On the Complexity of Some Packing and Covering Problems in Graphs and Hypergraphs
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1 On the Complexity of Some Packing and Covering Problems in Graphs and Hypergraphs Andreas Brandstädt University of Rostock, Germany (with C. Hundt, A. Leitert, M. Milanič, R. Mosca, R. Nevries, and D. Rautenbach)
2 Mostly based on results of
3 Mostly based on results of - > 100 papers on the Exact Cover, ED and EED problems - Some own results in: - [B., Hundt, Nevries LATIN 2010] - [B., Mosca ISAAC 2011; Algorithmica] - [B., Leitert, Rautenbach ISAAC 2012] - [B., Milanič, Nevries MFCS 2013]
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6 Exact Cover by 3-Sets (X3C) X3C problem [SP2] of [Garey, Johnson 1979]: INSTANCE: A finite set X with X 3q and a collection C of 3-element subsets of X. QUESTION: Does C contain an exact cover for X, i.e., a subcollection D C such that every element of X occurs in exactly one member of D?
7 a b c d e f
8 a b c d e f
9 Exact Cover by 3-Sets (X3C) Thm [Karp 1972] X3C is NP-complete. (reduction from 3DM) Remark. Exact Cover by 2-Sets corresponds to Perfect Matching.
10 Exact Cover Let H (V, E) be a hypergraph and w(e): e. Fact. M is an exact cover in H M is a max weight independent set in L(H) with w(m) V.
11 Efficient domination Let G (V, E) be a finite undirected graph. A vertex v dominates itself and its neighbors, i.e., v dominates N[v]. [Biggs 1973, Bange, Barkauskas, Slater 1988]: D is an efficient dominating set (e.d.) in G if (1) it is dominating in G and (2) every v V is dominated exactly once.
12 Efficient domination Not every graph has an efficient dominating set!
13 Efficient domination Not every graph has an efficient dominating set!
14 Efficient domination Not every graph has an efficient dominating set!
15 Efficient domination Not every graph has an efficient dominating set!
16 Efficient domination
17 Efficient domination Another name for e.d.: independent perfect dominating set. Let G 2 (V, E 2 ) with xy E 2 if d G (x,y) 2. Fact. Let N(G) be the closed neighborhood hypergraph of G (i.e., multiset of closed neighborhoods in G). Then: G 2 L(N(G)).
18 Efficient domination Fact. For D V, the following are equivalent: D is an e.d. in G. D dominating in G and independent in G 2. the closed neighborhoods N[v], v D, are an exact cover for V(G). Corollary. G has an e.d. N(G) has an exact cover for V(G).
19 Efficient domination ED problem INSTANCE: A finite graph G (V, E). QUESTION: Does G have an e.d.? Thm [Bange, Barkauskas, Slater 1988] ED is NP-complete.
20 Efficient domination Thm [Yen, Lee 1996] ED NP-complete for bipartite graphs and for chordal graphs. Corollary. For every k > 2, ED NP-complete for C k -free graphs.
21 Efficient domination Thm [Kratochvíl 1991] ED NP-complete for cubic planar bip. graphs. Thm [Lu, Tang 2002] ED NP-complete for chordal bip. graphs. (+ elegant proof by Haiko Müller in 2012)
22 Efficient domination Thm ED in pol. time for: - trees [Bange, Barkauskas, Slater 1988] - block graphs [Yen, Lee 1996] - series-parallel graphs [Yen, Lee 1994] - cocomparability gr. [Chang, Pandu Rangan, Coorg 1995] - circular-arc graphs [Chang, Liu 1994]
23 Efficient domination Thm ED in pol. time for: - bip. permutation graphs [Lu, Tang 2002] - trapezoid graphs [Liang, Lu, Tang 1997] - distance-hereditary graphs [Lu, Tang 2002] - split graphs [Chang, Liu 1993]
24 Efficient domination Open: [Lu, Tang 2002] ED for: - convex bipartite graphs - strongly chordal graphs
25 Complexity of Efficient Domination: claw-fr. perfect line gr. of planar cubic bipartite gr. bipartite chordal dist.-hered. cubic planar bipartite chordal bip. convex bip.? strongly chordal? block gr. trees permutation split gr. cocomparability trapezoid circular arc interval gr. bip. permutation P 4 fr.
26 Efficient edge domination [Grinstead, Slater, Sherwani, Holmes, 1993]: M E is an efficient edge dominating set (e.e.d.) in G if - M is dominating in L(G), and - every edge of E is dominated exactly once in L(G), that is, M is an e.d. in L(G).
