Counting Maximum Matchings in Planar Graphs Is Hard
Istvan Miklos, Miklos Kresz
备注：
The presented result is not new. A proof of the main theorem appeared in Vadhan (2001) THE COMPLEXITY OF COUNTING IN SPARSE, REGULAR, AND PLANAR GRAPHS
Here we prove that counting maximum matchings in planar, bipartite graphs is #P-complete. This is somewhat surprising in the light that the number of perfect matchings in planar graphs can be computed in polynomial time. We also prove that counting non-necessarily perfect matchings in planar graphs is already #P-complete if the problem is restricted to bipartite graphs. So far hardness was proved only for general, non-necessarily bipartite graphs.
We consider the problem of counting matchings in planar graphs. While perfect matchings in planar graphs can be counted by a classical polynomial-time algorithm, the problem of counting all matchings (possibly containing unmatched vertices, also known as defects) is known to be #P-complete on planar graphs. To interpolate between the hard case of counting matchings and the easy case of counting perfect matchings, we study the parameterized problem of counting matchings with exactly k unmatched vertices in a planar graph G, on input G and k. This setting has a natural interpretation in statisti…
We prove a complexity dichotomy theorem for counting planar graph homomorphisms of domain size 3. Given any 3 by 3 real valued symmetric matrix $H$ defining a graph homomorphism from all planar graphs $G \mapsto Z_H(G)$, we completely classify the computational complexity of this problem according to the matrix $H$. We show that for every $H$, the problem is either polynomial time computable or \#P-hard. The P-time computable cases consist of precisely those that are P-time computable for general graphs (a complete classification is known) or computable by Valiant's holographic algorithm via m…
We prove the #P-hardness of the counting problems associated with various satisfiability, graph and combinatorial problems, when restricted to planar instances. These problems include \begin{romannum} \item[{}] {\sc 3Sat, 1-3Sat, 1-Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Vertex Set, X3C, Partition Into Triangles, and Clique Cover.} \end{romannum} We also prove the {\sf NP}-completeness of the {\sc Ambiguous Satisfiability} problems \cite{Sa80} and the {\sf D$^P$}-completeness (with respect to random polynomial reducibility) of the unique satisfiability problems \…
We systematically investigate the complexity of counting subgraph patterns modulo fixed integers. For example, it is known that the parity of the number of $k$-matchings can be determined in polynomial time by a simple reduction to the determinant. We generalize this to an $n^{f(t,s)}$-time algorithm to compute modulo $2^t$ the number of subgraph occurrences of patterns that are $s$ vertices away from being matchings. This shows that the known polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry over into the setting of counting modulo $2^t$. Complementing our algorit…
We investigate the complexity of approximately counting stable matchings in the $k$-attribute model, where the preference lists are determined by dot products of "preference vectors" with "attribute vectors", or by Euclidean distances between "preference points" and "attribute points". Irving and Leather proved that counting the number of stable matchings in the general case is $#P$-complete. Counting the number of stable matchings is reducible to counting the number of downsets in a (related) partial order and is interreducible, in an approximation-preserving sense, to a class of problems tha…
We prove #P-completeness results for counting edge colorings on simple graphs. These strengthen the corresponding results on multigraphs from [4]. We prove that for any $κ\ge r \ge 3$ counting $κ$-edge colorings on $r$-regular simple graphs is #P-complete. Furthermore, we show that for planar $r$-regular simple graphs where $r \in \{3, 4, 5\}$ counting edge colorings with \k{appa} colors for any $κ\ge r$ is also #P-complete. As there are no planar $r$-regular simple graphs for any $r > 5$, these statements cover all interesting cases in terms of the parameters $(κ, r)$.
