Tuesday, November 3, 2015

Distributed interpretation of the constructive Lovász Local Lemma

I follow the paper A Kolmogorov complexity proof of the Lovász local lemma for satisfiability, by J. Messner and T. Thierauf. Let $\phi$ be a $k$-CNF formula with $n$ variables, and $m$ clauses, each clause containing exactly $k$ literals (i.e., a variable, or the negation of a variable). The goal is to explicitly build an assignment of truth values to the variables so that each clause contains at least one literal evaluating to true.

  1. Original algorithm

We define the graph $\Gamma'$, having the clauses of $\phi$ as nodes. In $\Gamma'$,  two clauses $C$ and $D$ define an edge if $C$ has a literal that occurs negated in $D$. We denote by $d$ the maximum degree of $\Gamma'$.

The symmetric version of the Lovász local lemma states that $\phi$ is satisfiable if the clauses do not "interact too much", i.e., more precisely
$$
  \frac{d^d}{(d-1)^{(d-1)}} \leq 2^k - 1
$$
Moser and Tardos gave a constructive proof of the previous result, in the sense that they defined a (randomized) algorithm that (efficiently) produces a satisfying assignment for $\phi$ within $O(m)$ time steps. This algorithm runs as follows:
  • Pick a random assignment for the variables of $\phi$
  • While some clause in $\phi$ is not satisfied
    • Choose (deterministically) an unsatisfied clause $C$
    • Reassign the variables in $C$ independently at random
  •  Output the satisfying assignment

      2. Reformulation

    We now proceed to the interpretation of the proof from the point of view of distributed algorithms. We define an event $e$ as a pair $(C,b)$ where $C$ is a clause, and $b$ is a bit assignment of the variables in $C$. Two events are said to be independent if their clauses are not neighbours in $\Gamma'$.

    Let $\gamma$ be a configuration, i.e., a bit assignment of the $n$ variables in the formula. The event $e = (C,b)$ is enabled in $\gamma$ if $\gamma$ does not satisfy  $C$, i.e., all the literals in $C$ evaluate to false. We say that the event  is applied to $\gamma$ when the variables of clause $C$ are reassigned according to $b$. If $\gamma'$ is the resulting configuration, we denote by $\gamma \xrightarrow{e} \gamma'$ this transition.

    We now introduce a normal form for executions inspired by the work of Cartier and Foata in trace monoids. The idea stems from the fact that if two events $e,e'$ are independent and enabled in $\gamma$, then these events can be applied in any order, and yield the same configuration. One can model a schedule of events as a word $e_0\cdot e_1 \dots e_{s-1}$ in the trace monoid generated by the events, i.e., the quotient of the free monoid over the event alphabet by the commutativity relations induced by event independence. Cartier and Foata showed that any word $w$ in the trace monoid can be uniquely represented by the following normal form
    $$
         V_0 | \dots |  V_{t-1}
    $$
    where each $V_i$ is composed of pairwise independent events, and each event in $V_{i+1}$ depends on some event in $V_i$. An execution can then be defined as a pair $(\gamma_0, S)$ where $\gamma_0$ is the initial configuration, and $S$ is a schedule given in the normal form above. If $S$ contains $s$ events, then the execution has consumed exactly $n + s\cdot k$ random bits. The normal form can be depicted as a forest:

    Fig. 1 Compact representation of a schedule.
    The dependence relations are subsumed by the graph $\Gamma'$. Each dot on a line $C_i$ represents an event with clause $C_i$. The bit assignments are omitted in the picture. The arrows represent dependences. In particular, there are at most $d$ arrows out of any node. Let $[S]$ denote the the schedule $S$ without the bit assignments, i.e., $[S]$ retains only the causal order of clauses, pretty much as depicted in Fig. 1. We will refer to $[S]$ as the causal structure of $S$.

    I don't know if Messner, Thierauf were aware of this, but it turns out that the forest construction they give in their paper corresponds almost (I have not checked it thoroughly) to the method of Cartier and Foata for computing a normal form of a word in a trace monoid.

      3. Final argument

    The crucial point in the proof of Messner and Thierauf consists in the following observation: a finite execution $(\gamma_0,S)$ is entirely determined by the final configuration $\gamma_{t-1}$ and the causal structure $[S]$. 

    Indeed, if you know $\gamma_{t-1}$ and the last group $U_{t-1}$ of pairwise independent clause in $[S]$, then the bit assignment associated with any clause $C \in U_{t-1}$ is simply the restriction of $\gamma_{t-1}$ to the variables of $C$ since they were lastly modified. In particular, $\gamma_{t-2}$ is obtained from $\gamma_{t-1}$ by simply inverting the value of every variable occurring in one of the clauses of $U_{t-1}$.

    The consequence of this is that the $n + s\cdot k$ (Kolmogorov) random bits can be computed from the pair $(\gamma_{t-1},[S])$. The classic compression method in algorithmic information theory yields
    $$
        n + s\cdot k \leq K(\gamma_{t-1},[S]) + O(1)
    $$
    where $K(\cdot)$ is the Kolmogorov complexity. An upper bound for the right-hand side is given by $n$ (number of bits to encode $\gamma_{t-1}$), and the logarithm of the number of causal structures $[S]$ with exactly $s$ (bit-free) events. Combinatorics yields
    $$
      n + s\cdot k \leq n + (d\cdot s + m) \cdot h\left(\frac{1}{d}\right)+ O(1)
    $$
    where $h(p) = - p\cdot \log p - (1-p)\cdot \log (1-p)$. The bound on $d$ gives $s = O(m)$.

    pb

    [1] Robin A. Moser, Gábor Tardos, A constructive proof of the Lovász local lemma, arxiv.org/abs/0903.0544, 2009
    [2] Jochen Messner, Thomas Thierauf, A Kolmogorov complexity proof of the Lovász local lemma for satisfiability, Theoretical Computer Science, volume 461, pages 55-64, 2012
    [3] Pierre Cartier, Dominique Foata, Problèmes combinatoire de commutation et de réarrangements, Lecture notes in Mathematics, 85, Springer Verlag, 1969 

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