1. Lower Bounds for Property Testing
Luca Trevisan
U.C. Berkeley
Joint work with Andrej Bogdanov and Kenji Obata

2. Sub-linear Time Algorithms
Want to design algorithms that run in less than linear time (and so cannot read entire input).
Must be probabilistic and approximate
For optimization problems:
Compute numerical apx of optimum cost (and implicit representation of apx solution?)
For decision problems:
What is approximation for decision problems?

3. (Graph) Property Testing
Testing a property P with accuracy e in adjacency matrix representation:
Given graph G that has property P, accept with probability >3/4
Given graph G that is e-far from property P accept with probability <1/4
e-far = must change e–fraction of adjacency matrix to get property P (add/remove > en2 edges)

4. Example [GGR,AK] Testing bipartiteness of a given graph G
Pick (1/e)polylog(1/e) vertices, and check if they induce a bipartite graph; if so accept otherwise reject
If G is bipartite then alg accepts with prob 1
If G is e-far from bipartite, then whp algorithm discovers an odd cycle (non-trivial to prove)
Running time: O ((1/e2)polylog(1/e))
We will discuss matching lower bound if time allows

5. Paleontologist’s approach

6. Bounded Degree Graphs
Testing a property P with accuracy e in adjacency lists representation:
Given graph G that has property P, accept with probability >3/4
Given graph G that is e-far from property P accept with probability <1/4
e-far = must change e–fraction of adjacency lists entries to get property P (add/remove > edn edges)

7. Bipartiteness [GR] Testing bipartiteness Repeat polylog n times:
Start at random point, and pick sqrt(n) random walks of length polylog n, if two of them combine to form an odd cycle reject, otherwise accept
Analysis:
in a graph where you need to remove constant fraction of edges to make it bipartite, algorithm finds odd cycle

8. Matching Lower Bound [GR]
Define two distributions of graphs:
Gfar: a random hamiltonian circuit, plus a random matching (whp 1/100-far from bipartite)
Gbip: a random hamiltonian circuit, plus a random matching conditioned on making the graph bipartite
Gfar and Gbip are indistinguishable to algorithms of query complexity o(sqrt(n)).

9. Sub-linear Time Approximation
Minimum spanning tree
given a connected weighted graph of degree d with weights in range {1,…,w}, can approximate MST weight within (1+e) in time about O(dw/e2) [Chazelle, Rubinfeld, T]
Max SAT
Given a CNF where every variable occurs at most d times, can approximate Max SAT optimum within .618, presumably also 2/3, in O(d) time [work in progress, hopefully will get 3/4-d]

10. Sublinear Time Approximation
Problems restricted to dense instances:
Max CUT and other graph problems can be approximated within (1+e) in graphs with at least an2 edges in time 2poly(1/ea) [GGR]
Max 3SAT can be approximated within (1+e) in instances with at least an3 clauses in time 2poly(1/ea) and similar results for other satisfiability problems [AFKK]

11. General Goals When looking for polynomial-time algorithms:
Several algorithmic techniques of general applicability
A general technique to “prove” impossibility (NP-completeness)
For sublinear-time algorithms:
General algorithmic techniques?
Impossibility results?

12. Testing 3-Colorability
Easy in adjacency matrix representation
NP-hard in adjacency list representation
Only for small enough e
Can find 3-coloring good for 80% of the edges in a 3-colorable graph using SDP
NP-hard to find 3-coloring good for 98% (?) fraction of edges
Non-tight, and conditional lower bound for query complexity

13. Other problems
The query complexity of following problems is equivalent to query complexity of testing 3col
Testing satisfiability of 3SAT instance
Every variable occurs in O(1) clauses, “adjacency list” representation
Approximating max cut, vertex cover, independent set, . . ., in bounded-degree graphs
Approximating Max SAT, Max 2SAT, . . .
Lower bound of sqrt(n) for all problems
Nothing better except with complexity assumptions

14. Our Results For one-sided error algorithms:
W(n) query complexity to distinguish 3-colorable graphs from graphs that are (1/3 – d)-far
Lower bound applies to testing problems that are solvable in polynomial time
For two-sided error algorithms:
For some e, W(n) query complexity to distinguish 3-colorable graphs from graphs that are e-far.

15. Additional Results
Unconditionally, algorithms running in time o(n) cannot:
Approximate Max 3SAT better than 7/8
Approximate Max Cut in bounded-degree graphs better than 16/17
. . .
Hastad’97 proved above problems are NP-hard

16. The 3-Coloring Lower Bound
Consider first one-sided error algorithms
It’s enough to find a graph G that is (1/3 – d)-far from 3-colorable, but every subgraph of size < an is 3-colorable
(for every d there is an a such that . . .)
Then an algorithm of query complexity < an either accepts G (which is wrong) or rejects some 3-colorable graph (which means the algorithm has not one-sided error)

17. The Graph
Pick a graph of degree O(1/d2) at random (pick so many random matchings)
Then it is (1/3 – d)-far whp
But, for some a, whp, every subgraph induced by k < an vertices contains <1.5k edges
In a minimal non-3-colorable graph, every vertex has degree at least 3
Every subgraph induced by < an vertices is 3-colorable
[Erdos]

18. Explicit Construction
Can the previous construction be derandomized?
For constants d, e, a, and for every suff large n, we can explicitly construct a graph on n vertices, max degree d, e-far from 3-colorable, and such that every subset of an vertices induces a 3-colorable subgraph.

