Minimum Latency Link Scheduling under Protocol Interference Model
|
|
|
- Rosalyn Horton
- 5 years ago
- Views:
Transcription
1 Minimum Latency Link Scheduling under Protocol Interference Model Peng-Jun Wan Peng-Jun Wan Minimum Latency Link Scheduling under Protocol Interference Model 1 / 31
2 Outline Problem Description Introduction to Graph Coloring First-Fit Link Scheduling Strip-wise Link Scheduling Peng-Jun Wan Minimum Latency Link Scheduling under Protocol Interference Model 2 / 31
3 Protocol Interference Model ρ(v) v r(v) u v u 1 v1 v2 u2 (a) (b) (c) Figure: (a) Communication range and interference range of each node; (b) a communication link; (c) a con icting pair of communication links. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 3 / 31
4 Link Schedule Given a set A of communication links A link schedule for A: a partition fa i : 1 i kg of A s.t. each A i is con ict-free Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 4 / 31
5 Link Schedule Given a set A of communication links A link schedule for A: a partition fa i : 1 i kg of A s.t. each A i is con ict-free k: latency (or length) of the schedule Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 4 / 31
6 Link Schedule Given a set A of communication links A link schedule for A: a partition fa i : 1 i kg of A s.t. each A i is con ict-free k: latency (or length) of the schedule Con ict graph H of A: a pair links in A are adjacent in H i they con ict with each other Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 4 / 31
7 Link Schedule Given a set A of communication links A link schedule for A: a partition fa i : 1 i kg of A s.t. each A i is con ict-free k: latency (or length) of the schedule Con ict graph H of A: a pair links in A are adjacent in H i they con ict with each other A link schedule for A corresponds to a vertex coloring of H Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 4 / 31
8 Minium-Latency Link Scheduling MLLS: Given a set A of communication links, nd a shortest link schedule for A Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 5 / 31
9 Minium-Latency Link Scheduling MLLS: Given a set A of communication links, nd a shortest link schedule for A NP-hard even restricted to the class of networks in which Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 5 / 31
10 Minium-Latency Link Scheduling MLLS: Given a set A of communication links, nd a shortest link schedule for A NP-hard even restricted to the class of networks in which all nodes have uniform (and xed) communication radii, Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 5 / 31
11 Minium-Latency Link Scheduling MLLS: Given a set A of communication links, nd a shortest link schedule for A NP-hard even restricted to the class of networks in which all nodes have uniform (and xed) communication radii, all nodes have uniform (and xed) interference radii, and Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 5 / 31
12 Minium-Latency Link Scheduling MLLS: Given a set A of communication links, nd a shortest link schedule for A NP-hard even restricted to the class of networks in which all nodes have uniform (and xed) communication radii, all nodes have uniform (and xed) interference radii, and the positions of all nodes are available Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 5 / 31
13 Summary on Scheduling Algorithms First- t link scheduling: applicable to any interference radii and any interference radii Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 6 / 31
14 Summary on Scheduling Algorithms First- t link scheduling: applicable to any interference radii and any interference radii Strip-wise link scheduling: applicable to uniform communication radii and uniform interference radii. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 6 / 31
15 Roadmap Problem Description Introduction to Graph Coloring First-Fit Link Scheduling Strip-wise Link Scheduling Peng-Jun Wan Minimum Latency Link Scheduling under Protocol Interference Model 7 / 31
16 Independence Number and Clique number v 4 v 3 v 5 v 1 v 2 v 6 v 7 Figure: α (G ) = 3 and fv 2, v 4, v 6 g is a maximum IS. ω (G ) = 4 and fv 1, v 2, v 3, v 7 g is a maximum clique. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 8 / 31
17 Vertex Coloring Vertex coloring: adj. nodes receive distinct colors partition of vertices into IS s Minimum Vertex Coloring: NP-hard in general chromatic number: χ (G ) χ (G ) max jv j α (G ), ω (G ). 2 v 4 3 v 3 v v 1 v v 6 v 7 Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 9 / 31
18 First-Fit Coloring hv 1, v 2,, v n i: a vertex ordering of V N (v i ) = fv j : 1 j < i, v j 2 N (v i )g. 2 v 4 3 v 3 v v 1 v v 6 v 7 Figure: First- t coloring in hv 1, v 2,, v 7 i, whose indictivity is 4. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 10 / 31
19 First-Fit Coloring hv 1, v 2,, v n i: a vertex ordering of V N (v i ) = fv j : 1 j < i, v j 2 N (v i )g. FF coloring in hv 1, v 2,, v n i: 2 v 4 3 v 3 v v 1 v v 6 v 7 Figure: First- t coloring in hv 1, v 2,, v 7 i, whose indictivity is 4. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 10 / 31
20 First-Fit Coloring hv 1, v 2,, v n i: a vertex ordering of V N (v i ) = fv j : 1 j < i, v j 2 N (v i )g. FF coloring in hv 1, v 2,, v n i: col (v 1 ) = f1g. 2 v 4 3 v 3 v v 1 v v 6 v 7 Figure: First- t coloring in hv 1, v 2,, v 7 i, whose indictivity is 4. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 10 / 31
21 First-Fit Coloring hv 1, v 2,, v n i: a vertex ordering of V N (v i ) = fv j : 1 j < i, v j 2 N (v i )g. FF coloring in hv 1, v 2,, v n i: col (v 1 ) = f1g. For i = 2 up to n, col (v i ) rst color /2 fcol (u) : u 2 N (v i )g. 2 v 4 3 v 3 v v 1 v v 6 v 7 Figure: First- t coloring in hv 1, v 2,, v 7 i, whose indictivity is 4. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 10 / 31
22 First-Fit Coloring hv 1, v 2,, v n i: a vertex ordering of V N (v i ) = fv j : 1 j < i, v j 2 N (v i )g. FF coloring in hv 1, v 2,, v n i: col (v 1 ) = f1g. For i = 2 up to n, col (v i ) rst color /2 fcol (u) : u 2 N (v i )g. number of colors 1 + max 1<in jn (v i )j. 2 v 4 3 v 3 v v 1 v v 6 v 7 Figure: First- t coloring in hv 1, v 2,, v 7 i, whose indictivity is 4. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 10 / 31
