Parameterized Edge-Hamiltonicity Valia Mitsou, JST ERATO
Joint with: Michael Lampis, Kazuhisa Makino, Yushi Uno. Osaka RIMS, Kyoto Prefecture University University
Edge-Hamiltonian Path Edge-Hamiltonian Path 1 2 “Ordering of the edges such that consecutive edges share a common vertex.” 5 3 4 1/24
Edge-Hamiltonian Path Edge-Hamiltonian Path 1 2 “Ordering of the edges such that consecutive edges share a common vertex.” 5 3 4 Edge-Hamiltonian Path: 14, 1/24
Edge-Hamiltonian Path Edge-Hamiltonian Path 1 2 “Ordering of the edges such that consecutive edges share a common vertex.” 5 3 4 Edge-Hamiltonian Path: 14, 42, 1/24
Edge-Hamiltonian Path Edge-Hamiltonian Path 1 2 “Ordering of the edges such that consecutive edges share a common vertex.” 5 3 4 Edge-Hamiltonian Path: 14, 42, 25, 1/24
Edge-Hamiltonian Path Edge-Hamiltonian Path 1 2 “Ordering of the edges such that consecutive edges share a common vertex.” 5 3 4 Edge-Hamiltonian Path: 14, 42, 25, 51, 1/24
Edge-Hamiltonian Path Edge-Hamiltonian Path 1 2 “Ordering of the edges such that consecutive edges share a common vertex.” 5 3 4 Edge-Hamiltonian Path: 14, 42, 25, 51, 53, 1/24
Edge-Hamiltonian Path Edge-Hamiltonian Path 1 2 “Ordering of the edges such that consecutive edges share a common vertex.” 5 3 4 Edge-Hamiltonian Path: 14, 42, 25, 51, 53, 34, 1/24
Edge-Hamiltonian Path Edge-Hamiltonian Path 1 2 “Ordering of the edges such that consecutive edges share a common vertex.” 5 3 4 Edge-Hamiltonian Path: 14, 42, 25, 51, 53, 34, 32 1/24
Edge-Hamiltonian Path 1 2 Edge-Hamiltonian Path is like Hamiltonian Path 25 on Line Graphs. 23 15 24 14 3 5 35 34 4 Edge-Hamiltonian Path: 14, 42, 25, 51, 53, 34, 32 2/24
Edge-Hamiltonian Path 1 2 Edge-Hamiltonian Path is like Hamiltonian Path 25 on Line Graphs. 23 15 24 14 3 5 35 34 4 Edge-Hamiltonian Path: 14, 42, 25, 51, 53, 34, 32 2/24
Edge-Hamiltonian Path as Dominating Eulerian Subgraph Dominating Eulerian Subgraph 1 2 “Find a connected subgraph G' of G, st: 1. G' is Eulerian; 5 3 2. All remaining edges of G are covered by a vertex in G'.” 4 Harary & Nash-Williams (1965): Edge-Hamiltonian Path is equivalent with Dominating Eulerian Subgraph. 3/24
Edge-Hamiltonian Path as Dominating Eulerian Subgraph Dominating Eulerian Subgraph 1 2 “Find a connected subgraph G' of G, st: 1. G' is Eulerian; 5 3 2. All remaining edges of G are covered by a vertex in G'.” 4 Eulerian path: 53, 34 3/24
Edge-Hamiltonian Path as Dominating Eulerian Subgraph Dominating Eulerian Subgraph 1 2 “Find a connected subgraph G' of G, st: 1. G' is Eulerian; 5 3 2. All remaining edges of G are covered by a vertex in G'.” 4 Eulerian path: 53, 34 Edge-Hamiltonian Path: 3/24
Edge-Hamiltonian Path as Dominating Eulerian Subgraph Dominating Eulerian Subgraph 1 2 “Find a connected subgraph G' of G, st: 1. G' is Eulerian; 5 3 2. All remaining edges of G are covered by a vertex in G'.” 4 Eulerian path: 53, 34 Edge-Hamiltonian Path: 51 3/24
Edge-Hamiltonian Path as Dominating Eulerian Subgraph Dominating Eulerian Subgraph 1 2 “Find a connected subgraph G' of G, st: 1. G' is Eulerian; 5 3 2. All remaining edges of G are covered by a vertex in G'.” 4 Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52 3/24
Edge-Hamiltonian Path as Dominating Eulerian Subgraph Dominating Eulerian Subgraph 1 2 “Find a connected subgraph G' of G, st: 1. G' is Eulerian; 5 3 2. All remaining edges of G are covered by a vertex in G'.” 4 Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52, 53 3/24
Edge-Hamiltonian Path as Dominating Eulerian Subgraph Dominating Eulerian Subgraph 1 2 “Find a connected subgraph G' of G, st: 1. G' is Eulerian; 5 3 2. All remaining edges of G are covered by a vertex in G'.” 4 Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52, 53, 32 3/24
Edge-Hamiltonian Path as Dominating Eulerian Subgraph Dominating Eulerian Subgraph 1 2 “Find a connected subgraph G' of G, st: 1. G' is Eulerian; 5 3 2. All remaining edges of G are covered by a vertex in G'.” 4 Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52, 53, 32, 34 3/24
Edge-Hamiltonian Path as Dominating Eulerian Subgraph Dominating Eulerian Subgraph 1 2 “Find a connected subgraph G' of G, st: 1. G' is Eulerian; 5 3 2. All remaining edges of G are covered by a vertex in G'.” 4 Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52, 53, 32, 34, 41 3/24
