Parameterized Complexity in Multiple-Interval Graphs: Partition, Separation, Irredundancy Minghui Jiang Utah State University Yong Zhang Kutztown University of Pennsylvania COCOON
August 14, 2011
Intersection Graphs The intersection graph GF of a family of sets F = {S1 , . . . , Sn } is the graph with F as the vertex set and with two different vertices Si and Sj adjacent if and only if Si ∩ Sj 6= ∅. The family F is called a representation of the graph GF .
Geometric Intersection Graphs The sets in the representation are natural geometric objects: • interval graph: intervals in a line • circular-arc graph: arcs of a circle • circle graphs: chords of a circle • disk intersection graphs: disks in the plane Natural representations has natural applications, for example, interval graphs for DNA assembly, and disk intersection graphs for wireless network.
Multiple-Interval Graphs Let t ≥ 2 be an integer. A t-interval is the union of t disjoint intervals in the real line. A t-interval graph is the intersection graph of a family of t-intervals. If a t-interval graph has a representation in which the t disjoint intervals of each t-interval have the same length (although the intervals from different t-intervals may have different lengths), then the graph is a balanced t-interval graph. If a t-interval graph has a representation in which all intervals have unit lengths, then the graph is a unit t-interval graph.
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Applications As generalizations of the ubiquitous interval graphs, multipleinterval graphs have wide applications, traditionally to scheduling and resource allocation and more recently to bioinformatics. In particular, 2-interval graphs are natural models for the similar regions of DNA sequences and for the helices of RNA secondary structures.
Approximation Complexity Studied by Bar-Yehuda et al. (SODA 2002) and Butman et al. (SODA 2007). Current best approximation ratios for optimization problems in t-interval graphs: • Maximum Independent Set: 2t • Minimum Vertex Cover: 2 − 1/t • Minimum Dominating Set: t 2 • Maximum Clique: (t 2 − t + 1)/2
Parameterized Complexity In general graphs, the following four optimization problems, parameterized by the optimal solution size k, are exemplary problems in parameterized complexity theory: • k-Vertex Cover: in FPT • k-Independent Set / k-Clique: W[1]-hard • k-Dominating Set: W[2]-hard Since t-interval graphs are a special class of graphs, all FPT algorithms for k-Vertex Cover in general graphs immediately carry over to t-interval graphs. The parameterized complexities of k-Independent Set, k-Clique, and k-Dominating Set in t-interval graphs, however, are not at all obvious.
In general graphs, k-Independent Set and k-Clique are essentially the same problem, but in t-interval graphs, they manifest different parameterized complexities. . . Fellows et al. (2009) recently initiated the study of the parameterized complexity of multiple-interval graph problems: • k-Independent Set in t-interval graphs is W[1]-hard for any constant t ≥ 2. • k-Dominating Set in t-interval graphs is also W[1]-hard for any constant t ≥ 2. • k-Clique in t-interval graphs admits an FPT algorithm parameterized by both k and t.
Four More Problems At the end of their paper, Fellows et al. (2009) listed four more problems that are W[1]-hard in general graphs, and suggested that a possibly prosperous direction for extending their work would be to investigate whether these problems become fixed-parameter tractable in multiple-interval graphs. • k-Vertex Clique Cover • k-Separating Vertices • k-Perfect Code • k-Irredundant Set We faithfully followed their suggestions. . .
The problem k-Vertex Clique Cover has a close relative called k-Edge Clique Cover. Given a graph G = (V, E) and an integer k, the problem k-Vertex Clique Cover asks whether the vertex set V can be partitioned into k disjoint subsets Vi , 1 ≤ i ≤ k, such that each subset Vi induces a complete subgraph of G, and the problem k-Edge Clique Cover asks whether there are k (not necessarily disjoint) subsets Vi of V , 1 ≤ i ≤ k, such that each subset Vi induces a complete subgraph of G and, moreover, for each edge {u, v } ∈ E, there is some Vi that contains both u and v .
The two problems k-Vertex Clique Cover and k-Edge Clique Cover are also known in the literature as k-Clique Partition and k-Clique Cover, respectively, and are both NP-complete; see Problems GT15 and GT17 in Garey and Johnson. To avoid possible ambiguity, we will henceforth use the term k-Vertex Clique Partition instead of k-Vertex Clique Cover or k-Clique Partition.