23 First-Fit Coloring hv 1, v 2,, v n i: a vertex ordering of V N (v i ) = fv j : 1 j < i, v j 2 N (v i )g. FF coloring in hv 1, v 2,, v n i: col (v 1 ) = f1g. For i = 2 up to n, col (v i ) rst color /2 fcol (u) : u 2 N (v i )g. number of colors 1 + max 1<in jn (v i )j. max 1<i n jn (v i )j: inductivity of hv 1, v 2,, v n i 2 v 4 3 v 3 v v 1 v v 6 v 7 Figure: First- t coloring in hv 1, v 2,, v 7 i, whose indictivity is 4. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 10 / 31
24 Smallest-Degree-Last Ordering Objective: nd a least-inductivity ordering hv 1, v 2,, v n i H G. For i = n down to 1, v i a vertex of H with least degree H H fv i g v 4 v 3 v 5 v 1 v 2 v 6 v 7 Figure: hv 1, v 7, v 6, v 5, v 4, v 3, v 2 i is a smallest-last ordering. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 11 / 31
25 Smallest-Degree-Last Ordering Inductivity of G: δ (G ) = max δ (G [U]) U V Theorem The smallest-degree-last ordering hv 1, v 2,, v n i achieves the smallest inductivity δ (G ) among all vertex orderings. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 12 / 31
26 Smallest-Degree-Last Ordering For each 1 < i n, jn (v i )j = δ (G [fv 1, v 2,, v i g]) δ (G ). Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 13 / 31
27 Smallest-Degree-Last Ordering For each 1 < i n, jn (v i )j = δ (G [fv 1, v 2,, v i g]) δ (G ). Let U be such that δ (G ) = δ (G [U]) and j = max f1 i n : v i 2 Ug. Then, δ (G ) = δ (G [U]) deg G [U ] (v j ) jn (v j )j. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 13 / 31
28 Inductive Local Independence Number (LIN) Inductive LIN of hv 1, v 2,, v n i: α = max 1<in fji j : I 2 I, I N (v i )g. v 4 v 3 v 5 v 1 v 2 v 6 v 7 Figure: Inductive indepence number of hv 1, v 2, v 3, v 4, v 5, v 6, v 7 i is 2, while the Inductive indepence number of hv 2, v 3, v 4, v 5, v 6, v 7, v 1 i is 3. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 14 / 31
29 Application of Inductive LIN Lemma Any vertex ordering with inductive LIN α has inductivity at most α (χ (G ) 1). Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 15 / 31
30 Application of Inductive LIN Lemma Any vertex ordering with inductive LIN α has inductivity at most α (χ (G ) 1). For any 1 < i n, χ (G ) jn (v i )j /α + 1 ) jn (v i )j α (χ (G ) 1) Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 15 / 31
31 Inward Local Independence Number (LIN) D = (V, A): an orientation of G v 4 v 3 v 4 v 3 v 5 v 2 v 5 v 2 v 1 v 1 v 6 v 7 (a) v 6 v 7 (b) Figure: Two graph orientations: (a) the local independence number is two, (b) the local independence number is one. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 16 / 31
32 Inward Local Independence Number (LIN) D = (V, A): an orientation of G Inward LIN of D: β = max u2v ji j : I 2 I, I N in D (u). v 4 v 3 v 4 v 3 v 5 v 2 v 5 v 2 v 1 v 1 v 6 v 7 (a) v 6 v 7 (b) Figure: Two graph orientations: (a) the local independence number is two, (b) the local independence number is one. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 16 / 31
33 Application of Inward LIN Lemma If G has an orientation D with inward LIN β, then δ (G ) 2β (χ (G ) 1). Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 17 / 31
34 Application of Inward LIN Lemma If G has an orientation D with inward LIN β, then δ (G ) 2β (χ (G ) 1). Consider a node v 2 V with largest in-degree in D. Then, χ (G ) in (D) /β + 1 ) in (D) β (χ (G ) 1). Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 17 / 31
35 Application of Inward LIN Lemma If G has an orientation D with inward LIN β, then δ (G ) 2β (χ (G ) 1). Consider a node v 2 V with largest in-degree in D. Then, χ (G ) in (D) /β + 1 ) in (D) β (χ (G ) 1). Let U be such that δ (G ) = δ (G [U]). D [U] contains at least one node u with in-degree out-degree. Thus, δ (G ) deg G [U ] (u) = deg in D [U ] (u) + degout D [U ] (u) 2 deg in D [U ] (u) 2 degin D (u) 2 in (D). Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 17 / 31
36 A Master Theorem on First-Fit Coloring Theorem Let G = (V, E ) be an undirected graph. 1 If G has an vertex ordering with inductive LIN α, then the rst- t coloring of G in the smallest-degree-last ordering uses at most α χ (G ) (α 1) colors and hence is a α -approximation. 2 If G has an orientation D with inward LIN β, then the rst- t coloring of G in the smallest-degree-last ordering uses at most 2β χ (G ) (2β 1) colors and hence is a 2β -approximation. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 18 / 31
37 Perfect Graphs A graph G = (V, E ) is perfect if it satis es any of the following three equivalent conditions: 1 For any U V, χ (G [U]) = ω (G [U]). 2 For any U V, α (G [U]) ω (G [U]) juj. 3 Neither G nor its complement contains an induced odd cycle of size larger than three. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 19 / 31
38 Perfect Graphs A graph G = (V, E ) is perfect if it satis es any of the following three equivalent conditions: 1 For any U V, χ (G [U]) = ω (G [U]). 2 For any U V, α (G [U]) ω (G [U]) juj. 3 Neither G nor its complement contains an induced odd cycle of size larger than three. Coloring of perfect graphs are solvable in poly. time. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 19 / 31
39 Cocomparability Graphs Cocomparability ordering hv 1, v 2,, v n i: i < j < k and v i v k 2 E ) either v i v j 2 E or v j v k 2 E. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 20 / 31
40 Cocomparability Graphs Cocomparability ordering hv 1, v 2,, v n i: i < j < k and v i v k 2 E ) either v i v j 2 E or v j v k 2 E. Perfect Minimum vertex coloring: reduced to maximum bipartite matching Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 20 / 31
41 Roadmap Problem Description Introduction to Graph Coloring First-Fit Link Scheduling Strip-wise Link Scheduling Peng-Jun Wan Minimum Latency Link Scheduling under Protocol Interference Model 21 / 31
