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

Distance Approximation Distance approximation is an extension of property testing. Property testing: distinguish between objects (e.g., graphs) that have property P and 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.

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].

Distance Approximation in Dense Graphs Theorem [Fischer, Newman] : Every property that has testing algorithm in dense-graphs model whose complexity is function only of 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)+  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 (n 2 ) entries in matrix that should be modified to get property 1 u v Complexity may have large dependence on 1/  (e.g., tower) but no dependence on size of graph. dist of G from P output of A [Alon, Shapira, Sudakov] give algorithm and direct analysis of additive approximation for all monotone properties.

Distance Approximation in Sparse Graphs 1 2 … d 1 n Bounded-Degree Graphs Model [Goldreich R]: (graph represented by n incidence lists of size d) Queries: Who is i’th neighbor of v? Distance: Fraction of (nd) entries in lists that should be modified to get property Suitable: (Almost)-regular sparse graphs (in particular, constant-degree graphs) Sparse Graphs Model [Parnas R]: (graph is represented by n incidence lists of varying size) 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 1 n

Distance Approximation in Sparse Graphs Cont’ Definition: Algorithm A is an - distance approximation algorithm (  ≥ 1) for property P, if for every graph G and any given 0<  <1, it outputs w.h.p an estimate  A s.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 sublinear time. Must allow additive error (or dependence on 1/  P (G) )

Our Results Definition: Algorithm A is an -distance approximation algorithm for property P, if outputs w.h.p an estimate  A s.t.  P (G) -    A    P (G)+   = 1 : distance approximation algorithm poly(k/(  d avg )) 1sparsek-Edge- Connectivity d O(log(d/)) 3bounded- degree Triangle- Freeness * O(1/(  d avg ) 4 ) 1sparseEulerian O(1/  3 ) 1bounded- degree Cycle- Freeness Let d avg = m/n (n: num of vertices, m: num of edges) Property Graph Model  Complexity * Extends to subgraph-freeness

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  (n 1/2 ) 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]. poly(k/( d avg )) 1sparsek-Edge-Conn d O(log(d/)) 3bounded - degree Triangle- Freeness * O(1/( d avg ) 4 ) 1sparseEulerian O(1/ 3 ) 1bounded - degree Cycle- Freeness

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

Min. Triangle-Cover Approx. Alg 1 (not sub-lin) 1)Let T be set of all triangles in G, TC= initial tri-cover. 2)For i=1 to r =  (log(d/  )) : (a) Select each triangle   T with prob 1/(cdeg(  )). (b) Unselect every 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. 3)Add to TC an edge from every remaining triangle in T. 4)Output TC.

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

Sublinear Min. Triangle-Cover Approx. Alg 1)Uniformly select s=  (1/ 2 ) vertices. 2)For each vertex v j selected construct (by BFS) subgraph induced by r-neighborhood of v j ( r =  (log(d/  )) ) 3)Run non-sublinear algorithm on union of subgraphs, and let e j be number of edges selected that are incident to v j. 4)Let Ĉ = (n/2s)  j e j and output (1/dn) Ĉ 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’s O(log n) rounds distributed algorithm for maximal independent set.) Implies that decision on whether or not to include edge in cover depends only on r-neighborhood of edge.

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

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

k-Connectivity, k >1 First attempt: Build directly on testing algorithm of [GR] – gives factor k multiplicative 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 ideas from [GR]. 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 X V\X

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 X Y X ` V ` ` ` V1V1 V2V2 V3V n 3 {1,2,3} ``

k-Connectivity, k >1, cont’ [NGM] showed: Can use extreme-sets tree to define partition of V into extreme sets ES(G)={X 1,X 2,…,X q } s.t.  kC (G) =  i  (X i )/m where  is some (easily computable) demand function. X1X1 X2X2 X3X3 X5X5 X4X4 X6X6 V

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

k-Connectivity, k >1, cont’ Let X(v) be (unique) set X i in partition ES (t) (G) s.t. vX i then Algorithm: 1. Uniformly select  ((k/  d avg ) 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)).

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-free algorithm 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)?

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

Some Notes on Our Results k-Edge-Connectivity: sparse, dist-approx, poly(k/( d avg )) Triangle-Freeness: bounded-degree, 3-dist-approx, d O(log(d/)) Eulerian: sparse, dist-approx, O(1/( d avg ) 4 ) Cycle-Freeness: bounded-degree, dist-approx, O(1/ 3 )  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  (n 1/2 ) 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].