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 Cover Label Cover (LC) G – Bipartite graph  1,  2 – labels (projection) constraint per edge c e:  1   2 Edge e is satisfied if c e (σ 1 )=σ 2 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 11 22

PCP Theorem Label Cover (LC) G – Bipartite graph  1,  2 – labels (projection) constraint per edge c e:  1   2 Edge e is satisfied if c e (σ 1 )=σ 2 PCP Theorem […., AS’92, ALMSS’92] Gap(1,0.9999)-LC is NP-hard 11 22

Stronger form of PCP Theorem Label Cover (LC) G – Bipartite graph  1,  2 – labels (projection) constraint per edge c e:  1   2 Edge e is satisfied if c e (σ 1 )=σ 2 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) 11 22

Proving tight hardness results 11 22 Outer Verifier Dictatorship Test [Fourier Analysis] + (composition) Inner Verifier

Proving tight hardness results 11 22 + 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 graph  1,  2 – labels (projection) constraint per edge c e:  1   2 Edge e is satisfied if c e (σ 1 )=σ 2 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) 11 22

Label Cover Constructions

Low Degree Test (LDT) [RS’92]  Given function f:F m  F (F – field), check if f is the evaluation of a low-degree polynomial without reading all of F f:F m  F Use fact that restriction of low-degree polynomial to a line is still low-degree

Label Cover for LDT Points table f:F m  F Lines table f lines Constraint: f lines (l)(x) = f(x) Large Alphabet Size

Label Cover -- LDT [AS’97, RS’97]  Completeness: If f:F m  F is a low-degree polynomial, there exists lines table f lines such that Pr[f lines (l)(x) = f(x)] = 1  Soundness: If f:F m  F is “far” from being low-degree polynomial, there for all lines table f lines we have Pr[f lines (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 | n polylog n Caveat: Large Alphabet Size Renders 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 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 NP Performing 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 x 1, x 2, …., x n  m = Cn clauses (x i v x j v x k ) 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 x 1, x 2, …., x n m ANDs of 3 literals (x i ∧ x j ∧ x k )  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 G 12, G 23, G 31 whose vertices are all 2n literals and edges as follows: G 12 :(x i, x j ) is there is a clause of the form (x i, v x j v x k ) 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 n 1-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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