42 First-Fit Link Scheduling First- t link scheduling: rst- t coloring of the con ict graph in the smallest-degree-last ordering Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 22 / 31
43 First-Fit Link Scheduling First- t link scheduling: rst- t coloring of the con ict graph in the smallest-degree-last ordering Assumption: c = min v 2V ρ (v) /r (v) > 1 Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 22 / 31
44 First-Fit Link Scheduling First- t link scheduling: rst- t coloring of the con ict graph in the smallest-degree-last ordering Assumption: c = min v 2V ρ (v) /r (v) > 1 Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 22 / 31
45 First-Fit Link Scheduling First- t link scheduling: rst- t coloring of the con ict graph in the smallest-degree-last ordering Assumption: c = min v 2V ρ (v) /r (v) > 1 Theorem The approximation bound of the rst- t link scheduling is at most 2 π/ arcsin c 1 2c 1. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 22 / 31
46 Orientation of The con ict Graph Orientation of a pair of con icting links a 1 = (u 1, v 1 ) and a 2 = (u 2, v 2 ) Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 23 / 31
47 Orientation of The con ict Graph Orientation of a pair of con icting links a 1 = (u 1, v 1 ) and a 2 = (u 2, v 2 ) if v 1 is in the interference range of u 2, take (a 2, a 1 ); otherwise (a 1, a 2 ). Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 23 / 31
48 Orientation of The con ict Graph Orientation of a pair of con icting links a 1 = (u 1, v 1 ) and a 2 = (u 2, v 2 ) if v 1 is in the interference range of u 2, take (a 2, a 1 ); otherwise (a 1, a 2 ). β : inward LIN of this orientation Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 23 / 31
49 Orientation of The con ict Graph Orientation of a pair of con icting links a 1 = (u 1, v 1 ) and a 2 = (u 2, v 2 ) if v 1 is in the interference range of u 2, take (a 2, a 1 ); otherwise (a 1, a 2 ). β : inward LIN of this orientation Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 23 / 31
50 Orientation of The con ict Graph Orientation of a pair of con icting links a 1 = (u 1, v 1 ) and a 2 = (u 2, v 2 ) if v 1 is in the interference range of u 2, take (a 2, a 1 ); otherwise (a 1, a 2 ). β : inward LIN of this orientation Lemma β π/ arcsin c 1 2c 1. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 23 / 31
51 Angle Separation q ρ o ρ p 1 v Figure: In a convex quadruple upvq with kpuk = kpqk = ρ 1 = kpvk and kuqk = kuvk, we have dquv 2 arcsin ρ 1 2ρ. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 24 / 31
52 Local Independence Number Consider three links a i = (u i, v i ) for 1 i 3 s.t. (1) v 1 is in the interference ranges of u 2 and u 3 (2) a 2 and a 3 are con ict-free. v 3 q v 1 u 2 ρ( ) ρ( u ) 2 u 2 p r( ) u 2 v 2 Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 25 / 31
53 Local Independence Number Consider three links a i = (u i, v i ) for 1 i 3 s.t. (1) v 1 is in the interference ranges of u 2 and u 3 (2) a 2 and a 3 are con ict-free. v 3 q \v 2 v 1 v 3 > [v 2 v 1 q v 1 u 2 ρ( ) ρ( u ) 2 u 2 p r( ) u 2 v 2 2 arcsin ρ (u 2) r (u 2 ) 2ρ (u 2 ) 2 arcsin c 1 2c. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 25 / 31
54 Local Independence Number Consider three links a i = (u i, v i ) for 1 i 3 s.t. (1) v 1 is in the interference ranges of u 2 and u 3 (2) a 2 and a 3 are con ict-free. v 3 q \v 2 v 1 v 3 > [v 2 v 1 q v 1 u 2 ρ( ) ρ( u ) 2 u 2 p r( ) u 2 v 2 Hence, 2 arcsin ρ (u 2) r (u 2 ) 2ρ (u 2 ) 2 arcsin c 1 2c. β π/ arcsin c 1 2c 1. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 25 / 31
55 Roadmap Problem Description Introduction to Graph Coloring First-Fit Link Scheduling Strip-wise Link Scheduling Peng-Jun Wan Minimum Latency Link Scheduling under Protocol Interference Model 26 / 31
56 Uniform Communication/Interference Radii Communication radii: normalized to one Peng-Jun Wan Minimum Latency Link Scheduling under Protocol Interference Model 27 / 31
57 Uniform Communication/Interference Radii Communication radii: normalized to one Interference radii: ρ > 1. Peng-Jun Wan Minimum Latency Link Scheduling under Protocol Interference Model 27 / 31
58 Uniform Communication/Interference Radii Communication radii: normalized to one Interference radii: ρ > 1. Better scheduling algorithm Peng-Jun Wan Minimum Latency Link Scheduling under Protocol Interference Model 27 / 31
59 A Height Function h (ρ) = (ρ 1) sin arccos ρ 1 2ρ arcsin 1 ρ, ρ 2 [1, ) Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 28 / 31
60 A Height Function h (ρ) = (ρ 1) sin arccos ρ 1 2ρ arcsin 1 ρ, ρ 2 [1, ) p u 1 q ρ ρ ρ 1 h(ρ) v Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 28 / 31
61 A Height Function h (ρ) = (ρ 1) sin arccos ρ 1 2ρ arcsin 1 ρ, ρ 2 [1, ) p h ρ ρ 1 h(ρ) 6 4 u 1 q ρ v rho Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 28 / 31
62 A Strip-wise Transitivity of Independence v 3 u 1 v 2 u 3 h(ρ) u 2 v 1 Figure: If both (u 1, v 1 ) and (u 3, v 3 ) are independent with (u 2, v 2 ), then they are independent with each other. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 29 / 31
63 Strip-wise Link Scheduling with ρ > 1 ρ+1 (ρ+1)/ f (ρ) Figure: Partition of the plane into strips of height (ρ + 1) /f (ρ) where f (ρ) = d(ρ + 1) /h (ρ)e. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 30 / 31
64 Approximation Bound approx. ratio 1 + f (ρ) = 1 + k if ρ 2 [ρ k, ρ k 1 ), where ρ 1 =, and ρ k with k 2 is the root of (ρ + 1) /h (ρ) = k rho Figure: Plot of (ρ + 1) /h (ρ). k ρ k k ρ k Table: Numberic values of ρ k for 2 k 11. Peng-Jun Wan (wan@cs.iit.edu) Minimum Latency Link Scheduling under Protocol Interference Model 31 / 31
CDS in Plane Geometric Networks: Short, Small, And Sparse
CDS in Plane Geometric Networks: Short, Small, And Sparse Peng-Jun Wan Peng-Jun Wan () CDS in Plane Geometric Networks: Short, Small, And Sparse 1 / 38 Outline Three Parameters of CDS Dominators Basic
Coloring. Basics. A k-coloring of a loopless graph G is a function f : V (G) S where S = k (often S = [k]).