19. Explicit Construction
We construct a 3SAT formula such that for constants k, e’, a’
Every variable occurs k times
No assignment satisfies more than 1-e’ fraction of clauses
Every a’ fraction of clauses is satisfiable
Then we use (slightly new) reduction from 3SAT to 3Coloring

20. The Formula
Fix a degree-d expander graph G=(V,E) such that for every cut (S,V-S) at least min{|S|,|V-S|} edges cross the cut (enough d=14)
Have two variables xuv and xvu for each egde (u,v)
For every vertex v have the (3SAT equivalent of) the constraint
Su xuv = 1 + Sw xvw

21. Structure of the Analysis
Impossible to satisfy more than a fraction 1/(d+1) of the constraints
Can always satisfy half of the constraint
define an auxiliary network
show that the auxiliary network has no small cut because of expansion
then there is a large flow
use large flow to find assignment for subset of constraint

22. Flow Argument
Want to satisfy constraints corresponding to vertices in C, with |C| < |V|/2
Construct flow network with new source s, sink t obtained by collapsing V-C, and vertices in C
V-C s t C

23. Flow Argument Every cut has size at least |C|
|A| edges A t
Every cut has size at least |C|
There is a 0/1 flow of cost at least |C|
Interpreted as an assignment, satisfies all constraints in C s
|C-A| edges
C-A

24. Two-Sided Error Algorithms
Need to define two distributions of graphs Gcol and Gfar such that
Graphs in Gcol are (almost) always 3-colorable
Graphs in Gfar are (almost) always far from 3-colorable
To an algorithm of bounded query complexity, Gcol and Gfar look (almost) the same

25. Main Step
Define two distributions Dsat and Dfar of instances of E3LIN-2 (systems over GF(2) with 3 variables per equation)
Systems in Dsat are always satisfiable
Systems in Dfar are (almost) always (1/2-d)-far from satisfiable
To an algorithm of bounded query complexity, Dsat and Dfar look the same
We get Gcol and Gfar using reduction from approximate E3LIN-2 to approximate 3-coloring

26. E3LIN-2 X1 + X3 + X10 = 0 mod 2 X2 + X3 + X4 = 1 mod 2
. . .

27. Main Building Block
We show that for every c there is a such that there exists a left-hand side with
n variables, cn equations, 3 variables per equations, every variable occurs in 3c equations
every an equations are linearly independent
Pick the left-hand side at random
repeat 3c times: pick at random a set of n/3 disjoint triples of variables
Explicit construction?

28. Distributions The left-hand side is always as before
In Dsat, we pick a random assignment to the variables, and set right-hand side consistently
always satisfiable
In Dfar, we pick the right-hand side uniformly at random
With high probability, (1/2 – O(1/sqrt c))-far

29. Indistinguishability
Two distributions differ only in right-hand side
In Dfar uniformly distributed
In Dsat, an-wise independent
Linear independence implies statistical independence
Look the same to algorithm that sees less than an equations

30. Conclusion of the Argument
No algorithm of “query complexity” o(n) can distinguish satisfiable instances of E3LIN-2 from instances that are (1/2-d)-far from satisfiable
For some e, no algorithm of query complexity o(n) can distinguish 3-colorable graphs from graphs that e–far from 3-col.
No algorithm of query complexity o(n) can approximate Max 3SAT better than 7/

31. Open Questions
Show that distinguishing 3-colorable graphs from (1/3-d)-far graphs requires query complexity W(n)
we can only prove it for one-sided error
Show that approximating Max SAT better than ¾ and Max CUT bettter than ½ requires query complexity W(n)
we only know W(sqrt(n)) [implicit in GR]
would “explain” why we need SDP

32. Back to Dense Graphs
Recall Alon-Krivelevich bipartiteness test for the adjacency matrix representation:
pick (1/e)polylog(1/e) vertices and look at induced subgraph
if see odd cycle reject, otherwise accept
Running time (1/e2)polylog(1/e)
We prove:
W(1/e2) for non-adaptive algorithms
W(1/e1.5) for adaptive algorithms

33. Two Distributions Gfar: every edge exists with probability e
whp it is e/3-far from bipartite
Gbip: pick a random partition, then every edge that crosses the partition exists with probability 2e
Thm1: look the same to non-adaptive algorithms making o(1/e2) queries
Thm2: look the same to adaptive algorithms making o(1/e1.5) queries

34. Proof of a Weaker Statement
Thm1 (weaker): a non-adaptive algorithm making q=o(1/e2) queries in Gfar is unlikely to see an odd cycle
Proof:
a non-adaptive algorithm asks about some subgraph with q edges.
There are at most about qt/2 cycles of length t, and each one exists with probability etqt/2, exponentially small in t.
Summing over all t, it’s still unlikely that there is a cycle

35. Proof of a Weaker Statement
Thm2 (weaker): an adaptive algorithm making q=o(1/e1.5) queries in Gfar is unlikely to see an odd cycle
Proof:
the algorithm sees an edge only once in 1/e queries
the algorithm sees a cycle only after querying a pair that it already sees as connects
It takes 1/e.5 edges to have 1/e pairs of connected vertices
It takes 1/e1.5 queries to have so many edges

36. Some more open questions
In adjacency matrix representation, most interesting problems solvable in constant (in e) time
For some problems (eg testing triangle-freeness) analysis uses Szemeredy’s regularity lemma, and constant is hyper-exponential in e
Lower bound (1/e)log 1/ e and only and for one-sided error
Alternative analysis / stronger lower bounds?