Graph Homomorphism Polynomials: Algorithms and Complexity

Abstract

We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph H to n-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new algorithms for counting and detecting graph patterns, and also for obtaining natural polynomial families which are complete for algebraic complexity classes VBP, VP, and VNP. We discover that, in the monotone setting, the formula complexity, the ABP complexity, and the circuit complexity of such polynomial families are exactly characterized by the treedepth, the pathwidth, and the treewidth of the pattern graph respectively.Furthermore, we establish a single, unified framework, using our characterization, to collect several known results that were obtained independently via different methods. For instance, we attain superpolynomial separations between circuits, ABPs, and formulas in the monotone setting, where the polynomial families separating the classes all correspond to well-studied combinatorial problems. Moreover, our proofs rediscover fine-grained separations between these models for constant-degree polynomials. The characterization additionally yields new space-time efficient algorithms for several pattern detection and counting problems.

Finding and Counting Patterns in Sparse Graphs

“…The best known algorithms can be expressed as divide-and-conquer algorithms that evaluate small formulas constructed by making use of the graph parameter treedepth. Komarath, Pandey, and Rahul [13] also showed that the running-time of these algorithms match the best possible formula size for all pattern graphs. These arithmetic circuit lower bounds serve as a technical motivation for considering sparse host graphs, in addition to the practical motivation mentioned earlier.…”

“…The graph parameter treedepth plays a crucial role in answering this question. It is well known that we can count the homomorphisms from a pattern of treedepth d in O(n d )-time while using only constant space (See Komarath, Rahul, and Pandey [13] for a construction of arithmetic formulas counting them. These arithmetic formulas can be implicitly constructed and evaluated in constant space.).…”

“…Can we improve these algorithms by finding more efficient ways to construct arithmetic circuits for homomorphism polynomials? Unfortunately, it is known that for the type of circuit that is constructed, i.e., circuits that do not involve cancellations, the existing constructions are the best possible for all pattern graphs, as shown by Komarath, Pandey, and Rahul [13]. The situation is similar for constant space algorithms.…”

Lower Bounds on Dynamic Programming for Maximum Weight Independent Set

“…As formulas can be thought of as bounded space analogies of circuits, Theorem 4 gives further evidence (in addition to e.g. [9,24,31]) supporting that while treewidth is the right parameter for CSP-like problems when equipped with unlimited space, treedepth is the right parameter when dealing with bounded space.…”

“…Multiple worst-case lower bounds of form n Ω(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 [4,24,25]. In particular, recently it was shown that the worst-case monotone arithmetic circuit complexity of homomorphism polynomial is Θ(n w+1 ), and the worst-case arithmetic formula complexity is Θ(n t ), where t is the treedepth of the pattern graph [24].…”