![Coloring. Basics. A k-coloring of a loopless graph G is a function f : V (G) S where S = k (often S = [k]).](/thumbs/83/87488957.jpg "Coloring. Basics. A k-coloring of a loopless graph G is a function f : V (G) S where S = k (often S = [k]).")
Coloring Basics A k-coloring of a loopless graph G is a function f : V (G) S where S = k (often S = [k]). For an i S, the set f 1 (i) is called a color class. A k-coloring is called proper if adjacent
On the Dynamic Chromatic Number of Graphs
On the Dynamic Chromatic Number of Graphs Maryam Ghanbari Joint Work with S. Akbari and S. Jahanbekam Sharif University of Technology m_phonix@math.sharif.ir 1. Introduction Let G be a graph. A vertex
Graph coloring, perfect graphs
Lecture 5 (05.04.2013) Graph coloring, perfect graphs Scribe: Tomasz Kociumaka Lecturer: Marcin Pilipczuk 1 Introduction to graph coloring Definition 1. Let G be a simple undirected graph and k a positive
Maximum-Life Routing Schedule
Maximum-Life Routing Schedule Peng-Jun Wan wan@cs.iit.edu Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 1 / 42 Outline Problem Description Min-Cost Routing Ellipsoid Algorithm Price-Directive
On Dominator Colorings in Graphs
On Dominator Colorings in Graphs Ralucca Michelle Gera Department of Applied Mathematics Naval Postgraduate School Monterey, CA 994, USA ABSTRACT Given a graph G, the dominator coloring problem seeks a
Massachusetts Institute of Technology 6.042J/18.062J, Fall 02: Mathematics for Computer Science Professor Albert Meyer and Dr.
Massachusetts Institute of Technology 6.042J/18.062J, Fall 02: Mathematics for Computer Science Professor Albert Meyer and Dr. Radhika Nagpal Quiz 1 Appendix Appendix Contents 1 Induction 2 2 Relations
d 2 -coloring of a Graph
d -coloring of a Graph K. Selvakumar and S. Nithya Department of Mathematics Manonmaniam Sundaranar University Tirunelveli 67 01, Tamil Nadu, India E-mail: selva 158@yahoo.co.in Abstract A subset S of
Cographs; chordal graphs and tree decompositions
Cographs; chordal graphs and tree decompositions Zdeněk Dvořák September 14, 2015 Let us now proceed with some more interesting graph classes closed on induced subgraphs. 1 Cographs The class of cographs
NP-complete Problems
NP-complete Problems HP, TSP, 3COL, 0/1IP Dimitris Diamantis µπλ November 6, 2014 Dimitris Diamantis (µπλ ) NP-complete Problems November 6, 2014 1 / 34 HAMILTON PATH is NP-Complete Definition Given an
Induced Subgraph Isomorphism on proper interval and bipartite permutation graphs
Induced Subgraph Isomorphism on proper interval and bipartite permutation graphs Pinar Heggernes Pim van t Hof Daniel Meister Yngve Villanger Abstract Given two graphs G and H as input, the Induced Subgraph
Multi-coloring and Mycielski s construction
Multi-coloring and Mycielski s construction Tim Meagher Fall 2010 Abstract We consider a number of related results taken from two papers one by W. Lin [1], and the other D. C. Fisher[2]. These articles
Equitable list colorings of planar graphs without short cycles
Theoretical Computer Science 407 (008) 1 8 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs Equitable list colorings of planar graphs
CS21 Decidability and Tractability
CS21 Decidability and Tractability Lecture 18 February 16, 2018 February 16, 2018 CS21 Lecture 18 1 Outline the complexity class NP 3-SAT is NP-complete NP-complete problems: independent set, vertex cover,
4-coloring P 6 -free graphs with no induced 5-cycles
4-coloring P 6 -free graphs with no induced 5-cycles Maria Chudnovsky Department of Mathematics, Princeton University 68 Washington Rd, Princeton NJ 08544, USA mchudnov@math.princeton.edu Peter Maceli,
Algorithms Design & Analysis. Approximation Algorithm
Algorithms Design & Analysis Approximation Algorithm Recap External memory model Merge sort Distribution sort 2 Today s Topics Hard problem Approximation algorithms Metric traveling salesman problem A
A taste of perfect graphs
A taste of perfect graphs Remark Determining the chromatic number of a graph is a hard problem, in general, and it is even hard to get good lower bounds on χ(g). An obvious lower bound we have seen before
INDUCED CYCLES AND CHROMATIC NUMBER
INDUCED CYCLES AND CHROMATIC NUMBER A.D. SCOTT DEPARTMENT OF MATHEMATICS UNIVERSITY COLLEGE, GOWER STREET, LONDON WC1E 6BT Abstract. We prove that, for any pair of integers k, l 1, there exists an integer
Algorithms and Theory of Computation. Lecture 22: NP-Completeness (2)
Algorithms and Theory of Computation Lecture 22: NP-Completeness (2) Xiaohui Bei MAS 714 November 8, 2018 Nanyang Technological University MAS 714 November 8, 2018 1 / 20 Set Cover Set Cover Input: a set
Preliminaries. Graphs. E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)}
Preliminaries Graphs G = (V, E), V : set of vertices E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) 1 2 3 5 4 V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)} 1 Directed Graph (Digraph)
FACTOR GRAPH OF NON-COMMUTATIVE RING
International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 9, Issue 6, November - December 2018, pp. 178 183, Article ID: IJARET_09_06_019 Available online at http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=9&itype=6
