Lower Bounds on Dynamic Programming for Maximum Weight Independent Set

We prove unconditional lower bounds for sizes of MWIS-circuits and MWIS-formulas parameterized by graph parameters treewidth and treedepth, respectively. The lower bounds are exponential in treewidth and treedepth, and therefore well-known algorithms yield matching upper bounds for them [3, 18]. We emphasize that our lower bounds are not worst-case bounds over graph classes, but instead hold for each individual graph.

Our main circuit complexity tool is an adaptation of a circuit decomposition lemma used in e.g. [22, 23, 37]. In particular, we show that this lemma can be adapted so that given an MWIS-circuit with $\tau$ gates of a graph with treewidth $w$ it extracts a family of $\tau$ vertex separators each of size $\Omega(w)$. Once this family has been extracted, the main challenge for proving Theorem 1.1 is to show that if this family of separators is too small, there exists an independent set that intersects all of the separators. For this we use the lopsided Lovász Local Lemma [15], though we note that more elementary arguments would suffice to prove the theorem with a worse dependency on $d$. To extend the result from bounded degree graphs to $H$-minor-free graphs we use the minor model of the bounded degree induced minor with high treewidth to further control the structure of these separators.

For MWIS-formulas parameterized by treedepth $t$ we similarly extract a family of $2\tau$ vertex sets each of size $\Omega(t)$ from a $\tau$-gate MWIS-formula, showing that if an independent set intersects all of these vertex sets the formula cannot compute it. The same application of the Local Lemma is used to prove that such an independent set indeed exists in low degree graphs if $\tau$ is too small. The argument for extracting the family from the formula is more ad-hoc than the argument for circuits.

The convention of modeling dynamic programming algorithms as tropical circuits originates from the recent works of Jukna [24, 25], although some earlier results in monotone arithmetic circuit complexity apply also to tropical circuits [23]. In general, tropical circuit lower bounds imply lower bounds for monotone arithmetic circuits, but not necessarily the other way around [24]. In addition to the works of Jukna, the other works explicitly giving lower bounds for tropical circuits or formulas that we are aware of are [30, 31]. We are not aware of prior works on lower bounds for tropical circuits or formulas considering maximum weight independent set or the graph parameters treewidth or treedepth.

There are multiple worst-case hardness results related to different formulations of the independent set polynomial. In [7] it was shown that the multivariate independent set polynomial is VNP-complete. The univariate independent set polynomial is #P-hard to evaluate at every non-zero rational point [5], and more fine-grainedly its evaluation has $2^{\Omega(n/\log^{3}n)}$ worst-case complexity assuming #ETH [21].

Chvátal has shown that a certain proof system for maximum independent set which naturally corresponds to branching algorithms requires exponential size proofs on almost all graphs that have the number of edges linear in the number of vertices [11].

Multiple worst-case lower bounds of form $n^{\Omega(w)}$ in limited models of computation for graph homomorphism problems of a pattern graph with treewidth $w$ to a graph with $n$ vertices are known [4, 26, 28]. In particular, recently it was shown that the worst-case monotone arithmetic circuit complexity of homomorphism polynomial is $\Theta(n^{w+1})$, and the worst-case monotone arithmetic formula complexity is $\Theta(n^{t})$, where $t$ is the treedepth of the pattern graph [26].

Recently, a lower bound of $2^{\Omega(w)}$ was shown for DNNF-compilation of monotone CNFs with primal treewidth $w$ and bounded degree and arity, applying to all such CNFs [2]. We note that after the acceptance of this paper, we became aware of a reduction from MWIS-circuits to DNNFs that allows to prove a weaker version of our Theorem 1.1 via the result of [2]. In particular, the techniques of [2] yield an exponent of form $\Omega(w/2^{d})$ instead of the best possible $\Omega(w/d)$ given in Theorem 1.1.

In Section 2 we present preliminaries on graph theory, define MWIS-circuits and prove simple lemmas on them, and discuss the lopsided Lovász Local Lemma and prove a lemma using it. In Section 3 we prove the lower bounds for MWIS-circuits parameterized by treewidth, i.e., Theorems 1.1 and 1.2. In Section 4 we prove the lower bound for MWIS-formulas parameterized by treedepth, i.e., Theorem 1.6. In Section 5 we give the construction that shows the optimality of Theorems 1.1 and 1.6 up to a factor $d$. We conclude and discuss future work in Section 6.

The vertex set of a graph $G$ is denoted by $V(G)$ and the edge set by $E(G)$. The set of neighbors of a vertex $v$ is denoted by $N(v)$ and the neighborhood of a vertex set $X$ by $N(X)=\bigcup_{v\in X}N(v)\setminus X$. Closed neighborhoods are denoted by $N[v]=N(v)\cup\{v\}$ and $N[X]=N(X)\cup X$. The subgraph $G[X]$ induced by a vertex set $X\subseteq V(G)$ has $V(G[X])=X$ and $E(G[X])=\{\{u,v\}\in E(G)\mid u\in X\wedge v\in X\}$. We also use $G\setminus X=G[V(G)\setminus X]$ to denote induced subgraphs. An independent set of $G$ is a vertex set $I$ such that $G[I]$ has no edges. In particular, an empty set is an independent set.

A tree decomposition of a graph $G$ is a tree $T$ whose each vertex $i\in V(T)$ corresponds to a bag $B_{i}\subseteq V(G)$, satisfying that

1. 1.