Edge-Hamiltonian Path as Dominating Eulerian Subgraph Dominating Eulerian Subgraph 1 2 “Find a connected subgraph G' of G, st: 1. G' is Eulerian; 5 3 2. All remaining edges of G are covered by a vertex in G'.” 4 Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52, 53, 32, 34, 41, 42 3/24
EHP – related work EHP is NP-Complete (Bertossi 1981); – NP-Complete even on bipartite graphs (Lai and Wei 1993) or on graphs with maximum degree 3 (Ryjacek et al. 2011). Demaine et. al. 2014: EHP on bipartite graphs parameterized by the size of the smaller part is in XP. Motivation: UNO game. 4/24
Solitaire UNO as Hamiltonicity Deck of cards ● c colors; ● b numbers. Matching rule: Cards agree either in number or in color . ➔ Given m cards, discard them one by one, following the matching rule. 5/24
Solitaire UNO as Hamiltonicity Deck of cards ● c colors; ● b numbers. Matching rule: Cards agree either in number or in color . ➔ Given m cards, discard them one by one, following the matching rule. 5/24
Solitaire UNO as Edge-Hamiltonicity Colors Numbers g 2 b 3 r 4 y 5 6/24
Solitaire UNO as Edge-Hamiltonicity Colors Numbers g 2 b 3 r 4 y 5 Solution: 2, 5, 5, 5, 4, 2, 2, 3, 4, 4 6/24
EHP – related work EHP is NP-Complete (Bertossi 1981); – NP-Complete even on bipartite graphs (Lai and Wei 1993) or on graphs with maximum degree 3 (Ryjacek et al. 2011). Demaine et. al. 2014: EHP on bipartite graphs parameterized by the size of the smaller part is in XP. ● Question: Is EHP on bipartite graphs FPT? 7/24
Our Results ● EHP parameterized by size of a vertex cover admits a cubic kernel; – (Independently shown by Dei et. al. in FUN 2014) ● EHP on arbitrary hypergraphs parameterized by size of a hitting set is FPT; ● EHP parameterized by treewidth or cliquewidth is FPT. – (Vertex HP is FPT for tw but W-hard for cw.) – Invoke theorem by Gursky & Wanke (2000): Graphs of bounded cw and no large bi-cliques have bounded tw. 8/24
Bipartite Graphs (one small part) Colors (c) Numbers (b) g 0 b 1 IS r 2 y 3 4 IS 5 6 7 8 9 9/24
Graphs of small Vertex Cover Colors (c) Numbers (b) g 0 b 1 r 2 y 3 4 IS 5 6 7 8 9 10/24
Kernelization algorithms Given an instance of the problem, we construct (in polynomial time) an equivalent new instance with size that depends only on the parameter. 11/24
EHP in bipartite graphs with one small part admits a cubic kernel Given an instance of EHP with c colors and b numbers, we construct (in polynomial time) an equivalent new instance where: – For each color, there will be at most O(c 2 ) numbers of that color. – Thus, there will be at most O(c 3 ) numbers. 12/24
EHP in bipartite graphs with one small part admits a cubic kernel In an EHP, each color-group appears at most c times. SOL: 1 5 5 3 … 2 8 8 7 13/24
EHP in bipartite graphs with one small part admits a cubic kernel In an EHP, each color-group appears at most c times. SOL':1 5 8 2 … 3 5 8 7 13/24
EHP in bipartite graphs with one small part admits a cubic kernel In an EHP, each color-group appears at most c times. At most c-1 SOL': … … … At most c 14/24
EHP in bipartite graphs with one small part admits a cubic kernel Intuition: EHP has a small backbone, which we need to identify. All other edges can move freely in and out of the EHP. SOL': … … … 5 At most 2c 15/24
EHP in bipartite graphs with one small part admits a cubic kernel Intuition: EHP has a small backbone, which we need to identify. All other edges can move freely in and out of the EHP. SOL': … … … At most 2c 15/24
EHP in bipartite graphs with one small part admits a cubic kernel Intuition: EHP has a small backbone, which we need to identify. All other edges can move freely in and out of the EHP. SOL': … … … 5 At most 2c 15/24
EHP in bipartite graphs with one small part admits a cubic kernel colors b Blue numbers > 4c 2 16/24
EHP in bipartite graphs with one small part admits a cubic kernel colors b N(S) o S v S e m Blue g r ≤ 4c a l a l numbers l p > 4c 2 16/24
EHP in bipartite graphs with one small part admits a cubic kernel colors b N(S) o S v S e m Blue r a l a l numbers l p > 4c 2 7 16/24
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