An approximate version of Hadwiger s conjecture for claw-free graphs
An approximate version of Hadwiger s conjecture for claw-free graphs Maria Chudnovsky Columbia University, New York, NY 10027, USA and Alexandra Ovetsky Fradkin Princeton University, Princeton, NJ 08544,
Topics on Computing and Mathematical Sciences I Graph Theory (6) Coloring I
Topics on Computing and Mathematical Sciences I Graph Theory (6) Coloring I Yoshio Okamoto Tokyo Institute of Technology May, 008 Last updated: Wed May 6: 008 Y. Okamoto (Tokyo Tech) TCMSI Graph Theory
Approximation algorithms for conflict-free channel assignment in wireless ad hoc networks
WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wirel. Commun. Mob. Comput. 2006; 6:201 211 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/wcm.380 Approximation algorithms
Minimization of Symmetric Submodular Functions under Hereditary Constraints
Minimization of Symmetric Submodular Functions under Hereditary Constraints J.A. Soto (joint work with M. Goemans) DIM, Univ. de Chile April 4th, 2012 1 of 21 Outline Background Minimal Minimizers and
On non-bipartite distance-regular graphs with valency k and smallest eigenvalue not larger than k/2
On non-bipartite distance-regular graphs with valency k and smallest eigenvalue not larger than k/2 J. Koolen School of Mathematical Sciences USTC (Based on joint work with Zhi Qiao) CoCoA, July, 2015
4/30/14. Chapter Sequencing Problems. NP and Computational Intractability. Hamiltonian Cycle
Chapter 8 NP and Computational Intractability Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 2 Hamiltonian Cycle 8.5 Sequencing Problems HAM-CYCLE: given an undirected
Upper Bounds of Dynamic Chromatic Number
Upper Bounds of Dynamic Chromatic Number Hong-Jian Lai, Bruce Montgomery and Hoifung Poon Department of Mathematics West Virginia University, Morgantown, WV 26506-6310 June 22, 2000 Abstract A proper vertex
The chromatic number of ordered graphs with constrained conflict graphs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 69(1 (017, Pages 74 104 The chromatic number of ordered graphs with constrained conflict graphs Maria Axenovich Jonathan Rollin Torsten Ueckerdt Department
This is a repository copy of Chromatic index of graphs with no cycle with a unique chord.
This is a repository copy of Chromatic index of graphs with no cycle with a unique chord. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/74348/ Article: Machado, RCS, de
Chapter 34: NP-Completeness
Graph Algorithms - Spring 2011 Set 17. Lecturer: Huilan Chang Reference: Cormen, Leiserson, Rivest, and Stein, Introduction to Algorithms, 2nd Edition, The MIT Press. Chapter 34: NP-Completeness 2. Polynomial-time
Dominator Colorings and Safe Clique Partitions
Dominator Colorings and Safe Clique Partitions Ralucca Gera, Craig Rasmussen Naval Postgraduate School Monterey, CA 994, USA {rgera,ras}@npsedu and Steve Horton United States Military Academy West Point,
NP and Computational Intractability
NP and Computational Intractability 1 Review Basic reduction strategies. Simple equivalence: INDEPENDENT-SET P VERTEX-COVER. Special case to general case: VERTEX-COVER P SET-COVER. Encoding with gadgets:
Complexity of locally injective homomorphism to the Theta graphs
Complexity of locally injective homomorphism to the Theta graphs Bernard Lidický and Marek Tesař Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Prague, Czech Republic
CS 301: Complexity of Algorithms (Term I 2008) Alex Tiskin Harald Räcke. Hamiltonian Cycle. 8.5 Sequencing Problems. Directed Hamiltonian Cycle
8.5 Sequencing Problems Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems: SAT, 3-SAT. Sequencing problems: HAMILTONIAN-CYCLE,
8.5 Sequencing Problems
8.5 Sequencing Problems Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems: SAT, 3-SAT. Sequencing problems: HAMILTONIAN-CYCLE,
Distinguishing Chromatic Number and NG-Graphs
Wellesley College Groups in Galway May 19, 2018 Table of contents 1 Distinguishing Number and Distinguishing Chromatic Number 2 Nordhaus-Gaddum Inequality 3 NG-Graphs Distinguishing Number and Distinguishing
Discrete Optimization 2010 Lecture 10 P, N P, and N PCompleteness
Discrete Optimization 2010 Lecture 10 P, N P, and N PCompleteness Marc Uetz University of Twente m.uetz@utwente.nl Lecture 9: sheet 1 / 31 Marc Uetz Discrete Optimization Outline 1 N P and co-n P 2 N P-completeness
Notes on Graph Theory
Notes on Graph Theory Maris Ozols June 8, 2010 Contents 0.1 Berge s Lemma............................................ 2 0.2 König s Theorem........................................... 3 0.3 Hall s Theorem............................................
Complexity Theory VU , SS The Polynomial Hierarchy. Reinhard Pichler
Complexity Theory Complexity Theory VU 181.142, SS 2018 6. The Polynomial Hierarchy Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien 15 May, 2018 Reinhard
Outline. Complexity Theory EXACT TSP. The Class DP. Definition. Problem EXACT TSP. Complexity of EXACT TSP. Proposition VU 181.
