Lower Bounds for Local Approximation Mika Göös, Juho Hirvonen & Jukka Suomela (HIIT) Göös et al. (HIIT) Local Approximation nd April 0 / 9
Lower Bounds for Local Approximation We prove: Local algorithms do not need unique IDs when computing approximations to graph optimization problems Göös et al. (HIIT) Local Approximation nd April 0 / 9
Input = Graph G = Communication Network Göös et al. (HIIT) Local Approximation nd April 0 3 / 9
Simple Graph Problems Old Classics Independent sets Göös et al. (HIIT) Local Approximation nd April 0 4 / 9
Simple Graph Problems Old Classics Independent sets Vertex covers Göös et al. (HIIT) Local Approximation nd April 0 4 / 9
Simple Graph Problems Old Classics Independent sets Vertex covers 3 Dominating sets Göös et al. (HIIT) Local Approximation nd April 0 4 / 9
Simple Graph Problems Old Classics Independent sets Vertex covers 3 Dominating sets 4 Matchings Göös et al. (HIIT) Local Approximation nd April 0 4 / 9
Simple Graph Problems Old Classics Independent sets Vertex covers 3 Dominating sets 4 Matchings 5 Edge covers Göös et al. (HIIT) Local Approximation nd April 0 4 / 9
Simple Graph Problems Old Classics Independent sets Vertex covers 3 Dominating sets 4 Matchings 5 Edge covers 6 Edge dom. sets 7 Etc... Göös et al. (HIIT) Local Approximation nd April 0 4 / 9
Local Algorithms Distributed algorithm A Deterministic, synchronous Göös et al. (HIIT) Local Approximation nd April 0 5 / 9
Local Algorithms Distributed algorithm A Deterministic, synchronous Göös et al. (HIIT) Local Approximation nd April 0 5 / 9
Local Algorithms Distributed algorithm A Deterministic, synchronous Göös et al. (HIIT) Local Approximation nd April 0 5 / 9
Local Algorithms Distributed algorithm A Deterministic, synchronous Göös et al. (HIIT) Local Approximation nd April 0 5 / 9
Local Algorithms Distributed algorithm A Deterministic, synchronous Göös et al. (HIIT) Local Approximation nd April 0 5 / 9
Local Algorithms Distributed algorithm A Deterministic, synchronous 3 Locality: running time r of A is independent of n = G may depend on maximum degree of G Göös et al. (HIIT) Local Approximation nd April 0 5 / 9
Local Algorithms Distributed algorithm A Deterministic, synchronous 3 Locality: running time r of A is independent of n = G may depend on maximum degree of G On bounded degree graphs ( = O()) running time is a constant: r N (e.g., r = 3) Göös et al. (HIIT) Local Approximation nd April 0 5 / 9
Local Algorithms Definition: A : { } {0, } Göös et al. (HIIT) Local Approximation nd April 0 6 / 9
Local Algorithms Definition: A : { } {0, } Output: Vertex set: Set of vertices v with A(G, v) = Göös et al. (HIIT) Local Approximation nd April 0 6 / 9
Local Algorithms Definition: A : { } {0, } Output: Vertex set: Set of vertices v with A(G, v) = Edge set: A(G, v) is a vector of length indicating which edges incident to v are included Göös et al. (HIIT) Local Approximation nd April 0 6 / 9
Two Network Models Unique Identifiers Anonymous Networks with Port Numbering Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Two Network Models Unique Identifiers Each node has a unique O(log n)-bit label: Anonymous Networks with Port Numbering V(G) {,,..., poly(n)} 5 8 3 Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Two Network Models Unique Identifiers Each node has a unique O(log n)-bit label: V(G) {,,..., poly(n)} Anonymous Networks with Port Numbering Node v can refer to its neighbours via ports,,..., deg(v) Edges are oriented 5 8 3 3 3 Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Example: Independent Sets on a Cycle ID-model PO-model 7 5 8 3 Göös et al. (HIIT) Local Approximation nd April 0 8 / 9
Example: Independent Sets on a Cycle ID-model PO-model 7 5 8 3 [Cole Vishkin 86, Linial 9]: Maximal independent set can be computed in Θ(log n) rounds Göös et al. (HIIT) Local Approximation nd April 0 8 / 9