27 Not every graph (not every tree!) has an e.e.d.:
28
29 G: e 1 e 2 e 3 e 4 e 5 e 1 e 2 e 3 e 4 L(G): e 5
30 Efficient edge domination EED problem INSTANCE: A finite graph G (V, E). QUESTION: Does G have an e.e.d.? Thm [Grinstead, Slater, Sherwani, Holmes 1993] EED is NP-complete. Corollary. ED is NP-c. for line graphs, and thus for claw-free graphs.
31 Efficient domination The ED problem for F-free graphs: Recall: If F contains a cycle or claw then ED is NP-complete on F-free graphs. ED on F-free graphs for linear forests F (i.e., disjoint unions of paths)? e.g. P 2 + P 3 + P 4
32 Efficient domination Fact. Every e.d. is a maximal independent set. Thm [Balas, Yu 1989], [Alexejev 1991], [Farber, Hujter, Tuza 1993], [Prisner 1995] kp 2 free graphs have at most n 2k-2 maximal independent sets, for any k > 0. Corollary. For every k, ED in pol. time for kp 2 free graphs.
33 Efficient domination Recall: Thm [Yen, Lee 1996] ED is NP-c. for bipartite graphs and for chordal graphs. Proof by simple standard reduction from X3C:
34 q r s a b c d e f
35 q r s a b c d e f
36 q r s a b c d e f
37 a b c d e f
38 a b c d e f
39 Efficient domination P 3 + P 3 = 2P 3 and P 7
40 Efficient domination From the standard reduction, it follows: Corollary. ED is NP-c. for 2P 3 -free graphs, and thus also for P 7 -free graphs.
41 Efficient domination From the standard reduction, it follows: Corollary. ED is NP-c. for 2P 3 -free graphs, and thus for P 7 -free graphs. Thm [B., Milanič, Nevries MFCS 2013] ED in time O(n m) for (P 2 + P 4 ) -free graphs and for P 5 -free graphs.
42 Efficient domination Summarizing: - For all F with < 6 vertices, the complexity of ED for F-free graphs is known. - For F with 6 vertices, the only open case is P 6. Thus, a main open problem is: Problem. Complexity of ED for P 6 free graphs.
43 Complexity of Efficient Domination: claw-fr. perfect P 7 fr. line gr. of planar cubic bipartite gr. 2P 3 fr. P 6 fr.? P 5 fr. P 2 +P 5 fr.? P 2 +P 4 fr. bipartite chordal kp 2 fr. cubic planar bip. strongly chordal? 2P 2 fr. P 4 fr. convex bip.? chordal bip. trees interval gr. split gr.
44 Efficient domination G is a split graph if V(G) is partitionable into a clique and an independent set. Thm [Földes, Hammer 1977] G is a split graph G is (2P 2,C 4,C 5 ) free. Thm [M.-S. Chang, Liu 1993] ED in time O(n + m) for split graphs.
45 thin spider
46 Efficient domination Lemma [B., Milanič, Nevries MFCS 2013] A prime 2P 2 -free graph has an e.d. it is a thin spider. Thm [B., Milanič, Nevries MFCS 2013] ED is solvable in time O(n + m) for 2P 2 -free graphs.
47 Efficient domination Recall: D is an e.d. in G D is dominating in G and independent in G 2. Let w(v): N [v]. Then: (i) D dominating in G V w(d). (ii) D independent in G 2 w(d) V.
48 Efficient domination Fact [Leitert; Milanič 2012] The following are equivalent: 1. D is an e.d. in G 2. D is a max weight independent set in G 2 with w(d) V. 3. D is a min weight dominating set in G with w(d) V.
49 Efficient domination Corollary. Let C be a graph class. If MWIS is solvable in pol. time for G 2, for all G C ED is solvable in pol. time on C.
50 Efficient domination Examples: Thm [B., Dragan, Chepoi, Voloshin 1994] G is dually chordal G 2 is chordal and N(G) is Helly. (recall G 2 L(N(G)).) Thm [Chang, Ho, Ko 2003] G AT-free G 2 cocomparability.
51 Complexity of Efficient Domination: claw-fr. perfect P 7 fr. line gr. of planar 2P 3 fr. P 6 fr.? P 2 +P 5 fr.? cubic bipartite gr. P 5 fr. P 2 +P 4 fr. bipartite dually chordal chordal kp 2 fr. chordal bipartite strongly chordal AT-fr. cocomparability 2P 2 fr. trees cubic planar bip. trapezoid convex bip. permutation circular arc split gr. bip. permutation interval gr. P 4 fr.