Complexity Theory Complexity Theory Outline Complexity Theory VU 181.142, SS 2018 6. The Polynomial Hierarchy Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität
1. (a) Explain the asymptotic notations used in algorithm analysis. (b) Prove that f(n)=0(h(n)) where f(n)=0(g(n)) and g(n)=0(h(n)).
Code No: R05220502 Set No. 1 1. (a) Explain the asymptotic notations used in algorithm analysis. (b) Prove that f(n)=0(h(n)) where f(n)=0(g(n)) and g(n)=0(h(n)). 2. (a) List some of the relative advantages
Math 5707: Graph Theory, Spring 2017 Midterm 3
University of Minnesota Math 5707: Graph Theory, Spring 2017 Midterm 3 Nicholas Rancourt (edited by Darij Grinberg) December 25, 2017 1 Exercise 1 1.1 Problem Let G be a connected multigraph. Let x, y,
Even Pairs and Prism Corners in Square-Free Berge Graphs
Even Pairs and Prism Corners in Square-Free Berge Graphs Maria Chudnovsky Princeton University, Princeton, NJ 08544 Frédéric Maffray CNRS, Laboratoire G-SCOP, University of Grenoble-Alpes, France Paul
Dominating Set Counting in Graph Classes
Dominating Set Counting in Graph Classes Shuji Kijima 1, Yoshio Okamoto 2, and Takeaki Uno 3 1 Graduate School of Information Science and Electrical Engineering, Kyushu University, Japan kijima@inf.kyushu-u.ac.jp
Fractional and circular 1-defective colorings of outerplanar graphs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 6() (05), Pages Fractional and circular -defective colorings of outerplanar graphs Zuzana Farkasová Roman Soták Institute of Mathematics Faculty of Science,
Shortest Link Scheduling with Power Control under Physical Interference Model
2010 Sixth International Conference on Mobile Ad-hoc and Sensor Networks Shortest Link Scheduling with Power Control under Physical Interference Model Peng-Jun Wan Xiaohua Xu Department of Computer Science
4/20/11. NP-complete problems. A variety of NP-complete problems. Hamiltonian Cycle. Hamiltonian Cycle. Directed Hamiltonian Cycle
A variety of NP-complete problems NP-complete problems asic genres. Packing problems: SE-PACKING, INDEPENDEN SE. Covering problems: SE-COVER, VEREX-COVER. Constraint satisfaction problems: SA, 3-SA. Sequencing
Strongly chordal and chordal bipartite graphs are sandwich monotone
Strongly chordal and chordal bipartite graphs are sandwich monotone Pinar Heggernes Federico Mancini Charis Papadopoulos R. Sritharan Abstract A graph class is sandwich monotone if, for every pair of its
On the oriented chromatic index of oriented graphs
On the oriented chromatic index of oriented graphs Pascal Ochem, Alexandre Pinlou, Éric Sopena LaBRI, Université Bordeaux 1, 351, cours de la Libération 33405 Talence Cedex, France February 19, 2006 Abstract
Triangle-free graphs that do not contain an induced subdivision of K 4 are 3-colorable
Triangle-free graphs that do not contain an induced subdivision of K 4 are 3-colorable Maria Chudnovsky Princeton University, Princeton, NJ 08544 Chun-Hung Liu Princeton University, Princeton, NJ 08544
On the chromatic number of the lexicographic product and the Cartesian sum of graphs
On the chromatic number of the lexicographic product and the Cartesian sum of graphs Niko Čižek University of Maribor TF, Smetanova ulica 17 62000 Maribor, Slovenia Sandi Klavžar University of Maribor
The Waldschmidt constant for squarefree monomial ideals
The Waldschmidt constant for squarefree monomial ideals Alexandra Seceleanu (joint with C. Bocci, S. Cooper, E. Guardo, B. Harbourne, M. Janssen, U. Nagel, A. Van Tuyl, T. Vu) University of Nebraska-Lincoln
Induced Saturation of Graphs
Induced Saturation of Graphs Maria Axenovich a and Mónika Csikós a a Institute of Algebra and Geometry, Karlsruhe Institute of Technology, Englerstraße 2, 76128 Karlsruhe, Germany Abstract A graph G is
Exact Algorithms for Dominating Induced Matching Based on Graph Partition
Exact Algorithms for Dominating Induced Matching Based on Graph Partition Mingyu Xiao School of Computer Science and Engineering University of Electronic Science and Technology of China Chengdu 611731,
Zero-Sum Flows in Regular Graphs
Zero-Sum Flows in Regular Graphs S. Akbari,5, A. Daemi 2, O. Hatami, A. Javanmard 3, A. Mehrabian 4 Department of Mathematical Sciences Sharif University of Technology Tehran, Iran 2 Department of Mathematics
PERFECT GRAPHS AND THE PERFECT GRAPH THEOREMS
PERFECT GRAPHS AND THE PERFECT GRAPH THEOREMS PETER BALLEN Abstract. The theory of perfect graphs relates the concept of graph colorings to the concept of cliques. In this paper, we introduce the concept
1 The Well Ordering Principle, Induction, and Equivalence Relations
1 The Well Ordering Principle, Induction, and Equivalence Relations The set of natural numbers is the set N = f1; 2; 3; : : :g. (Some authors also include the number 0 in the natural numbers, but number
Some results on the reduced power graph of a group
Some results on the reduced power graph of a group R. Rajkumar and T. Anitha arxiv:1804.00728v1 [math.gr] 2 Apr 2018 Department of Mathematics, The Gandhigram Rural Institute-Deemed to be University, Gandhigram
Equitable coloring of random graphs
Michael Krivelevich 1, Balázs Patkós 2 1 Tel Aviv University, Tel Aviv, Israel 2 Central European University, Budapest, Hungary Phenomena in High Dimensions, Samos 2007 A set of vertices U V (G) is said