   $V(G)=\bigcup_{i\in V(T)}B_{i}$,

2. 2.

   for each $\{u,v\}\in E(G)$ there is a bag $B_{i}$ with $\{u,v\}\subseteq B_{i}$, and

3. 3.

   for each $v\in V(G)$ the subtree of $T$ induced by bags containing $v$ is connected.

The width of a tree decomposition is $\max|B_{i}|-1$ and the treewidth $tw(G)$ of a graph $G$ is the minimum width over its tree decompositions.

A treedepth decomposition of a graph $G$ is a rooted forest $F$ with vertex set $V(F)=V(G)$, satisfying for each $\{u,v\}\in E(G)$ that $u$ and $v$ have an ancestor-descendant relation in $F$. The depth of $F$ is the maximum number of vertices on a simple path from a root to a leaf in $F$. The treedepth $td(G)$ of a graph $G$ is the minimum depth over its treedepth decompositions. Note that $tw(G)+1\leq td(G)$.

We start by giving a formal definition of a tropical circuit. Our definition is non-standard in that it does not allow any other input constants than $0$, which we can w.l.o.g. assume in the context of maximum weight independent set. For a comprehensive treatment of tropical circuits and their relations to monotone Boolean and monotone arithmetic circuits see [24].

With an assignment of real numbers to the variables $X$, a tropical circuit outputs a number computed by the output gate by natural semantics, i.e., a gate labeled with a variable $x_{i}$ computes the value of $x_{i}$, a gate labeled with $0$ computes $0$, a gate labeled with $+$ computes the sum of the values computed by its children, and a gate labeled with $\max$ computes the maximum of the values computed by its children. In particular, a tropical circuit computes a tropical polynomial in the variables $X$ over the tropical $(\mathbb{R}\cup\{-\infty\},\max,+)$ semiring. In the tropical semiring $\max$ corresponds to addition and $+$ corresponds to multiplication, with $-\infty$ as the zero and $0$ as the unit. We will refer to $\max$ as addition and to $+$ as multiplication.

We define an MWIS-polynomial with the following simple lemma.

An MWIS-polynomial of a graph $G$ is a polynomial $f$ satisfying (1) and (2) in Lemma 2.2. An MWIS-circuit of $G$ is a tropical circuit that computes an MWIS-polynomial of $G$. An MWIS-formula of $G$ is an MWIS-circuit of $G$ that is a tropical formula.

We note that requiring the circuit to work for all real weights is not a strong assumption: Any MWIS-circuit that works for weights $\{0,1\}$ can be turned into an MWIS-circuit that works for weights $\mathbb{R}\cup\{-\infty\}$ by replacing each input variable $v_{i}$ by $\max(v_{i},0)$. In particular, the weight of an empty independent set is $0$, so negative weights will never be used. Our assumption that the only constant available to the circuit is $0$ can be justified by noting that if an output monomial would contain a positive constant the circuit would be incorrect on the all-zero input, and that if an output monomial would contain a negative constant it should also occur without the constant. In particular, any other constants than $0$ could be replaced by $0$.

Next we make some simple observations on the structure of MWIS-circuits.

Note that by monotonicity of $(\max,+)$ computations we can assume that each gate of an MWIS-circuit computes a partial MWIS-polynomial and therefore each subcircuit is a partial MWIS-circuit.

We also use $\text{Sup}(g)$ for a gate $g$ to denote the support of the polynomial computed by the gate. Note that each monomial of $f$ corresponds to an independent set of $G[\text{Sup}(f)]$.

The following property is the basis for proving lower bounds for MWIS-circuits.

We will say that a partial MWIS-polynomial $f$ or a circuit computing $f$ computes an independent set $I$ if $f$ contains the monomial $\prod_{v_{i}\in I}v_{i}$. In particular, an MWIS-polynomial computes every independent set.

The lopsided Lovász Local Lemma [15] (see [1] for the general version) is a method for showing that there is a non-zero probability that none of the events in a collection of events hold. In particular, we use it to show that independent sets satisfying certain requirements exist.

In words, the negative dependency graph should capture all negative correlations between the events.

We prove a lemma which captures our use of the Local Lemma in Theorems 1.1 and 1.6. We spell out the constants to emphasize that they are not particularly high, although noting that a more careful proof could improve them a bit.

Now we complete the proof of Theorem 1.1 by putting Lemmas 2.10, 3.1, and 3.2 together.

An induced minor model of a graph $H$ in a graph $G$ is a function $f:V(H)\to 2^{V(G)}\setminus\{\emptyset\}$, where $2^{V(G)}$ denotes the power set of $V(G)$, satisfying that

1. 1.

   the sets $f(u)$ and $f(v)$ are disjoint for $u\neq v$,

2. 2.

   for each $v\in V(H)$ the induced subgraph $G[f(v)]$ is connected, and

3. 3.

   $\{u,v\}\in E(H)$ if and only if $N(f(u))$ intersects $f(v)$.

A graph $G$ contains a graph $H$ as an induced minor if and only if there is an induced minor model of $H$ in $G$. For $v\in V(H)$ we call the induced subgraphs $G[f(v)]$ clusters.

First, we ensure that the maximum degree of each cluster is bounded.

We also need the following lemma.

Next we finish the proof with similar arguments as in the proof of Theorem 1.1, but with a different kind of construction of the independent set with the Local Lemma. In this case the constants involved appear to be impractical.