Scribed by Luowen Qian
In which we use spectral techniques to find certificates of unsatisfiability for random -SAT formulas.
Given a random -SAT formula with clauses and variables, we want to find a certificate of unsatisfiability of such formula within polynomial time. Here we consider as fixed, usually equal to 3 or 4. For fixed , the more clauses you have, the more constraints you have, so it becomes easier to show that these constraints are inconsistent. For example, for 3-SAT,
The algorithm for finding such certificate is shown below.
We know that we can solve 2-SATs in linear time, and approximately
clauses contains . Similarly when is sufficiently large, the 2-SATs will almost surely be unsatisfiable. When a subset of the clauses is not satisfiable, the whole 3-SAT formula is not satisfiable. Therefore we can certify unsatisfiability for 3-SATs with high probability.
In general for -SAT,
Since for every fixed assignments to the first variables, approximately
portion of the clauses remains, we expect the constant and the running time is .
So what about ‘s that are in between? It turns out that we can do better with spectral techniques. And the reason that spectral techniques work better is that unlike the previous method, it does not try all the possible assignments and fails to find a certificate of unsatisfiability.
2. Reduce certifying unsatisfiability for k-SAT to finding largest independent set
2.1. From 3-SAT instances to hypergraphs
Given a random 3-SAT formula , which is an and of random 3-CNF-SAT clauses over variables (abbreviated as vector ), i.e.
where , and no two are exactly the same. Construct hypergraph , where
is a set of vertices, where each vertex means an assignment to a variable, and
is a set of 3-hyperedges. The reason we’re putting in the negation of is that a 3-CNF clause evaluates to false if and only if all three subclauses evaluate to false. This will be useful shortly after.
First let’s generalize the notion of independent set for hypergraphs.
An independent set for hypergraph is a set that satisfies .
If is satisfiable, has an independent set of size at least . Equivalently if the largest independent set of has size less than , is unsatisfiable. Proof: Assume is satisfiable, let be a satisfiable assignment, where . Then is an independent set of size . If not, it means some hyperedge , so and the -th clause in evaluates to false. Therefore evaluates to false, which contradicts the fact that is a satisfiable assignment.
We know that if we pick a random graph that’s sufficiently dense, i.e. the average degree , by spectral techniques we will have a certifiable upper bound on the size of the largest independent set of with high probability. So if a random graph has random edges, we can prove that there’s no large independent set with high probability.
But if we have a random hypergraph with random hyperedges, we don’t have any analog of spectral theories for hypergraphs that allow us to do this kind of certification. And from the fact that the problem of certifying unsatisfiability of random formula of clauses is considered to be hard, we conjecture that there doesn’t exist a spectral theory for hypergraphs able to replicate some of the things we are able to do on graphs.
However, what we can do is possibly with some loss, to reduce the hypergraph to a graph, where we can apply spectral techniques.
2.2. From 4-SAT instances to graphs
Now let’s look at random 4-SATs. Similarly we will write a random 4-SAT formula as:
where , and no two are exactly the same. Similar to the previous construction, but instead of constructing another hypergraph, we will construct just a graph , where
is a set of vertices and
is a set of edges.
If is satisfiable, has an independent set of size at least . Equivalently if the largest independent set of has size less than , is unsatisfiable. Proof: The proof is very similar to the previous one. Assume is satisfiable, let be a satisfiable assignment, where . Then is an independent set of size . If not, it means some edge , so and the -th clause in evaluates to false. Therefore evaluates to false, which contradicts the fact that is a satisfiable assignment.
From here, we can observe that is not a random graph because some edges are forbidden, for example when the two vertices of the edge has some element in common. But it’s very close to a random graph. In fact, we can apply the same spectral techniques to get a certifiable upper bound on the size of the largest independent set if the average degree , i.e. if , we can certify unsatisfiability with high probability, by upper bounding the size of the largest independent set in the constructed graph.
We can generalize this results for all even ‘s. For random -SAT where is even, if , we can certify unsatisfiability with high probability, which is better than the previous method which requires . The same is achievable for odd , but the argument is significantly more complicated.
2.3. Certifiable upper bound for independent sets in modified random sparse graphs
Despite odd ‘s, another question is that in this setup, can we do better and get rid of the term? This term is coming from the fact that spectral norm break down when the average degree . However it’s still true that random graph doesn’t have any large independent sets even when the average degree is constant. It’s just that the spectral norm isn’t giving us good bounds any more, since the spectral norm is at most . So is there something tighter than spectral bounds that could help us get rid of the term? Could we fix this by removing all the high degree vertices in the random graph?
This construction is due to Feige-Ofek. Given random graph , where the average degree is some large constant. Construct by taking and removing all edges incident on nodes with degree higher than where is the average degree of . We denote for the adjacency matrix of and for that of . And it turns out,
With high probability, .
It turns out to be rather difficult to prove. Previously we saw spectral results on random graphs that uses matrix traces to bound the largest eigenvalue. In this case, it’s hard to do so because the contribution to the trace of a closed walk is complicated by the fact that edges have dependencies. The other approach is that given random matrix , we will try to upper bound . A standard way for this is to that for every solution, count the instances of in which the fixed solution is good, and argue that the number of the fixed solutions is small, which tells us that there’s no good solution. The problem here is that the set of solutions is infinitely large. So Feige-Ofek discretize the set of vectors, and then reduce the bound on the quadratic form of a discretized vector to a sum of several terms, each of which has to be carefully bounded.
