Approximating the Distance to Properties in Bounded-Degree and Sparse Graphs

1. Approximating the Distance to Properties in Bounded-Degree and Sparse Graphs Sharon Marko, Weizmann InstituteDana Ron, Tel Aviv University

2. Distance Approximation Distance approximation is an extension of property testing. Property testing: distinguish between objects (e.g., graphs) that have propertyPand objects that are far from having property P. Distance approximation: estimate distance of object from having property P. In both cases, algorithm is allowed a small failure probability, and task is performed by querying object, where query complexity should be sublinear in size of object.

3. Distance Approximation - Background First explicitly studied in [Parnas, R, Rubinfeld] together with related notion of tolerant property testing (distinguish between objects that are 1-close and 2-far from property). • Problems studied for these extensions: • Monotonicity of functions [PRR],[Ailon, Chazelle, Comandur, Liue]. • Clustering of points [PRR] . • Tolerant vs. intolerant (standard) testing [Fischer, Fortnow] . • Local tolerant testing of codes[Guruswami, Rudra] . • Graph properties in dense graphs model [Fischer, Newman] .

4. v u 1 Distance Approximation in Dense Graphs • Dense Graphs Model[Goldreich Goldwasser R]: • (graph represented by n x n adjacency matrix) • Queries: Is (u,v)  E ? (probe into matrix) • Distance: Fraction of (n2) entries in matrix that should be modified to get property Theorem [Fischer, Newman] : Every property that has testing algorithm in dense-graphs model whose complexity is function onlyof distance parameter , has distance approximation algorithm A with additive error in model, whose complexity is function only of  . That is, P(G)-   A  P(G)+  output of A dist of G from P Complexity may havelargedependence on 1/ (e.g., tower) but nodependence on size of graph. [Alon, Shapira, Sudakov]give algorithm and direct analysis of additive approximation for all monotone properties.

5. 1 2 … d 1 2 … d 1 1 n n Distance Approximation in Sparse Graphs • Bounded-Degree Graphs Model[Goldreich R]: • (graph represented byn incidence lists of size d) • Queries: Who is i’th neighbor ofv? • Distance: Fraction of (nd) entries in lists that should be modified to get property • Suitable: (Almost)-regularsparse graphs(in particular, constant-degree graphs) • Sparse Graphs Model[Parnas R]: • (graph is represented byn incidence lists of varyingsize) • Queries: Who is i’th neighbor of v? • Distance: Fraction of (m)edges in graph that should be modified to get property. • Suitable: General Sparse Graphs

6. Distance Approximation in Sparse Graphs Cont’ Definition:Algorithm Ais an -distance approximationalgorithm ( ≥ 1) for property P,if for every graph G and any given 0<<1 , it outputs w.h.p an estimate As.t.P(G) -   A   P(G)+  where P(G) is the distance of graph G from property P. If  = 1 then algorithm is distance approximation algorithm Note: Cannot get only multiplicative error in general in sublineartime. Must allow additive error (or dependence on 1/P(G) )

7. Definition:Algorithm Ais an -distance approximation algorithm for property P,if outputs w.h.p an estimate As.t. P(G) -   A   P(G)+   = 1 : distance approximation algorithm Our Results Letdavg = m/n(n: num of vertices,m:num of edges)  Property Graph Model Complexity * Extends to subgraph-freeness

8. Some Notes on Our Results • Complexity of all algs but tri(sub)-free are poly in complexity of testing algs of [Goldreich R]. • All algs but tri(sub)-free have only additive error. Cannot obtain such result for tri-free in poly-time/sublin-queries. • Tri(sub)-free and cycle-free algs are in bounded-degree and not (general) sparse-graphs model. Have (n1/2)lower bound for them in latter model. • Case of k=1 for connectivity was addressed in [Chazelle, Rubinfeld, Trevisan]as central part of min-span-tree weight approx alg. • Can adapt tri-free alg to get sublinear approx for min-VC size, improving on [Parnas R].