The problem k-Vertex Clique Partition in any graph G is the same as the problem k-Vertex Coloring in the complement graph G. • 3-Vertex Coloring of planar graphs of maximum degree 4 is NP-hard (Garey et al. 1976). • k-Vertex Coloring in circular-arc graphs is NP-hard if k is part of the input (Garey et al. 1980). • Graphs of maximum degree 4 are unit 3-track interval graphs (Jiang 2010). • Circular-arc graphs are obviously 2-track interval graphs (by a simple cutting argument).
Thus we immediately have the following easy theorem on the complexity of k-Vertex Clique Partition in the complements of multiple-interval graphs: Theorem 1. 3-Vertex Clique Partition in co-unit 3-track interval graphs is NP-hard; thus, unless NP = P, k-Vertex Clique Partition in co-unit 3-track interval graphs does not admit any FPT algorithms with parameter k. Also, k-Vertex Clique Partition in co-2-track interval graphs is NP-hard if k is part of the input. For the complexity of k-Vertex Clique Partition in multipleinterval graphs, we obtain the following theorem: Theorem 2. k-Vertex Clique Partition in unit 2-interval graphs is W[1]-hard with parameter k.
Although the two problems k-Vertex Clique Partition and k-Edge Clique Cover are both NP-complete, they have very different parameterized complexities. k-Edge Clique Cover is fixed-parameter tractable in general graphs (Gramm et al. 2009), hence it is also fixed-parameter tractable in multiple-interval graphs and their complements.
Given a graph G = (V, E) and two integers k and l , the problem k-Separating Vertices is that of deciding whether there is a partition V = X ∪ S ∪ Y of the vertices such that |X| = l , |S| ≤ k, and there is no edge between X and Y ? In other words, is it possible to cut l vertices off the graph by deleting k vertices? The problem k-Separating Vertices is one of several closely related graph separation problems considered by Marx (2004) in terms of parameterized complexity. Marx showed that k-Separating Vertices is W[1]-hard in general graphs with two parameters k and l , but is fixedparameterized tractable with three parameters k, l , and the maximum degree d of the graph.
In the following two theorems, we show that with two parameters k and l , k-Separating Vertices remains W[1]-hard in multiple-interval graphs and their complements: Theorem 3. k-Separating Vertices in balanced 2-track interval graphs is W[1]-hard with parameters k and l . Theorem 4. k-Separating Vertices in co-balanced 3-track interval graphs is W[1]-hard with parameters k and l .
The problem k-Separating Vertices was studied under the name Cutting l Vertices by Marx (2004), who also studied two closely related variants called Cutting l Connected Vertices and Cutting into l components: • In Cutting l Connected Vertices, the l vertices that are separated from the rest of G must induce a connected subgraph of G. • In Cutting into l components, the objective is to delete at most k vertices such that the remaining graph is broken into at least l connected components. Marx showed that Cutting l Connected Vertices is W[1]hard when parameterized by either k or l , and that Cutting into l components is W[1]-hard when parameterized by both k and l .
We observe that Marx’s proof of W[1]-hardness of Cutting l Connected Vertices with parameter l involves only line graphs, which are obviously a subclass of unit 2-interval graphs. We extend the other two W[1]-hardness results to multipleinterval graphs and their complements: Theorem 5. Cutting l Connected Vertices in balanced 2track interval graphs and co-balanced 3-track interval graphs is W[1]-hard with parameter k. Theorem 6. Cutting into l components in balanced 2-track interval graphs and co-balanced 3-track interval graphs is W[1]hard with parameters k and l .
Both k-Perfect Code and k-Irredundant Set are very important problems in the development of parameterized complexity theory. The problem k-Perfect Code was shown to be W[1]-hard as early as 1995, but its membership in W[1] was proved much later in 2002. Indeed this problem was once conjectured by Downey and Fellows (1999) either to represent an intermediate between W[1] and W[2], or is complete for W[2]. Similarly, the problem k-Irredundant Set was shown to be in W[1] in 1992, and was once conjectured as an intermediate between FPT and W[1] before it was finally proved to be W[1]hard in 2000.