We always have
and so, with high probability, we get an polynomial time upper bound certificate to the size of the independent set for a random graph. This removes the extra term from our analysis of certificates of unsatisfiability for random -SAT when is even.
3. SDP relaxation of independent sets in random sparse graphs
In order to show a random graph has no large independent sets, a more principled way is to argue that there is some polynomial time solvable relaxation of the problem whose solution is an upper bound of the problem.
Let SDPIndSet be the optimum of the following semidefinite programming relaxation of the Independent Set problem, which is due to Lovász:
Since it’s the relaxation of the problem of finding the maximum independent set, for any graph . And this relaxation has a nice property.
Proof: First we note that SDPIndSet is at most
and this is equal to
which is at most
Finally, the above optimization is equivalent to the following
which is at most the unconstrained problem
Recall from the previous section that we constructed by removing edges from , which corresponds to removing constraints in our semidefinite programming problem, so , which is by theorem 3 at most with high probability.
4. SDP relaxation of random k-SAT
From the previous section, we get an idea that we can use semidefinite programming to relax the problem directly and find a certificate of unsatisfiability for the relaxed problem.
Given a random -SAT formula :
The satisfiability of is equivalent of the satisfiability of the following equations:
Notice that if we expand the polynomial on the left side, there are some of the monomials having degree higher than 2 which prevents us relaxing these equations to a semidefinite programming problem. In order to resolve this, and we introduce . Then we can relax all variables to be vectors, i.e.
For example, if we have a 4-SAT clause
we can rewrite it as
For this relaxation, we have:
(Photo from facebook.com)
Michael Cohen, one the most brilliant young minds of our field, recently passed away in Berkeley.
After going to MIT for college, Michael worked for Facebook and was a graduate student at MIT. This semester, he was at Berkeley as Simons Fellow in connection with the program on optimization at the Simons Institute.
In a few short years, Michael left his mark on a number of problems that are close to the heart of in theory‘s readers.
He was part of the team that developed the fastest algorithm for solving systems of linear equations in which the matrix of constraints is a graph Laplacian (or, more generally, is symmetric and diagonally dominated), running in time where is the number of non-zero entries of the matrix and is the number of variables.
He also worked on matrix approximation via subsampling, on algorithms that approximate random walk properties, on algorithms for flow and shortest paths, and on geometric algorithms.
My favorite result is his single-author paper giving a polynomial time construction of bipartite Ramanujan graphs of all degree and all sizes, making the approach of Marcus, Spielman and Srivastava constructive.
Michael was a unique person, who gave a lot to our community and had touched several lives. His loss is an unspeakable tragedy that I still find very hard to process.
“Art imitates life, but life imitates bad TV” (Woody Allen)
The mention for a major alumni award given by U.C. Berkeley is for excellence in achievement.
Meanwhile, in the episode “Brother, can you spare two dimes?”, Mr. Burns has to come up on the spot with the name for a fake awards, and he comes up with an award for outstanding achievement in the field of excellence.
(You’ll note that the dancers in the video are wearing gold and blue)
From an interview for New York Magazine:
It’s like if you were on a plane and you wanted to choose a pilot. You have one person, Hillary, who says, “Here’s my license. Here’s all the thousands of flights that I’ve flown. Here’s planes I’ve flown in really difficult situations. I’ve had some good flights and some bad flights, but I’ve been flying for a very long time, and I know exactly how this plane works.” Then you’ve got Bernie, who says, “Everyone should get a ride right to their house with this plane.” “Well, how are you going to do that?” “I just think we should. It’s only fair that everyone gets to use the plane equally.” And then Trump says, “I’m going to fly so well. You’re not going to believe how good I’m going to fly this plane, and by the way, Hillary never flew a plane in her life.” “She did, and we have pictures.” “No, she never did it.”
Having spent some time in Japan, I have learnt of the tradition of holding a Bōnenkai, literally a party to forget the year. Held either as a company end-of-year party, or by groups of friends, it’s a get-together in which people drink a lot and forget the bad things that happened to them during the year.
It occurred to me that this is the complement of Thanksgiving, in which you get together to remember the good things that happened during the year.
I don’t think there is anything else left to say about the difference between Japanese and American culture.
Interestingly, there are a couple more possibilities. One could remember the bad things that happened during the year, as in the airing of grievances during Festivus.
Finally, one could forget the good things, which is very much the Italian attitude.
Edited to add: I don’t know how I forgot (ah!) but there is a famous Neapolitan folk song that goes
Chi ha avuto, ha avuto, ha avuto
Chi ha dato, ha dato, ha dato,
Scurdammuce ‘o passato,
simm’e Napule, paisa’
which is roughly
Who has received, has received
Who has given, has given,
Let’s forget the past
We are [all] from Naples
Today we show how to construct an inefficient (but efficiently verifiable) signature scheme starting from a one-time signature scheme.
Next time we shall see how to make it efficient using a pseudorandom function.
Today we finish the analysis of a construction of a pseudorandom permutation (block cipher) given a pseudorandom function.