Counting and Constructing Minimal Spanning Trees. Perrin Wright. Department of Mathematics. Florida State University. Tallahassee, FL
Counting and Constructing Minimal Spanning Trees Perrin Wright Department of Mathematics Florida State University Tallahassee, FL 32306-3027 Abstract. We revisit the minimal spanning tree problem in order
On shredders and vertex connectivity augmentation
On shredders and vertex connectivity augmentation Gilad Liberman The Open University of Israel giladliberman@gmail.com Zeev Nutov The Open University of Israel nutov@openu.ac.il Abstract We consider the
1 Integer Decomposition Property
CS 598CSC: Combinatorial Optimization Lecture date: Feb 2, 2010 Instructor: Chandra Chekuri Scribe: Siva Theja Maguluri Material taken mostly from [1] (Chapter 19). 1 Integer Decomposition Property A polyhedron
ACO Comprehensive Exam March 17 and 18, Computability, Complexity and Algorithms
1. Computability, Complexity and Algorithms (a) Let G(V, E) be an undirected unweighted graph. Let C V be a vertex cover of G. Argue that V \ C is an independent set of G. (b) Minimum cardinality vertex
Combinatorial Optimization
Combinatorial Optimization Problem set 8: solutions 1. Fix constants a R and b > 1. For n N, let f(n) = n a and g(n) = b n. Prove that f(n) = o ( g(n) ). Solution. First we observe that g(n) 0 for all
UC Berkeley CS 170: Efficient Algorithms and Intractable Problems Handout 22 Lecturer: David Wagner April 24, Notes 22 for CS 170
UC Berkeley CS 170: Efficient Algorithms and Intractable Problems Handout 22 Lecturer: David Wagner April 24, 2003 Notes 22 for CS 170 1 NP-completeness of Circuit-SAT We will prove that the circuit satisfiability
Hanna Furmańczyk EQUITABLE COLORING OF GRAPH PRODUCTS
Opuscula Mathematica Vol. 6 No. 006 Hanna Furmańczyk EQUITABLE COLORING OF GRAPH PRODUCTS Abstract. A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a
Circular choosability of graphs
Circular choosability of graphs Xuding Zhu Department of Applied Mathematics National Sun Yat-sen University Kaohsiung, Taiwan zhu@math.nsysu.edu.tw Abstract This paper discusses the circular version of
Blockers for the stability number and the chromatic number
Blockers for the stability number and the chromatic number C. Bazgan C. Bentz C. Picouleau B. Ries Abstract Given an undirected graph G = (V, E) and two positive integers k and d, we are interested in
Equitable Coloring On Mycielskian Of Wheels And Bigraphs
Applied Mathematics E-Notes, 1(201), 174-182 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Equitable Coloring On Mycielskian Of Wheels And Bigraphs Kaliraj Kalimuthu,
Graph homomorphism into an odd cycle
Graph homomorphism into an odd cycle Hong-Jian Lai West Virginia University, Morgantown, WV 26506 EMAIL: hjlai@math.wvu.edu Bolian Liu South China Normal University, Guangzhou, P. R. China EMAIL: liubl@hsut.scun.edu.cn
EPTAS for Maximum Clique on Disks and Unit Balls
EPTAS for Maximum Clique on Disks and Unit Balls Édouard Bonnet joint work with Panos Giannopoulos, Eunjung Kim, Paweł Rzążewski, and Florian Sikora and Marthe Bonamy, Nicolas Bousquet, Pierre Chabit,
A New Facet Generating Procedure for the Stable Set Polytope
1 / 22 A New Facet Generating Procedure for the Stable Set Polytope Álinson S. Xavier a Manoel Campêlo b a Mestrado e Doutorado em Ciência da Computação Universidade Federal do Ceará Fortaleza, Brazil
The Signless Laplacian Spectral Radius of Graphs with Given Degree Sequences. Dedicated to professor Tian Feng on the occasion of his 70 birthday
The Signless Laplacian Spectral Radius of Graphs with Given Degree Sequences Xiao-Dong ZHANG Ü À Shanghai Jiao Tong University xiaodong@sjtu.edu.cn Dedicated to professor Tian Feng on the occasion of his
arxiv: v2 [math.co] 19 Jun 2018
![arxiv: v2 [math.co] 19 Jun 2018](/thumbs/96/129281759.jpg "arxiv: v2 [math.co] 19 Jun 2018")
arxiv:1705.06268v2 [math.co] 19 Jun 2018 On the Nonexistence of Some Generalized Folkman Numbers Xiaodong Xu Guangxi Academy of Sciences Nanning 530007, P.R. China xxdmaths@sina.com Meilian Liang School
1. Introduction Recap
1. Introduction Recap 1. Tractable and intractable problems polynomial-boundness: O(n k ) 2. NP-complete problems informal definition 3. Examples of P vs. NP difference may appear only slightly 4. Optimization
Preliminaries and Complexity Theory
Preliminaries and Complexity Theory Oleksandr Romanko CAS 746 - Advanced Topics in Combinatorial Optimization McMaster University, January 16, 2006 Introduction Book structure: 2 Part I Linear Algebra
More NP-Complete Problems
CS 473: Algorithms, Spring 2018 More NP-Complete Problems Lecture 23 April 17, 2018 Most slides are courtesy Prof. Chekuri Ruta (UIUC) CS473 1 Spring 2018 1 / 57 Recap NP: languages/problems that have
Perfect Zero-Divisor Graphs of Z_n
Rose-Hulman Undergraduate Mathematics Journal Volume 17 Issue 2 Article 6 Perfect Zero-Divisor Graphs of Z_n Bennett Smith Illinois College Follow this and additional works at: http://scholar.rose-hulman.edu/rhumj
A characterization of graphs by codes from their incidence matrices
A characterization of graphs by codes from their incidence matrices Peter Dankelmann Department of Mathematics University of Johannesburg P.O. Box 54 Auckland Park 006, South Africa Jennifer D. Key pdankelmann@uj.ac.za