Theorem 7 (Downey, Fellows, and Raman, 2000). k-Irredundant Set in general graphs is W[1]-hard with parameter k. The celebrated proof of Downey et al. (2000) was a major breakthrough in parameterized complexity theory, but it is rather complicated, spanning seven pages. In this paper, we give a very simple alternative proof (less than two pages) of Theorem 7. The problem k-Perfect Code, also known as k-Perfect Dominating Set, is a variant of k-Dominating Set. In a sibling paper (to appear in IPEC 2011) that focuses on the parameterized complexities of variants of k-Dominating Set in multiple-interval graphs and their complements, we showed that k-Perfect Code in unit 2-track interval graphs is W[1]hard with parameter k.
So Many Theorems, So Little Time. . . We have to focus on only one proof in this talk: k-Vertex Clique Partition in unit 2-interval graphs is W[1]-hard with parameter k.
k-Multicolored Clique Given a graph G and a vertex-coloring κ : V (G) → {1, 2, . . . , k}, decide whether G has a clique of k vertices containing exactly one vertex of each color. • Proved to be W[1]-complete by Fellows et al. (2009).
• Quickly becoming a standard tool for proving W[1]-hardness. • Simplifies FPT reductions in the same spirit as the colorcoding technique facilitates FPT algorithms. • Three conceptual components: vertex selection, edge selection, validation.
Overview Let (G, κ) be an instance of k-Multicolored Clique. We will construct a family F of unit 2-intervals such that G has a clique of k vertices containing exactly one vertex of each color if and only if the vertices of the intersection graph GF of F can be partitioned into k ′ cliques, where k ′ = 3k + 2 k2 .
A Technical Lemma Denote by Cn the cycle graph of n vertices c1 , . . . , cn and n edges ci ci +1 , 1 ≤ i ≤ n − 1, and cn c1 . Lemma 1. For each integer n ≥ 1, the cycle graph C4n+1 satisfies the following four properties: 1. bzzz. . . 2. chirp. . . chirp. . . 3. meow. . . meow. . . meow. . . 4. oink. . . oink. . . oink. . . oink. . .
Property 1 The chromatic number of C4n+1 is 3. C4n+1 is an odd cycle; hence it is not bipartite and has chromatic number at least 3. To achieve the chromatic number 3, we can assign each vertex ci the color 1 if i is odd but not equal to 4n + 1, the color 2 if i is even, and the color 3 if i is equal to 4n + 1.
Property 2 The chromatic number of the graph obtained from C4n+1 by deleting at least 1 and at most 2n vertices, is 2. With any vertex deleted from C4n+1 , the resulting graph does not have any cycles and hence is bipartite, with chromatic number at most 2. Note that the number of edges in C4n+1 is 4n + 1, and that each vertex is incident to 2 edges. With at most 2n vertices deleted from C4n+1 , the resulting graph has at least one edge remaining, and hence has chromatic number at least 2.
Property 3 In any partition of the vertices of C4n+1 into 3 independent sets, at most one independent set can have size one. Let I1 ∪ I2 ∪ I3 be any partition of the vertices of C4n+1 into 3 independent sets. Again note that the number of edges in C4n+1 is 4n+1 ≥ 5, and that each vertex is incident to 2 edges. If both I1 and I2 have size one, then the 2 vertices in I1 ∪I2 are together incident to at most 4 edges, and there must be at least one edge remaining between two vertices in I3 , which contradicts our assumption that it is an independent set.
Property 4 The complement graph C4n+1 is a unit 2-interval graph. Moreover, there exists a 2-partition An ∪ B3n+1 of the vertices such that the graph can be represented by one unit interval for each vertex ai ∈ An , 1 ≤ i ≤ n, and two unit intervals for each vertex bj ∈ B3n+1 , 1 ≤ j ≤ 3n + 1.
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Consider 4n + 1 vertices spread evenly on a circle of unit perimeter. Connect each vertex to the two farthest vertices by two edges. Then we obtain the cycle graph C4n+1 .
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The complement graph C4n+1 is clearly a circular-arc graph, i.e., the intersection graph of a set of circular-arcs, where each 2n . vertex is represented by an open circular arc of length 4n+1
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Let An be any n consecutive vertices along the circle and let B3n+1 be the remaining 3n + 1 vertices. Then the circular-arc representation of C4n+1 can be easily “cut” and “stretched” into a 2-interval representation.
