Inapproximability from different

hardness assumptions

Prahladh Harsha TIFR

2011 School on Approximability

Hardness of approximation

Worst case hardness

PCP theorem, Hardness of Label cover Unique Games Conjecture

Average case hardness

Feige’s Random-3-SAT assumption

Label CoverLabel Cover (LC)G – Bipartite graph1, 2 – labels(projection) constraint per edge ce:1 2

Edge e is satisfied if ce(σ1)=σ2

Goal: Find an assignment to vertices that satisfies the most edges

Gap(α,β)-LC: Distinguish between instances

• At least α fraction of constraints satisfied

• At most β fraction of constraints satisfied

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PCP TheoremLabel Cover (LC)G – Bipartite graph1, 2 – labels(projection) constraint per edge ce:1 2

Edge e is satisfied if ce(σ1)=σ2

PCP Theorem […., AS’92, ALMSS’92]

Gap(1,0.9999)-LC is NP-hard

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Stronger form of PCP Theorem

Label Cover (LC)G – Bipartite graph1, 2 – labels(projection) constraint per edge ce:1 2

Edge e is satisfied if ce(σ1)=σ2

PCP Theorem + Repetition Theorem [Raz’95]

For every constant δ there exists alphabets 1,2, Gap(1,δ)-LC is NP-hard

(starting point for all tight hardness of approximation reductions)

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Proving tight hardness results

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Outer Verifier

Dictatorship Test

[Fourier Analysis]+(composition)

Inner Verifier

Proving tight hardness results

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+direction reduction SETCOVER

Lattice probs.

Fourier analysis MAX3LIN, MAX3SAT,

CLIQUE

Fourier analysis MAXCUT, VERTEX-COVER

(with unique constraints)Unique Games Conjecture

Stronger form of PCP Theorem

Label Cover (LC)G – Bipartite graph1, 2 – labels(projection) constraint per edge ce:1 2

Edge e is satisfied if ce(σ1)=σ2

PCP Theorem + Repetition Theorem [Raz’95]

For every constant δ there exists alphabets 1,2, Gap(1,δ)-LC is NP-hard

(mother of all tight hardness of approximation reductions)

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Label Cover Constructions

Low Degree Test (LDT) [RS’92] Given function f:Fm F (F – field), check if f

is the evaluation of a low-degree polynomial without reading all of F

f:Fm F

Use fact that restriction of low-degree polynomial to a line is still low-degree

Label Cover for LDT

Points tablef:Fm F Lines table

flines

Constraint: flines(l)(x) = f(x)

Large Alphabet

Size

Label Cover -- LDT [AS’97, RS’97] Completeness: If f:Fm F is a low-degree polynomial, there

exists lines table flines such that

Pr[flines(l)(x) = f(x)] = 1 Soundness: If f:Fm F is “far” from being low-degree

polynomial, there for all lines table flines we have

Pr[flines(l)(x) = f(x)] ≤ δ

Label Cover for NP Encode problem in NP using polynomials

to lift the label cover for LDT to all of NP

Label Cover for NP [RS’97, AS’97]:

For every alphabet and error δ=1/log||, Gap(1,δ)-LC is NP-hard, if || > npolylog n

Caveat: Large Alphabet SizeRenders result “useless” for hardness results

Alphabet Reduction [MR’08, DH’09]

Alphabet Reduction: Label Cover instance with large alphabet size

Label Cover instance with small alphabet size

Idea: Recurse!![in the style of AS’92]

Use an “Inner” Label Cover to reduce alphabet of outer label cover

Alphabet Reduction

Alphabet Reduction

Alphabet Reduction Alphabet Size

Reduced

However

3-partite graph instead of bipartite

Idea: [2-query composition DH’09]

Combine leftmost and rightmost components by identifying nodes in left partition

(combine all left-neighbours of a right vertex)

Label Cover for NPPerforming alphabet reduction repeatedly:

Label Cover for NP [MR’08, DH’09]:

For every alphabet and error δ=1/log||, Gap(1,δ)-LC is NP-hard.

Advantages:• Sub-constant error achievable• Nearly linear sized reduction

Label Cover variants Some hardness reductions require more

structure of the label cover instance

[KH’04] Hardness of Balanced homogenous linear equations MAXBISECTION

Mixing property Bipartite graph is a good sampler

Smoothness

Open Questions Sliding Scale Conjecture [BGLR’93]

For every alphabet and error δ=1/poly||, Gap(1,δ)-LC is NP-hard

(current results only obtain δ=1/log|| )

Obtain polynomial sized mixing and smooth PCPs (current constructions require subexponential sized proofs)

Average Case Hardness

Assumptions

Inapproximability results based on Worst case hardness assumptions (so far)

Average case hardness assumptions Cryptographic assumptions Random 3SAT hardness (Feige)

Random 3SAT n variables x1, x2, …., xn

m = Cn clauses (xi v xj v xk) Chosen randomly and independently

C – small, satisfiable w.h.p C – large, unsatisfiable w.h.p

Random 3SAT – large C

For large C Typical – unsatisfiable In fact at most (7/8 + δ) clauses satisfiable Rare – satisfiable Rare even for (1-δ) clauses to be satisfied

Random 3SAT – large C Proofs of unsatisfiability (coNP proof)

When C > √n, can find short of proof of unsatisfiability w.h.p.

