tdk(G)) =kiµ(G) +pk(k−1) (4.3)
It is also straightforward to verify that, if k ∈ D∪ {d+ 1}, d 6= k, then Tdk(G)
is a bipartite (D\ {d} ∪ {d+ 1})-graph. Since G∪Kd,d and Tdk(G) can be performed
in polynomial time, relations (4.2) and (4.3) imply that the related reduction preserves polynomial approximation schema.
The rst of these is an L-reduction withα= 2, β= 1, wheret1mapsGtoG∪Kd,d
and t2 maps (G∪Kd,d, M) to (G, M0), where M0 is the set of edges in M that do not
occur in theKd,d. SinceGis a D-graph, we know thatiµ(G)≥1, so the rst inequality
at most one edge in the Kd,d can occur in M.
The second reduction is also an L-reduction withα=k(1+2(2∆(∆−1)+1)(k−1))
and β = 1/k, where∆ = max(D). We dene tot1 to mapG to Tdk(G) and dene t2 to map(Tdk(G), M)to(G, M0), whereM0 is the set of edges inM that belong to one of the
copies ofGinTdk(G), where this copy ofGis chosen so that the size ofM0 is maximised.
Indeed, observe that since the degree inGis bounded above by∆, any edge inGis
linked to at most2∆(∆−1)edges. SinceDis made up of positive integers, the minimum
degree inGis at least 1, so there are at least n2 edges inG. Thus iµ(G)≥ n
2(2∆(∆−1)+1). Note that p≤nfor this transformation. This means that
iµ(Tdk(G)) = kiµ(G) +pk(k−1)
≤ kiµ(G) +nk(k−1)
≤ k(1 + 2(2∆(∆−1) + 1)(k−1))iµ(G)
as required by the rst inequality in the denition of L-reduction. For the second in- equality, consider(G, M0) =t2(Tdk(G), M). Let M00 consist of the edges inTdk(G) of the form (3i,4i) along those edges in in every copy of G in Tdk(G) that correspond to the
edges inM0. Then |M00|= k|M|+pk(k−1). Using Lemma 40 as before, we conclude
that|M| ≤ |M00|, so iµ(G)− |M0| = 1 k(kiµ(G) +pk(k−1)− |M 00| ≤ 1 k(iµ(T k d(G))− |M|) as required.
Consequently if the Maximum Induced Matching problem is APX-complete in bipartiteD-graphs, then for any positive integerdit is also APX-complete in bipartite
(D∪ {d})-graphs and, using the transformation Tdd+1 for any d ∈ D, d ≥ 2, it is also
APX-complete for bipartite(D\ {d} ∪ {d+ 1})-graphs.
The problem is shown to be APX-complete for bipartite{2,3}-graphs [Duckworth et al., 2005]. (More precisely, for any ε > 0, the problem of approximating Maximum
Induced Matching within a factor of 9570
9569 −ε is NP-hard for graphs in this class.) Then, using the above remarks successively for d= 2,3, . . . we deduce that it is APX-
complete in bipartite{3}-,{4}-, . . . , {k}-graphs for anyk≥3and consequently, that it
is APX-complete in bipartiteD-graphs for any niteDwith at least one elementk≥3.