NCG Group New Results and Open Problems

Transcription

1 NCG Group Ne Results and Open Problems

2 Table of Contents NP-Completeness 1 NP-Completeness / 19

3 NP-Completeness 3 / 19

4 (P)NSP OPT - Problem Definition An instance I of (P)NSP opt : I = (G, F ), here G = (V, E) is a graph and F is a set of friendships. n := V Metric costs f G (v) of v V can be max d G (u, v) or u F (v) d G (u, v) 1 F (v) u F (v) v V f G (v) has to be minimized in an Nash Equilibrium (NE) An instance I of (P)NSP dec : I = (G, k, F ), hereat G, F and n are the same as for (P)NSP opt. The question: v V f G (v) k possible? 4 / 19

5 (P)NSP OPT - Reduction What e sho: CLIQUE dec p NSP dec Reduction function: r((g, k)) := (G, k, F ), hereat F is a set of friendships hich precisely guarantees k pairise friended nodes. I CLIQUE dec r(i) NSP dec 5 / 19

6 SC OPT - Reduction NP-Completeness Metric costs f G (v) of v V can be max d G (u, v) or u F (v) d G (u, v) u F (v) SC opt (SC dec ) analogously to (P)NSP opt ((P)NSP dec ) For a boolean 3 CNF function ψ: ψ C is the number of clauses and ψ A is the number of atoms What e sho: 3 CNF SAT p SC dec Reduction function: r(ψ) := (G, F, 18 ψ C + 26 ψ A ) or r(ψ) := (G, F, 34 ψ C + 54 ψ A ) 6 / 19

7 SC OPT - Graphical Reduction - Shortcut fixation c c c c v c c c 7 / 19

8 SC OPT - Graphical Reduction C1 Ci 1 2 Ci 2 Ci+1 C ψ C A t j A f j A t m A f m A t l A f l 5 A1 Aj Am Al A ψ A C 1 C i 1 2 C i 2 C i+1 C ψ C 8 / 19

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10 Friendships as subgraphs: NE reachable? start 10 / 19

11 Average metric NP-Completeness 1 n 3 1 n 3 is maximum distance beteen friends reachable friendship example tight 2 orst-case PoA upper bounded by n 3 reachable friendship example PoA = Θ(n) 1 11 / 19

12 Maximum metric NP-Completeness 1 k 1 k k n 1 is maximum distance beteen friends (non-reachable) friendship example tight 2 orst-case PoA upper bounded by n 1 (non-reachable) friendship example PoA = Θ( n) 3 Reachable friendship example for Θ( n) tightness (both statements) layer 1 layer n 12 / 19

13 Maximum metric NP-Completeness 1 k 1 k k n 1 is maximum distance beteen friends (non-reachable) friendship example tight 2 orst-case PoA upper bounded by n 1 (non-reachable) friendship example PoA = Θ( n) 3 Reachable friendship example for Θ( n) tightness (both statements) layer 1 layer n 12 / 19

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15 Average metric Convergence speed NP-Completeness At most F (diam(g) 1) improving moves until next NE Non-reachable friendship example tight reachable friendship example tight in Θ( F (diam(g) 1)) k k 14 / 19

16 Average metric Convergence speed NP-Completeness At most F (diam(g) 1) improving moves until next NE Non-reachable friendship example tight reachable friendship example tight in Θ( F (diam(g) 1)) k k 14 / 19

17 Average metric Convergence speed NP-Completeness At most F (diam(g) 1) improving moves until next NE Non-reachable friendship example tight reachable friendship example tight in Θ( F (diam(g) 1)) k k 14 / 19

18 Average metric PoA NP-Completeness Worst case PoA upper bounded by diam(g) reachable friendship example tight in Θ(diam(G)) 15 / 19

19 Average metric PoA NP-Completeness Worst case PoA upper bounded by diam(g) reachable friendship example tight in Θ(diam(G)) 15 / 19

20 2: Maximum metric Convergence 2: for sapping, both nodes need to improve Lemma For every instance of the 2 ith an arbitrary sequence of improving moves and the maximum of all distances to friends as metric, an equilibrium exists and the game converges. Proof. Potential function ( ) Φ : G N 0, G sort max d G (u, v) v V (G) u F (v) ( ) VF + diam(g) 1 At most improving moves until next NE V F (V F := {v V F (v) }) 16 / 19

21 2: Maximum metric Convergence 2: for sapping, both nodes need to improve Lemma For every instance of the 2 ith an arbitrary sequence of improving moves and the maximum of all distances to friends as metric, an equilibrium exists and the game converges. Proof. Potential function ( ) Φ : G N 0, G sort max d G (u, v) v V (G) u F (v) ( ) VF + diam(g) 1 At most improving moves until next NE V F (V F := {v V F (v) }) 16 / 19

22 2: Maximum metric Convergence 2: for sapping, both nodes need to improve Lemma For every instance of the 2 ith an arbitrary sequence of improving moves and the maximum of all distances to friends as metric, an equilibrium exists and the game converges. Proof. Potential function ( ) Φ : G N 0, G sort max d G (u, v) v V (G) u F (v) ( ) VF + diam(g) 1 At most improving moves until next NE V F (V F := {v V F (v) }) 16 / 19

23 2: Maximum metric PoA Worst case PoA upper bounded by diam(g) reachable friendship example tight in Θ(diam(G)) 17 / 19

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25 PoA for NSP in general graphs Tightness for convergence of 2 PoS Characterization of graphs/friendships for convergence/good NE Shortcut problem anyone? 19 / 19