For large constant < C < √n, though unsatisfiable, current techniques do not prove unsatisfiability

Feige’s Random 3SAT hypothesis For all 0<δ<1/8, there exists a large

constant C, and there does NOT exist a polynomial time algorithm that

INPUT: Random 3CNF formula with n variables and Cn clauses

OUTPUT: “typical” or “rare” on most inputs (> 50%) output typical but

never outputs “typical” on a rare instance (i.e, (1-δ) satisfiable )

MAX3SAT approximability

Feige’s Random 3SAT hypothesis

MAX3SAT is inapproximable (within

polynomial time) to a factor better than (7/8 +δ), for all δ>0.

MAX3AND

INPUT Boolean formula on n variables x1, x2, …., xn

m ANDs of 3 literals (xi ∧ xj ∧ xk)

OUTPUT Assignment

OBJECTIVE maximizes number of ANDs being satisfied

MAX3AND - approximability Feige’s Random 3SAT hypothesis

Not possible to approximate better than a factor (1/2 +δ) In particular, can’t distinguish between

MAX3AND instances > (1/4 - δ) satisfiable < (1/8 + δ ) satisfiable

(even for random MAX3AND instances)

Why MAX3AND? Gives inapproximability results

MAX-COMPLETE BIPARITITE GRAPH MINBISECTION DENSE k-SUBGRAPH 2-CATALOG SEGMENTATION

(not approximable beyond a particular constant)Previously, no known inapproximability results

Max Complete Bipartite Graph INPUT

n x n bipartite graph OUTPUT

k x k complete bipartite subgraph OBJECTIVE

Maximize k

Feige’s Hypothesis implies can’t approximate better than (1/2 + δ) (reduction from hardness of randomMAX3AND)

Reduction from MAX3AND Given a random MAX3AND with m ANDs construct bipartite graph

• Vertices on each side• m ANDS

• Edges• If two ANDS can

be satisfied simulatenously

Reduction from MAX3AND (contd) MAX3AND instance – (1/4 -δ ) satsifiable

Corresponding vertices from a kxk complete bipartite graph with k = (1/4 -δ )m

random MAX3AND instance – (1/8+δ ) satisfiable Easy to check that for a random MAX3AND

instance, whp every (1/8 +δ)m ANDs involve at least (n+1) literals

Any kxk bipartite graph with k > (1/8+δ)m involves a variable and its negation and hence not complete

Hardness of random MAX3AND Algorithm for random 3SAT (refuting

Feige’s hypothesis)

Idea: View input random 3CNF formula as a 3AND formula and use algorithm for random MAX3AND 3CNF - (7/8 +δ)-satisfiable

3AND formula – (1/8 +δ)-satisfiable 3CNF – (1-δ) satisfiable

3AND formula – (1/4 -δ)-satisfiable

Not exactly true, instead will use simple checks and SDPs to detect non-typical behavior

Hardness of MAX3AND (contd) Input: random 3SAT instance Algorithm

1. If any literal does not occur (3C/2±δ) times, output “rare”

2. Construct graphs G12, G23, G31whose vertices are all 2n literals and edges as follows:G12 :(xi,xj) is there is a clause of the form (xi,v xj v xk)

3. Run MAXCUT SDP on all 3 graphs, if SDP outputs larger than (1/2+δ)m, output “rare”

4. Run MAX3AND algorithm on instance If output > (1/4-δ), output “rare” If output < (1/8+δ), output “typical”

Hardness of MAX3AND (contd) Typical Instances ( < (7/8+δ)-satisfiable)

Easy to check algorithm outputs “typical” on most typical instances

Rare instance (> (1-δ)-satisfiable) Literal-occurrence and SDP checks ensure that

when viewed as a NOT-ALL-EQUAL-SAT instance, no assignment satisfies > (3/4+δ) clauses

Hence, at least (1/4-δ)-clauses are satisfied as ANDs for which the algorithm outputs “rare”

Random 3SAT assumption Feige’s hypothesis

MAX3AND inapproximable to better than ½ (even on random instances)

Inapproximability results COMPLETE-BIPARTITE-GRAPH, MINBISECTION,

DENSEST k-SUBGRAPH, ….

Other assumptions? Maximal number of equations satisfiable in

a random linear system [Ale’03]

Implies Feige’s hypothesis

Inapproximability of nearest-codeword-problem to within n1-δ

Hard to distinguish low-rigidity matrices and random matrices

Quasirandom PCPs [Kho’04] Suffices to having following quasi-

randomess of 3SAT For any set of half of the variables, (1/8±δ)-

fraction of clauses have all 3 variables from this set

Khot constructed PCPs with this quasirandom property leading to inapproximability results for earlier problems (based on worst case hardness)

Quasirandom PCPs PCPs which exhibit very different query

behaviour on YES and NO instance PCP verifier makes d queries

NO instances: For any set of half the proof locations, the probability that all the d queries are in the set ≈ 2-d

YES instances: There is a set of half the proof location, which the verifier queries more frequently ( > 2-(d-1))

Open Problems (Dis)prove Feige’s hypothesis

Connections between average complexity and approximation complexity

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