9. Triangle (Subgraph) Freeness Testing algorithm is a simple brute-force algorithm. Its adaptation gives a multiplicative factor ofd(degree) error. Hence need different approach. Def1. A triangle-cover of graph G is a set of edges whose removal leaves G triangle free. Let CM(G) denote min-size of triangle cover. Our goal: Estimate CM(G)/(dn) (in sublinear time) Def2. Two triangles are neighbors if they share an edge. For a triangle, degree of triangle: deg() = numberof neighboring triangles

10. Min. Triangle-Cover Approx. Alg 1 (not sub-lin) • LetT be set of all triangles in G, TC= initial tri-cover. • For i=1 to r = (log(d/)) :(a) Select each triangle   T withprob 1/(cdeg()).(b) Unselectevery two neighboring triangles that were selected.(c) Add all edges of selected triangles to TC.(d) Remove from T all selected triangles and their neighbors and update degrees. • Add to TC an edge from every remaining triangle in T. • Output TC.

11. Min. Triangle-Cover Approx. Alg (not sub-lin) • Theorem:TC is a triangle cover s.t. w.h.p|TC|  3 CM(G) + (/2)m • Proof: • A triangle-cover by construction. • During the loop: chosen triangles are edge-disjointthus at most 3 CM(G) edges added to TC in loop. • After the loop? Let Ti betriangles left in T after i’th iteration, ni = |Ti|. Lemma: Exp[ ni | ni-1]  (1- 1/c’) ni-1 Corollary: After r = (log(d/)) iterations, w.h.p. |Tr| (/2)m .Since add to TC one edge from each triangle in Tr , Theorem follows.

12. Sublinear Min. Triangle-Cover Approx. Alg Build on approach from [PR] (for min-VC sublinear alg).Algorithm (non sub-lin) can be viewed as distributed algorithm with r = (log(d/)) rounds. (Indeed similar to Luby’sO(log n) rounds distributed algorithm for maximal independent set.) • Uniformly select s=(1/2) vertices. • For each vertex vjselected construct (by BFS) subgraph induced byr-neighborhoodof vj ( r = (log(d/)) ) • Run non-sublinear algorithm on union of subgraphs, and let ejbe number of edges selected that are incident to vj . • Let Ĉ = (n/2s) j ej and output (1/dn)Ĉ Implies thatdecision on whether or not to include edge in cover depends only on r-neighborhood of edge.

13. Sublinear Min. Triangle-Cover Approx. Alg 1) Uniformly select s=(1/2) vertices.2) For each vertex vjselected construct (by BFS) subgraph induced byr-neighborhoodof vj ( r = (log(d/)) )3) Run non-sublinear algorithm on union of subgraphs, and let ejbe number of edges selected that are incident to vj .4) Let Ĉ = (n/2s) j ej and output (1/dn)Ĉ Theorem: Algorithm is a 3-distance-approximation algorithm for triangle freeness. Its complexity is dO(log(d/ )) . Proof combines error bound of non-sublinear algorithm with sampling error.

14. k=1 k-Connectivity Step 1. Let C(G) = num of connected component in G.Then 1C(G) = (C(G)-1)/m . Step 2. Let nvbe num of vertices in connected component of vertex v. Then vV(1/nv) = C(G). Step 3. Let t = 4/( davg) and V’  V be vertices that belong to connected components of size at most t.ThenvV’(1/nv) ≥ C(G) - n. Algorithm: 1. Uniformly select (1/(davg)2) vertices. For each v in sample S finds nvor discovers that v not in V’ (by BFS). 2. Output

15. V\X X k-Connectivity, k >1 Recall: A graph G is k-edge-connected if there are k edge-disjoint paths between every pair of verticesall cuts (X,V\X) of size at least k First attempt: Build directly on testing algorithm of [GR] – gives factor kmultiplicative error (in addition to additive error). The source of multiplicative error: k-connectivity structure used in [GR] (cactus structure [Dinitz]). Instead: Use different k-connectivity structure of extreme-sets tree/partition[Naor,Gusfield,Martel] + adapt & extend ideasfrom[GR] .