52 E net
53 Efficient domination Thm [Milanič 2013] G E and net free G 2 claw free.
54 Efficient domination Thm [Milanič 2013] G E and net free G 2 claw free. Thm [B., Milanič, Nevries MFCS 2013] G P 5 free and has an e.d. G 2 P 4 free. Corollary. ED is solvable in time O(MM) for P 5 free graphs.
55 Complexity of Efficient Domination: claw-fr. perfect P 7 fr. line gr. of planar 2P 3 fr. P 6 fr.? P 2 +P 5 fr.? cubic bipartite gr. P 5 fr. P 2 +P 4 fr. bipartite dually chordal chordal kp 2 fr. chordal bipartite strongly chordal AT-fr. cocomparability 2P 2 fr. trees cubic planar bip. trapezoid convex bip. permutation circular arc split gr. bip. permutation interval gr. P 4 fr.
56 Efficient domination Recall: [Leitert; Milanič 2012] The following are equivalent: 1. D is an e.d. in G 2. D is a max weight independent set in G 2 with w(d) V. 3. D is a min weight dominating set in G with w(d) V.
57 Efficient domination Thm [Damaschke, Müller, Kratsch 1990] Minimum dominating set for convex bip. graphs in pol. time. Thm [Gamble, Pulleyblank, Reed, Shepherd 1995] Min weight dominating set for convex bip. graphs in pol. time. Corollary. ED pol. for convex bip. graphs.
58 Efficient domination Thm [Müller 1997] convex bip. graphs interval bigraphs chordal bipartite. Thm [Keil 2012] Boolean width of 1. convex bipartite graphs is at most log n, 2. interval bigraphs is at most 2 log n.
59 Efficient domination Thm [Bui-Xuan, Telle, Vatshelle 2011] If for a graph class, boolean width O(log n) min weight dominating set in pol. time. Corollary. ED pol. for interval bigraphs.
60 k-colorability Thm [Holyer 1981] 3-Col is NP-c. for line graphs. Thm [Leven, Galil 1983] For all k > 3, k-col is NP-c. for line graphs. Corollary. For all k > 2, k-col is NP-c. for H-free graphs whenever H contains claw.
61 k-colorability Thm [Kamiński, Lozin 2007] For all k > 2, k-col is NP-c. for H-free graphs whenever H contains a cycle.
62 k-colorability Thm [Kamiński, Lozin 2007] For all k > 2, k-col is NP-c. for H-free graphs whenever H contains a cycle. Complexity of k-col for H-free graphs when H is a linear forest?
63 k-colorability Thm [Hoàng, Kamiński, Lozin, Sawada, Shu 2008] For all k, k-col in pol. time for P 5 -free graphs.
64 k-colorability Thm [Hoàng, Kamiński, Lozin, Sawada, Shu 2008] For all k, k-col in pol. time for P 5 -free graphs. Thm [Shenwei Huang MFCS 2013] 4-Col is NP-c. for P 7 -free graphs.
65 4-Col P 4 free P 5 free P 6 free P 7 free LIN POL? NP-c.
66 4-Col P 4 free P 5 free P 6 free P 7 free LIN POL? NP-c. ED LIN O(n m)? NP-c.
67 Efficient edge domination Recall: M E is an efficient edge dominating set (e.e.d.) in G if - M is dominating in L(G), and - every edge of E is dominated exactly once in L(G), that is, M is an e.d. in L(G).
68 Efficient edge domination
69 Efficient edge domination Thm [Lu, Tang 1998, Lu, Ko, Tang 2002] EED is NP-c. for bipartite graphs, and is solvable in linear time for - bipartite permutation graphs, - generalized series-parallel graphs and for - chordal graphs.
70 Efficient edge domination Open [Lu, Ko, Tang 2002] Complexity of EED for weakly chordal graphs and for permutation graphs.
71 Efficient edge domination Thm [Cardoso, Lozin 2008] EED is NP-c. for (very special) bipartite graphs, and is solvable in - linear time for chordal graphs, - pol. time for claw-free graphs.
72 Efficient edge domination Open [Cardoso, Korpelainen, Lozin 2011] Complexity of EED for - P k free graphs, k > 4, - chordal bipartite graphs, - weakly chordal graphs.