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Let a1 . . . an b1 . . . b3n+1 be the 4n + 1 vertices along the circle. Then C4n+1 can be represented by one unit interval for each ai and two unit intervals for each bj in the order b1 . . . b3n+1 a1 . . . an b1 . . . b3n+1 .
Vertex Selection For each color i , 1 ≤ i ≤ k, let Vi be the set of vertices of color i . Let ni = |Vi |. We will construct one vertex gadget for each color i .
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Construct a graph C4ni +1 on the ni vertices in Vi and 3ni +1 additional dummy vertices, represented (using Property 4) by one unit interval for each vertex in Vi , and two unit intervals for each dummy vertex.
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This leaves one free interval for each vertex in Vi . Put these ni free intervals aside, pairwise-disjoint. Thus we have ni unit 2-intervals including one unit 2interval hui for each vertex u ∈ Vi , and 3ni + 1 additional dummy unit 2-intervals.
Edge Selection For each pair of distinct colors i and j, 1 ≤ i < j ≤ k, let Ei j be the set of edges uv such that u has color i and v has color j. Let mi j = |Ei j |. We will construct one edge gadget for each color pair i j.
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Construct a graph C4mij +1 on mi j vertices (one for each edge in Ei j ) and 3mi j + 1 additional dummy vertices, represented (using Property 4) by one unit interval for each edge in Ei j , and two unit intervals for each dummy vertex.
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For each edge uv = e ∈ Ei j , we construct two unit 2intervals huei and hv ei. Let hei be the unit interval in the representation of C4mij +1 that corresponds to the edge e. The two unit 2-intervals huei and hv ei share hei as one unit interval, and each of them has one more free interval. Thus we have 2mi j unit 2-intervals including two unit 2intervals huei and hv ei for each edge uv = e ∈ Ei j , and 3mi j + 1 additional dummy unit 2-intervals.
Validation
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For each edge uv = e ∈ Ei j , place the free interval of huei to coincide with the free interval of hui, and place the free interval of hv ei to coincide with the free interval of hv i.
Summary Let F be the following family of n+2m+(3n+3m+k + unit 2-intervals: F = hui | u ∈ Vi , 1 ≤ i ≤ k ∪ huei, hv ei | uv = e ∈ Ei j , 1 ≤ i < j ≤ k ∪ DUMMIES,
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If and Only If It remains to prove this lemma: Lemma 2. G has a k-multicolored clique if and only if GF has a k ′ -vertex clique partition. Note that the parameter k ′ = 3k + 2 not n.
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Direct Implication Suppose that G has a k-multicolored clique K. We will partition GF into k ′ = 3k + 2 steps: • 3 cliques for each color i , • 2 cliques for each color pair i j.
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For each color i , 1 ≤ i ≤ k, let Si be the subgraph of GF represented by the 4ni + 1 2-intervals for the ni vertices in Vi and the 3ni + 1 additional dummy vertices.
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Let ui be the vertex of color i in K. Put the 2-interval hui i, together with the 2-intervals hui ei for all edges e incident to ui , into one clique.
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The chromatic number of the graph obtained from C4n+1 by deleting at least 1 and at most 2n vertices, is 2. Since Si is isomorphic to C4ni +1 , it follows by Property 2 that the remaining 4ni 2-intervals in Si can be partitioned into two cliques. Thus we have three cliques for each color.
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For each pair of distinct colors i and j, 1 ≤ i < j ≤ k, let Si j be the subgraph of GF represented by the 5mi j + 1 2intervals including the two 2-intervals huei and hv ei for each edge uv = e ∈ Ei j and the 3mi j +1 additional dummy vertices.
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Let Si′j be the graph obtained from Si j by contracting each pair of vertices represented by huei and hv ei for some edge e (they have the same open neighborhood in Si j ) into a single vertex represented by hei. Then Si′j is isomorphic to C4mij +1 .
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Let ui vj = ei j be the edge in K such that ui has color i and vj has color j. The two 2-intervals hui ei j i and hvj ei j i have already been included in two cliques containing hui i and hvj i, respectively.