16. X Y X V V1 V2 V3 . . . {1,2,3} ` ` ` ` 1 3 2 n ` ` k-Connectivity, k >1, cont’ Def1: The degree of a set X,d(X) = num of edges with one end-point in X (size of cut (X,V\X)) Def2: A set X is j-extreme if d(X)=j and YX, d(Y)>j Def3: The extreme-sets tree of G: - a leaf for every vertex v (a d(v)-extreme set), - root is V (a 0-extreme set), - if j-extreme set Y is node in tree then parent X is minimal extreme set X  Y

17. V X1 X3 X2 X4 X6 X5 k-Connectivity, k >1, cont’ [NGM] showed: Can use extreme-sets tree to define partition of V into extreme sets ES(G)={X1,X2,…,Xq} s.t.kC(G) = i(Xi)/m where  is some (easily computable) demand function.

18. [NGM] showed: Extreme-sets tree defines partition of V into extreme sets ES(G)={X1,X2,…,Xq} s.t. kC(G) = i(Xi)/m k-Connectivity, k >1, cont’ Note1: Can refine ES(G) and get partition ES(t)(G) s.t.(1) |Xi| t for every Xiin ES(t)(G) and (2) |kC(G) - i(Xi)/m |/2 for t=4k/( davg) Note2: Let X(v) be (unique) set Xiin partition ES(t)(G) s.t. vXi then

19. Let X(v) be (unique) set Xiin partition ES(t)(G) s.t. vXi then k-Connectivity, k >1, cont’ Algorithm: 1. Uniformly select ((k/davg)2) vertices. For each v in sample S find (w.h.p.) X(v) and computes (X(v))2. Output Step 1 in Algorithm : Find extreme set of T size at most t that contains X(v)(“random search process” similar to [GR]), construct “extreme-sets sub-tree” of T, which determines X(v) and (X(v)) .

20. Summary and Open Problems • Give distance approximation algorithms for all properties studied in [GR] (testing of bounded-degree graphs). With exception of triangle(subgraph)-freeness, complexity is polynomial in that of testing algorithms, and have only additive error. • Can complexity of tri-freealgorithm be improved?Can we decrease constant multiplicative factor in approximation? • Sublinear distance approximation for bipartiteness in bounded-degree/sparse graphs? • Is there any general relation between testing and distance approximation in bounded-degree/sparse graphs (as there is in dense graphs)?

22. Definition:Algorithm Ais an -distance approximation algorithm for property P,if outputs w.h.p an estimate As.t. P(G) -   A   P(G)+   = 1 : distance approximation algorithm Our Results • Letdavg = m/n(n: num of vertices,m:num of edges) • k-Edge-Connectivity in sparse model: dist-approx, complexity poly(k/( davg)) • Triangle-Freenessin bounded-degree model: 3-dist-approx, complexity dO(log(d/)) (extends to subgraph-freeness) • Eulerian in sparse model: dist-approx, complexity O(1/( davg)4) • Cycle-Freeness in bounded-degree model: dist-approx, complexity O(1/3)

23. k-Edge-Connectivity: sparse, dist-approx, poly(k/( davg))Triangle-Freeness:bounded-degree, 3-dist-approx, dO(log(d/))Eulerian: sparse, dist-approx, O(1/( davg)4)Cycle-Freeness:bounded-degree, dist-approx, O(1/3) Some Notes on Our Results • Complexity of all algs but tri(sub)-free are poly in complexity of testing algs of [Goldreich R]. • All algs but tri(sub)-free have only additive error. Cannot obtain such result in poly-time / sublinear queries. • Tri(sub)-free and cycle-free algs are in bounded-degree and not (general) sparse-graphs model. Have (n1/2)lower bound for them in latter model. • Case of k=1 for connectivity was addressed in [Chazelle, Rubinfeld, Trevisan]as central part of min-span-tree weight approx alg. • Can adapt tri-free alg to get sublinear approx for min-VC size, improving on [Parnas R].