Example: Independent Sets on a Cycle ID-model PO-model 7 5 8 3 [Cole Vishkin 86, Linial 9]: Maximal independent set can be computed in Θ(log n) rounds Above PO-network is fully symmetric Göös et al. (HIIT) Local Approximation nd April 0 8 / 9
Example: Independent Sets on a Cycle ID-model PO-model 7 5 8 3 [Cole Vishkin 86, Linial 9]: Maximal independent set can be computed in Θ(log n) rounds Above PO-network is fully symmetric All nodes give same output Göös et al. (HIIT) Local Approximation nd April 0 8 / 9
Example: Independent Sets on a Cycle ID-model PO-model 7 5 8 3 [Cole Vishkin 86, Linial 9]: Maximal independent set can be computed in Θ(log n) rounds Above PO-network is fully symmetric All nodes give same output Empty set is computed! Göös et al. (HIIT) Local Approximation nd April 0 8 / 9
Example: Independent Sets on a Cycle 5 ID-model 8 7 3 in general PO-model [Cole Vishkin 86, Linial 9]: Maximal independent set can be computed in Θ(log n) rounds Above PO-network is fully symmetric All nodes give same output Empty set is computed! Göös et al. (HIIT) Local Approximation nd April 0 8 / 9
Example: Independent Sets on a Cycle ID-model PO-model 7 5 8 3 [Lenzen Wattenhofer 08, Czygrinow et al. 08]: MIS cannot be approximated to within a constant factor in O() rounds! Above PO-network is fully symmetric All nodes give same output Empty set is computed! Göös et al. (HIIT) Local Approximation nd April 0 8 / 9
Example: Independent Sets on a Cycle 5 ID-model 8 7 3 for Local Approximation PO-model [Lenzen Wattenhofer 08, Czygrinow et al. 08]: MIS cannot be approximated to within a constant factor in O() rounds! Above PO-network is fully symmetric All nodes give same output Empty set is computed! Göös et al. (HIIT) Local Approximation nd April 0 8 / 9
Known Approximation Ratios ID PO Max Independent Set Max Matching Min Vertex Cover Min Edge Cover Min Dominating Set + + ( := / ) Göös et al. (HIIT) Local Approximation nd April 0 9 / 9
Known Approximation Ratios ID PO Max Independent Set Max Matching Min Vertex Cover Min Edge Cover Min Dominating Set + + ( := / ) Min Edge Dominating Set α 4 / 3 α 4 /??? Göös et al. (HIIT) Local Approximation nd April 0 9 / 9
Our Result (informally) Main Thm: When Local Algorithms compute constant factor approximations, ID = PO for a general class of graph problems Göös et al. (HIIT) Local Approximation nd April 0 0 / 9
Proof! Göös et al. (HIIT) Local Approximation nd April 0 / 9
OI: Order Invariant Algorithms [Naor Stockmeyer 95]: Ramsey s theorem implies that local ID-algorithms can only compare identifiers ID = OI? = PO Göös et al. (HIIT) Local Approximation nd April 0 / 9
OI: Order Invariant Algorithms [Naor Stockmeyer 95]: Ramsey s theorem implies that local ID-algorithms can only compare identifiers ID = OI? = PO Order invariant algorithm A: Input: Ordered graph (G, ) Output: A(G,, v) depends only on order type of the radius-r neighbourhood of v Göös et al. (HIIT) Local Approximation nd April 0 / 9
Example: Ordered Cycle Göös et al. (HIIT) Local Approximation nd April 0 3 / 9
Example: Ordered Cycle Göös et al. (HIIT) Local Approximation nd April 0 3 / 9
Example: Ordered Cycle Göös et al. (HIIT) Local Approximation nd April 0 3 / 9
Example: Ordered Cycle Göös et al. (HIIT) Local Approximation nd April 0 3 / 9
Example: Ordered Cycle Göös et al. (HIIT) Local Approximation nd April 0 3 / 9
Example: Ordered Cycle Göös et al. (HIIT) Local Approximation nd April 0 3 / 9
Example: Ordered Cycle Göös et al. (HIIT) Local Approximation nd April 0 3 / 9
Example: Ordered Cycle On a large enough cycle ( ɛ)-fraction of neighbourhoods are isomorphic Göös et al. (HIIT) Local Approximation nd April 0 3 / 9
Example: Ordered Cycle On a large enough cycle ( ɛ)-fraction of neighbourhoods are isomorphic OI-algorithm outputs the same almost everywhere! Göös et al. (HIIT) Local Approximation nd April 0 3 / 9
