Exact Algorithms for MinimumEdge Dominating Set and Lowest

Edge Dominating Set

Discrete Mathematics Lab.Ken Iwaide

February 18, 2016

2016 2 18

平成28年度数理工学専攻説明会

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京都大学大学院 情報学研究科 数理工学専攻

修士課程，博士課程の学生募集

Edge Dominating Set (EDS)An undirected graph

· Each edge dominates all adjacent edges & itself· edge is dominates by edge EDS

1

Outline

1. An algorithm solving Minimum EDS in time & Parameterized EDS in time

2. An algorithm solving Lowest EDS in time

2

NP-Hard Problems for EDS

Previous Time BoundsMinimum EDS

Input: An -vertex graph

Output: A minimum EDS of [Xiao & Nagamochi 2012]

Parameterized EDS

Input: An -vertex graph & an integer

Output: Does have an EDS of size ?

· Fast for small

[Iwaide & Nagamochi 2015]

3

Previous AlgorithmsMinimum EDS

Authors Time Bound YearRaman et al. 2007Fomin et al. (exp. space) 2009Van Rooij & Bodlaender 2008Xiao & Nagamochi 2012

Parameterized EDS

Fernau 2006Fomin et al. (exp. space) 2009Binkele-Raible & Fernau 2012Xiao et al. 2013Iwaide & Nagamochi 2015

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Branching AlgorithmSelect a vertex to decide

Branch on

used as an endpoint of EDS

1.2. not used

All neighbors of are used as endpoints· All vertices are decided (leaf node)

“minimum” EDS can be found in polynomial time[van Rooij & Bodlaender 2008] 5

Branching AlgorithmSelect a vertex to decide

Branch on

used as an endpoint of EDS

1.2. not used

All neighbors of are used as endpoints· All vertices are decided (leaf node)

“minimum” EDS can be found in polynomial time[van Rooij & Bodlaender 2008]

: measure Time bound

6

NP-Hard Problems for EDS

Previous Time BoundsMinimum EDSInput: An -vertex graph Output: A minimum EDS of [Xiao & Nagamochi 2012]

Parameterized EDSInput: An -vertex graph & an integer Output: Does have an EDS of size ?

· Fast for small [Iwaide & Nagamochi 2015]

How fast can solve?

7

NP-Hard Problems for EDS

Previous Time BoundsMinimum EDSInput: An -vertex graph Output: A minimum EDS of [Xiao & Nagamochi 2012]

Parameterized EDSInput: An -vertex graph & an integer Output: Does have an EDS of size ?

· Fast for small [Iwaide & Nagamochi 2015]

How fast can solve?

Our Purpose· Design an algorithm solving both problems with currently best time  time & time 8

Additional Selection Criteria: measure for

    Parameterized EDSNote: #· Criteria to select a vertex· Flexibility

· Keep decrement of measure time· Introduce further criteria for time

: measure for      Minimum EDSNote: #Better decrement of ?

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Outline

1. An algorithm solving Minimum EDS in time & Parameterized EDS in time

2. An algorithm solving Lowest EDS in time

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Lowest SolutionVertex Cover (VC) and Maximum Matching (MM)

Note: MM can be foundin polynomial time

VC MM

· To find a minimum VC NP-hard· VC with VC MM ? in polynomial time [Gavril 1977]

Lowest case is easy for some NP-hard problem

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Lower Bound on EDS· is a lower bound on EDSMM

EDS Endpoints of EDS become VC EDS VC MMLowest EDS

Input: An -vertex graph

Output: Does have an EDS of size ?MM

Note: special case of Parameterized EDS with MM

· Solvable in polynomial time? Prove the NP-completeness

· Solvable faster than time of PEDS?

  Design an -time algorithm 12

Properties of Lowest EDSWe revealed the following properties· Case MM is “odd”: reducible into “even” cases

“odd”   “even”   “even”

#edges

· Case MM is “even”:

edge LEDS dominates exactly two edges MM

edge MM is dominated by exactly one egde LEDS

New Reduction rules

Better time bound

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Conclusions & Future Work· Designed a polynomial-space algorithm solving

Minimum EDS in time & Parameterized EDS in time

· Proved the NP-completeness of Lowest EDS· Designed a polynomial-space algorithm solving

Lowest EDS in time

· For , efficiently solvable whether  EDS of size MM ?

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Minimum EDS by PEDS Algorithm· Minimum Independent EDS Minimum EDS

[Yannakakis & Gavril 1980]

: minimum size of EDS of found by a PEDS algorithm in poly. time

Case 1. minimum EDS that contains

Case 2. minimum EDS that contains

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NP-Completeness of Lowest EDS1-In-3 3SAT NP-hard [Garey & Johnson 1979]

Input: A set of variables & a set of clauses on s.t. each

clause has exactly three literals

Output: assignment on s.t. each clause has exactly one

true literal?

· Polynomial-time reduction to Lowest EDS

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Application· Evaluating the worst-case performance of trunking

Switch

Shared lines

Nodes

Occupied / Vacant lines

Bipartite graph

· Line is busy Occupied lines become a maximal matching of

· Minimum maximal matching Minimum EDS

[Yannakakis & Gavril 1980]18