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Excluding hei j i, the remaining 4mi j 2-intervals in Si′j can be partitioned into two cliques by Property 2. Now expand each contracted vertex back into two vertices. The two cliques in Si′j remain two cliques in Si j . Thus we have two cliques for each pair of distinct colors.
Reverse Implication Suppose that GF has a k ′ -vertex clique partition. We will find a k-multicolored clique in G.
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Define the subgraphs Si , 1 ≤ i ≤ k, and the subgraphs Si j and Si′j , 1 ≤ i < j ≤ k, as before.
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The chromatic number of C4n+1 is 3. By Property 1, each subgraph Si of GF can be partitioned into 3 but no less than 3 cliques.
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Define Si′′j , 1 ≤ i < j ≤ k, as the subgraph of Si j (and of induced by the 3mi j + 1 dummy vertices.
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The chromatic number of the graph obtained from C4n+1 by deleting at least 1 and at most 2n vertices, is 2. Since Si′′j can be obtained from C4mij +1 by deleting mi j vertices, it follows by Property 2 that Si′′j can be partitioned into 2 but no less than 2 cliques.
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Observe that the k subgraphs Si and the k2 subgraphs Si′′j do not have edges in between. Since k ′ = 3k + 2 k2 , we must partition each subgraph Si into exactly 3 cliques, and partition each subgraph Si′′j into exactly 2 cliques. The remaining 2-intervals huei and hv ei for the edges e are then added to these cliques.
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For each pair of distinct colors i and j, 1 ≤ i < j ≤ k, since Si′j is isomorphic to C4mij +1 , it follows by Property 1 that there exists at least one edge uv = e ∈ Ei j such that neither huei nor hv ei is included in the two cliques for Si′′j .
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Then huei must be included in one of the three cliques for Si that includes hui, and hv ei must be included in one of the three cliques for Sj that includes hv i.
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Since huei intersects hui but not the other 2-intervals in Si , this clique includes only one 2-interval hui from Si . In any partition of the vertices of C4n+1 into 3 independent sets, at most one independent set can have size one. By Property 3, at most one of the three cliques for Si can include only one 2-interval from Si .
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Now for each color i , 1 ≤ i ≤ k, find the unique vertex ui such that the 2-interval hui i appears in a clique without any other 2-intervals from Si . Then the set of k vertices ui corresponds to a k-multicolored clique in G.
Multiple-Interval Graphs Let t ≥ 2 be an integer. A t-interval is the union of t disjoint intervals in the real line. A t-interval graph is the intersection graph of a family of t-intervals. If a t-interval graph has a representation in which the t disjoint intervals of each t-interval have the same length (although the intervals from different t-intervals may have different lengths), then the graph is a balanced t-interval graph. If a t-interval graph has a representation in which all intervals have unit lengths, then the graph is a unit t-interval graph.
Multiple-Track Interval Graphs Let t ≥ 2 be an integer. A t-track interval is the union of t disjoint intervals on t disjoint parallel lines called tracks, one interval on each track. A t-track interval graph is the intersection graph of a family of t-track intervals. If a t-track interval graph has a representation in which the t disjoint intervals of each t-track interval have the same length (although the intervals from different t-track intervals may have different lengths), then the graph is a balanced ttrack interval graph. If a t-track interval graph has a representation in which all intervals have unit lengths, then the graph is a unit t-track interval graph.
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Graph Hierarchy The t disjoint tracks for a t-track interval graph can be viewed as t disjoint “host” intervals in the real line for a tinterval graph. t-track interval graphs ⊂ t-interval graphs t-interval graphs ⊂ (t + 1)-interval graphs t-track interval graphs ⊂ (t + 1)-track interval graphs unit t-interval graphs ⊂ t-interval graphs unit t-track interval graphs ⊂ t-track interval graphs The most basic subclass: unit 2-track interval graphs
An Open Question In a recent paper, Jiang (2010) proved that the two problems k-Independent Set and k-Dominating Set remain W[1]hard even in unit 2-track interval graphs, which answers an open question of Fellows et al. (2009). Here is a question in the same spirit: Does k-Vertex Clique Partition remain W[1]-hard with parameter k in unit 2-track interval graphs?