A graph $H$ is single-crossing if it can be drawn in the plane with at most one crossing. For any single-crossing graph $H$, we give an $O(n^4)$ time algorithm for counting perfect matchings in graphs excluding $H$ as a minor. The runtime can be lowered to $O(n^{1.5})$ when $G$ excludes $K_5$ or $K_{3,3}$ as a minor. This is the first generalization of an algorithm for counting perfect matchings in $K_{3,3}$-free graphs (Little 1974, Vazirani 1989). Our algorithm uses black-boxes for counting perfect matchings in planar graphs and for computing certain graph decompositions. Together with an in…
In this paper, we build on the idea of Valiant \cite{Val79a} and Ben-Dor/Halevi \cite{Ben93}, that is, to count the number of satisfying solutions of a boolean formula via computing the permanent of a specially constructed matrix. We show that the Desnanot-Jacobi identity ($\dji$) prevents Valiant's original approach to achieve a parsimonious reduction to the permanent over a field of characteristic two. As the next step, since the computation of the permanent is $#\classP$-complete, we make use of the equality of the permanent and the number of perfect matchings in an unweighted graph's bipar…
In this paper we rigorously prove the validity of the cavity method for the problem of counting the number of matchings in graphs with large girth. Cavity method is an important heuristic developed by statistical physicists that has lead to the development of faster distributed algorithms for problems in various combinatorial optimization problems. The validity of the approach has been supported mostly by numerical simulations. In this paper we prove the validity of cavity method for the problem of counting matchings using rigorous techniques. We hope that these rigorous approaches will finall…
We study the approximation complexity of the partition function of the eight-vertex model on general 4-regular graphs. For the first time, we relate the approximability of the eight-vertex model to the complexity of approximately counting perfect matchings, a central open problem in this field. Our results extend those in arXiv:1811.03126 [cs.CC]. In a region of the parameter space where no previous approximation complexity was known, we show that approximating the partition function is at least as hard as approximately counting perfect matchings via approximation-preserving reductions. In ano…
Is perfect matching in NC? That is, is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in theoretical computer science for over three decades, ever since the discovery of RNC matching algorithms. Within this question, the case of planar graphs has remained an enigma: On the one hand, counting the number of perfect matchings is far harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs, counting has long been known to be in NC whereas finding one has resisted a solution. In this paper, we g…
Recently, perfect matching in bounded planar cutwidth bipartite graphs (\BGGM) was shown to be in ACC$^0$ by Hansen et al.. They also conjectured that the problem is in AC$^0$. In this paper, we disprove their conjecture by showing that the problem is not in AC$^0[p^α]$ for every prime $p$. Our results show that the previous upper bound is almost tight. Our techniques involve giving a reduction from Parity to BGGM. A further improvement in lower bounds is difficult since we do not have an algebraic characterization for AC$^0[m]$ where $m$ is not a prime power. Moreover, this will also imply a …
Consider a planar graph $G=(V,E)$ with polynomially bounded edge weight function $w:E\to [0, poly(n)]$. The main results of this paper are NC algorithms for the following problems: - minimum weight perfect matching in $G$, - maximum cardinality and maximum weight matching in $G$ when $G$ is bipartite, - maximum multiple-source multiple-sink flow in $G$ where $c:E\to [1, poly(n)]$ is a polynomially bounded edge capacity function, - minimum weight $f$-factor in $G$ where $f:V\to [1, poly(n)]$, - min-cost flow in $G$ where $c:E\to [1, poly(n)]$ is a polynomially bounded edge capacity function and…
We consider the $\#\mathsf{W}[1]$-hard problem of counting all matchings with exactly $k$ edges in a given input graph $G$; we prove that it remains $\#\mathsf{W}[1]$-hard on graphs $G$ that are line graphs or bipartite graphs with degree $2$ on one side. In our proofs, we use that $k$-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of $k$ length-$2$ paths into (arbitrary) host graphs. Here, a homomorphism from $H$ to $G$ is edge-injective if it maps any two distinct edges of $H$ to distinct edges in $G$. We show that edge-injective h…
