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 (n2) entries in matrix that should be modified to get property

1u

v

Complexity may have large dependence on 1/ (e.g., tower) but no dependence on size of graph.

dist of G from Poutput 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 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 (nd) 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 approximationalgorithm ( ≥ 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

k-Edge-Connectivity

sparse1poly(k/( davg))

Triangle-Freeness*

bounded- degree

3dO(log(d/))

Euleriansparse1O(1/( davg)4)

Cycle-Freeness

bounded- degree

1O(1/3)

Let davg = 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 (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].

k-Edge-Connsparse1poly(k/( davg))

Triangle-Freeness*

bounded- degree

3dO(log(d/))

Euleriansparse1O(1/( davg)4)

Cycle-Freeness

bounded- degree

1O(1/3)

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 CM(G)/(dn) (in sublinear time)

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.

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/(cdeg()).(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 CM(G) + (/2)m

Proof:

A triangle-cover by construction.

During the loop: chosen triangles are edge-disjoint thus at most 3 CM(G) edges added to TC in loop.

After the loop?

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.

Let Ti be triangles left in T after i’th iteration, ni = |Ti|.

Sublinear Min. Triangle-Cover Approx. Alg

1) Uniformly select s=(1/2) vertices.

2) For each vertex vj selected construct (by BFS) subgraph induced by r-neighborhood of vj ( r = (log(d/)) )

3) Run non-sublinear algorithm on union of subgraphs, and let ej be number of edges selected that are incident to vj .

4) Let Ĉ = (n/2s) j ej 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 dO(log(d/ )) .

Proof combines error bound of non-sublinear algorithm with sampling error.

1) Uniformly select s=(1/2) vertices.2) For each vertex vj selected construct (by BFS) subgraph induced by r-neighborhood of vj ( r = (log(d/)) ) 3) Run non-sublinear algorithm on union of subgraphs, and let ej be number of edges selected that are incident to vj .4) Let Ĉ = (n/2s) j ej 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 nv be num of vertices in connected component of

vertex v. Then vV(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 vV’(1/nv) ≥ C(G) - n.

Algorithm: 1. Uniformly select (1/(davg)

2) vertices. For each v in sample S finds nv or discovers that v not in V’ (by BFS). 2. Output

11

||

1

'VSv vnS

n

m

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 kX 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 YX, 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

YX

`

V

` ` `

V1 V2 V3

. . .

1 2 n3

{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)={X1,X2,…,Xq} s.t.

kC(G) = i(Xi)/m where is some (easily computable) demand function.

X1

X2 X3

X5

X4

X6

V

k-Connectivity, k >1, cont’

[NGM] showed: Extreme-sets tree defines partition of V into extreme sets ES(G)={X1,X2,…,Xq} s.t. kC(G) = i(Xi)/m

Note2: Let X(v) be (unique) set Xi in partition ES(t)(G) s.t. vXi then

Note1: Can refine ES(G) and get partition ES(t)(G) s.t.(1) |Xi| t for every Xi in ES(t)(G) and

(2) | kC(G) - i(Xi)/m | /2 for t=4k/( davg)

2|)(|

))((1)(

VvkC vX

vX

mG

k-Connectivity, k >1, cont’

Let X(v) be (unique) set Xi in partition ES(t)(G) s.t. vXi then

2|)(|

))((1)(

VvkC vX

vX

mG

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

Sv vX

vX

s

n

m |)(|

))((1

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 davg = m/n (n: num of vertices, m: num of edges)

k-Edge-Connectivity in sparse model: dist-approx, complexity poly(k/( davg))

Triangle-Freeness in 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)

Some Notes on Our Results

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)

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].