Two Advantages of OI over PO. Global linear order vs. Local port numbering Linear order introduces symmetry breaking Main challenge: How to control this? Göös et al. (HIIT) Local Approximation nd April 0 4 / 9
Two Advantages of OI over PO. Global linear order vs. Local port numbering Linear order introduces symmetry breaking Main challenge: How to control this?. PO-algorithms cannot detect small cycles Göös et al. (HIIT) Local Approximation nd April 0 4 / 9
Two Advantages of OI over PO. Global linear order vs. Local port numbering Linear order introduces symmetry breaking Main challenge: How to control this?. PO-algorithms cannot detect small cycles? = Göös et al. (HIIT) Local Approximation nd April 0 4 / 9
Two Advantages of OI over PO. Global linear order vs. Local port numbering Linear order introduces symmetry breaking Main challenge: How to control this?. PO-algorithms cannot detect small cycles This disadvantage disappears on high girth graphs Göös et al. (HIIT) Local Approximation nd April 0 4 / 9
Two Advantages of OI over PO. Global linear order vs. Local port numbering Linear order introduces symmetry breaking Main challenge: How to control this?. PO-algorithms cannot detect small cycles This disadvantage disappears on high girth graphs We assume: Feasibility of a solution can be checked by a PO-algorithm Göös et al. (HIIT) Local Approximation nd April 0 4 / 9
Two Advantages of OI over PO + = Need to understand linear orders on high girth graphs Göös et al. (HIIT) Local Approximation nd April 0 5 / 9
Order Homogeneity Definition: We say that an ordered graph (G, ) is (α, r)-homogeneous if α-fraction of nodes have isomorphic radius-r neighbourhoods Göös et al. (HIIT) Local Approximation nd April 0 6 / 9
Order Homogeneity Definition: We say that an ordered graph (G, ) is (α, r)-homogeneous if α-fraction of nodes have isomorphic radius-r neighbourhoods Example: Large cycles are ( ɛ, r)-homogeneous (α = not possible) Göös et al. (HIIT) Local Approximation nd April 0 6 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph } Main Technical Result: Graphs (H ɛ, ɛ ) with properties 4 exist Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph } Main Technical Result: Graphs (H ɛ, ɛ ) with properties 4 exist Proof. We use Cayley graphs of soluble groups Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph } Main Technical Result: Graphs (H ɛ, ɛ ) with properties 4 exist? F(α, β) Z Z Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph } Main Technical Result: Graphs (H ɛ, ɛ ) with properties 4 exist Want: Invariant order: x < y gx < gy F(α, β) Z Z Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph } Main Technical Result: Graphs (H ɛ, ɛ ) with properties 4 exist Want: Invariant order Polynomial growth [Gromov s theorem] F(α, β) Z Z Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph } Main Technical Result: Graphs (H ɛ, ɛ ) with properties 4 exist Want: Invariant order Polynomial growth Large girth [Gamburd et al. 09] F(α, β) Z Z Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph } Main Technical Result: Graphs (H ɛ, ɛ ) with properties 4 exist Want: Invariant order Polynomial growth Large girth F(α, β) G = Z, G i+ = (G i G i ) Z Z Z Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Homogeneous High Girth Graphs ( ɛ, r)-homogeneous k-regular 3 Large girth 4 Finite graph } Main Technical Result: Graphs (H ɛ, ɛ ) with properties 4 exist Proof of Main Thm: Form graph products (H ɛ, ɛ ) G Göös et al. (HIIT) Local Approximation nd April 0 7 / 9
Conclusion Our result: For Local Approximation, ID = OI = PO Open problems: Planar graphs? Applications elsewhere descriptive complexity? Göös et al. (HIIT) Local Approximation nd April 0 8 / 9
Cheers! Göös et al. (HIIT) Local Approximation nd April 0 9 / 9