73 Complexity of Efficient Edge Domination: claw-fr. perfect bipartite weakly chordal? P 8 fr.? P 7 fr.? chordal bipartite? permutation? chordal P 6 fr.? P 5 fr.? bip. permutation trees P 4 fr.
74 Efficient edge domination Recall: D is an e.e.d. in G D is dominating in L(G) and independent in L(G) 2 (i.e., D is an e.d. in L(G)) Let w(e): N [e] (neighborhood w.r.t. L(G)). Fact. M is an e.e.d. in G M is a max weight independent set in L(G) 2 with w(m) E.
75 Efficient edge domination
76 Squares of Line Graphs G chordal L(G) 2 chordal [Cameron 1989] G circular-arc L(G) 2 circular-arc [Golumbic, Laskar 1993] G cocomparability L(G) 2 cocomparability [Golumbic, Lewenstein 2000] G weakly chordal L(G) 2 weakly chordal [Cameron, Sritharan, Tang 2003] stronger result for AT-free graphs [J.-M. Chang 2004]
77 Efficient edge domination Thm [B., Hundt, Nevries LATIN 2010] EED is solvable in - linear time for chordal bipartite graphs, - pol. time for hole-free graphs, and - is NP-c. for planar bipartite graphs with max degree 3.
78 Efficient edge domination Thm [B., Mosca ISAAC 2011; Algorithmica, online] EED in linear time for P 7 -free graphs in a robust way.
79 Complexity of Efficient Edge Domination: claw-fr. hole-fr. perfect P 9 fr.? P 8 fr. bipartite weakly chordal P 7 fr. chordal bipartite AT-fr. chordal P 6 fr. cubic planar bip. cocomparability trees P 5 fr. trapezoid split gr. permutation circular arc bip. permutation P 4 fr. interval gr.
80 Maximum Independent Set Thm [Lokshtanov, Vatshelle, Villanger 2013] Maximum Independent Set is polynomial for P 5 free graphs.
81 ED P 4 free P 5 free P 6 free P 7 free LIN O(n m)? NP-c. EED LIN LIN LIN LIN MIS LIN POL??
82 ED for hypergraphs H = (V,E) - a finite hypergraph. D V is an e.d. in H if D is an e.d. in 2sec(H). Thm [B., Leitert, Rautenbach ISAAC 2012] ED is NP-complete for acyclic hypergraphs, and is solvable in pol. time for hypertrees.
83 EED for hypergraphs
84 EED for hypergraphs Thm [B., Leitert, Rautenbach ISAAC 2012] EED is solvable in pol. time for acyclic hypergraphs, and is NP-c. for hypertrees.
85 Maximum induced matching H = (V,E) - a finite hypergraph. M E is an induced matching in H if M is an independent node set in L(H) 2. Thm [B., Leitert, Rautenbach ISAAC 2012] Max Induced Matching is - pol. time for acyclic hypergraphs, - NP-c. for hypertrees.
86 Exact Cover Thm [B., Leitert, Rautenbach ISAAC 2012] Exact Cover is - NP-c. for acyclic hypergraphs, and - pol. time for hypertrees.
87 [B., Leitert, Rautenbach ISAAC 2012]: chordal dually chordal acyclic hyp. hypertrees ED NP-c. [ ] lin. NP-c. pol. EED lin. [ ] lin. pol. NP-c. MIM pol. [ ] NP-c. pol. NP-c. XC NP-c. pol.
88 Thank you for your attention!
89 Thank you for your attention!
90 Thank you for your attention!
91 Efficient domination Corollary. For interval bigraphs, ED can be solved in polynomial time. Recall: G convex bipartite G interval bigraph
92 Efficient domination ED in Monadic Second Order Logic: Fact. G = (V, E) has an e.d. V V v V! v V (v N[v ])
93 Efficient edge domination EED in Monadic Second Order Logic: Fact. G = (V, E) has an e.e.d. E E e E! e E (e e )
94 T x 3 y 2 x 1 y 1 y 2 y 3 y 1 x 1 y 3 x 2 x 3 x 4 x 2 x 4 T is an interval bigraph; T 2 contains net x 2 x 3 x 4 y 1 y 2 y 3 (and thus AT) B y 1 x 2 y 3 x 4 x 1 y 2 x 4 y 4 x 3 y 2 y 1 y 3 x 1 x 2 x3 y 4 B is an interval bigraph; B 2 contains C 5 x 1 x 2 x 3 y 4 y 2
95 University of Rostock was founded in 1419
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