The Chromatic Number of Ordered Graphs With Constrained Conflict Graphs
The Chromatic Number of Ordered Graphs With Constrained Conflict Graphs Maria Axenovich and Jonathan Rollin and Torsten Ueckerdt September 3, 016 Abstract An ordered graph G is a graph whose vertex set
Minimum Cost Homomorphisms to Semicomplete Bipartite Digraphs
Minimum Cost Homomorphisms to Semicomplete Bipartite Digraphs Gregory Gutin Arash Rafiey Anders Yeo Abstract For digraphs D and H, a mapping f : V (D) V (H) is a homomorphism of D to H if uv A(D) implies
Extremal Graphs Having No Stable Cutsets
Extremal Graphs Having No Stable Cutsets Van Bang Le Institut für Informatik Universität Rostock Rostock, Germany le@informatik.uni-rostock.de Florian Pfender Department of Mathematics and Statistics University
Hardness of Approximation
Hardness of Approximation We have seen several methods to find approximation algorithms for NP-hard problems We have also seen a couple of examples where we could show lower bounds on the achievable approxmation
P P P NP-Hard: L is NP-hard if for all L NP, L L. Thus, if we could solve L in polynomial. Cook's Theorem and Reductions
Summary of the previous lecture Recall that we mentioned the following topics: P: is the set of decision problems (or languages) that are solvable in polynomial time. NP: is the set of decision problems
Chih-Hung Yen and Hung-Lin Fu 1. INTRODUCTION
TAIWANESE JOURNAL OF MATHEMATICS Vol. 14, No. 1, pp. 273-286, February 2010 This paper is available online at http://www.tjm.nsysu.edu.tw/ LINEAR 2-ARBORICITY OF THE COMPLETE GRAPH Chih-Hung Yen and Hung-Lin
On the Tractability of (k, i)-coloring
On the Tractability of (, i)-coloring Saurabh Joshi, Subrahmanyam Kalyanasundaram, Anjeneya Swami Kare, and Sriram Bhyravarapu Department of Computer Science and Engineering, IIT Hyderabad {sbjoshi,subru,cs14resch01002,cs16resch11001}@iith.ac.in
On subgraphs of large girth
1/34 On subgraphs of large girth In Honor of the 50th Birthday of Thomas Vojtěch Rödl rodl@mathcs.emory.edu joint work with Domingos Dellamonica May, 2012 2/34 3/34 4/34 5/34 6/34 7/34 ARRANGEABILITY AND
Partial characterizations of clique-perfect graphs I: subclasses of claw-free graphs
Partial characterizations of clique-perfect graphs I: subclasses of claw-free graphs Flavia Bonomo a,1, Maria Chudnovsky b,2 and Guillermo Durán c,3 a Departamento de Computación, Facultad de Ciencias
3-Chromatic Cubic Graphs with Complementary Connected Domination Number Three
Vol.3, Issue.1, Jan-Feb. 2013 pp-231-239 ISSN: 2249-6645 3-Chromatic Cubic Graphs with Complementary Connected Domination Number Three Selvam Avadayappan, 1 S. Kalaimathy, 2 G. Mahadevan 3 1, 2 Department
ON THE NP-COMPLETENESS OF SOME GRAPH CLUSTER MEASURES
ON THE NP-COMPLETENESS OF SOME GRAPH CLUSTER MEASURES JIŘÍ ŠÍMA AND SATU ELISA SCHAEFFER Academy of Sciences of the Czech Republic Helsinki University of Technology, Finland elisa.schaeffer@tkk.fi SOFSEM
NP-Completeness. ch34 Hewett. Problem. Tractable Intractable Non-computable computationally infeasible super poly-time alg. sol. E.g.
NP-Completeness ch34 Hewett Problem Tractable Intractable Non-computable computationally infeasible super poly-time alg. sol. E.g., O(2 n ) computationally feasible poly-time alg. sol. E.g., O(n k ) No
Equitable Colorings of Corona Multiproducts of Graphs
Equitable Colorings of Corona Multiproducts of Graphs arxiv:1210.6568v1 [cs.dm] 24 Oct 2012 Hanna Furmańczyk, Marek Kubale Vahan V. Mkrtchyan Abstract A graph is equitably k-colorable if its vertices can
Multiflows in Multi-Channel Multi-Radio Multihop Wireless Networks
This paper was presented as part of the main technical program at IEEE INFOCOM 2011 Multiflows in Multi-Channel Multi-Radio Multihop Wireless Networks Peng-Jun Wan, Yu Cheng, Zhu Wang and Frances Yao Department
GRAPH MINORS AND HADWIGER S CONJECTURE
GRAPH MINORS AND HADWIGER S CONJECTURE DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Eliade
3. Linear Programming and Polyhedral Combinatorics
Massachusetts Institute of Technology 18.453: Combinatorial Optimization Michel X. Goemans April 5, 2017 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the introductory
Homomorphisms of sparse graphs to small graphs
LaBRI Université Bordeaux I France Graph Theory 2009 Fredericia, Denmark Thursday November 26 - Sunday November 29 Flow Let G be a directed graph, A be an abelian group. A-flow A-flow f of G is a mapping
Introduction to Semidefinite Programming I: Basic properties a
Introduction to Semidefinite Programming I: Basic properties and variations on the Goemans-Williamson approximation algorithm for max-cut MFO seminar on Semidefinite Programming May 30, 2010 Semidefinite
Uniquely 2-list colorable graphs
Discrete Applied Mathematics 119 (2002) 217 225 Uniquely 2-list colorable graphs Y.G. Ganjali a;b, M. Ghebleh a;b, H. Hajiabolhassan a;b;, M. Mirzazadeh a;b, B.S. Sadjad a;b a Institute for Studies in
The Clique Problem with Multiple-Choice Constraints under a Cycle-Free Dependency Graph
The Clique Problem with Multiple-Choice Constraints under a Cycle-Free Dependency Graph Andreas Bärmann 1, Patrick Gemander 1 and Maximilian Merkert 2 1 Andreas.Baermann@math.uni-erlangen.de Patrick.Gemander@fau.de