We establish the expressibility in fixed-point logic with counting (FPC) of a number of natural polynomial-time problems. In particular, we show that the size of a maximum matching in a graph is definable in FPC. This settles an open problem first posed by Blass, Gurevich and Shelah, who asked whether the existence of perfect matchings in general graphs could be determined in the more powerful formalism of choiceless polynomial time with counting. Our result is established by showing that the ellipsoid method for solving linear programs can be implemented in FPC. This allows us to prove that l…
We show that an effective version of Siegel's Theorem on finiteness of integer solutions and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the k >= 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. This dichotomy holds even when restricted to planar graphs. A special case of this result is that counting edge k-colorings is #P-hard over planar 3-regular graphs for k…
We consider a weighted counting problem on matchings, denoted $\textrm{PrMatching}(\mathcal{G})$, on an arbitrary fixed graph family $\mathcal{G}$. The input consists of a graph $G\in \mathcal{G}$ and of rational probabilities of existence on every edge of $G$, assuming independence. The output is the probability of obtaining a matching of $G$ in the resulting distribution, i.e., a set of edges that are pairwise disjoint. It is known that, if $\mathcal{G}$ has bounded treewidth, then $\textrm{PrMatching}(\mathcal{G})$ can be solved in polynomial time. In this paper we show that, under some ass…
We present an algorithm that takes as input an $n$-vertex planar graph $G$ and a $k$-vertex pattern graph $P$, and computes the number of (induced) copies of $P$ in $G$ in $2^{O(k/\log k)}n^{O(1)}$ time. If $P$ is a matching, independent set, or connected bounded maximum degree graph, the runtime reduces to $2^{\tilde{O}(\sqrt{k})}n^{O(1)}$. While our algorithm counts all copies of $P$, it also improves the fastest algorithms that only detect copies of $P$. Before our work, no $2^{O(k/\log k)}n^{O(1)}$ time algorithms for detecting unrestricted patterns $P$ were known, and by a result of Bodla…
Given an edge-weighted graph G, let PerfMatch(G) denote the weighted sum over all perfect matchings M in G, weighting each matching M by the product of weights of edges in M. If G is unweighted, this plainly counts the perfect matchings of G. In this paper, we introduce parity separation, a new method for reducing PerfMatch to unweighted instances: For graphs G with edge-weights -1 and 1, we construct two unweighted graphs G1 and G2 such that PerfMatch(G) = PerfMatch(G1) - PerfMatch(G2). This yields a novel weight removal technique for counting perfect matchings, in addition to those known fro…
We discover new P-time computable six-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. We further prove that there are no more: Together, they exhaust all P-time computable six-vertex models on planar graphs, assuming #P is not P. This leads to the following exact complexity classification: For every parameter setting in ${\mathbb C}$ for the six-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) #P-hard for general graphs but computable in P-time for planar graphs, or (3) #P-hard even for planar gra…
Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in statistical physics community for decades. They capture precisely those problems which are #P-hard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the …
Let $A,B$ with $|A| = m$ and $|B| = n\ge m$ be two sets. We assume that every element $a\in A$ has a reference list over all elements from $B$. We call an injective mapping $τ$ from $A$ to $B$ a matching. A blocking coalition of $τ$ is a subset $A'$ of $A$ such that there exists a matching $τ'$ that differs from $τ$ only on elements of $A'$, and every element of $A'$ improves in $τ'$, compared to $τ$ according to its preference list. If there exists no blocking coalition, we call the matching $τ$ an exchange stable matching (ESM). An element $b\in B$ is reachable if there exists an exchange st…
University of Notre Dame（圣母大学）；Utrecht University（乌得勒支大学）；TU Eindhoven（埃因霍温理工大学）；KTH Royal Institute of Technology（皇家理工学院）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）
We prove a complexity dichotomy theorem for a class of Holant problems on planar 3-regular bipartite graphs. The complexity dichotomy states that for every weighted constraint function $f$ defining the problem (the weights can even be negative), the problem is either computable in polynomial time if $f$ satisfies a tractability criterion, or \#P-hard otherwise. One particular problem in this problem space is a long-standing open problem of Moore and Robson on counting Cubic Planar X3C. The dichotomy resolves this problem by showing that it is \numP-hard. Our proof relies on the machinery of si…
We prove a complexity dichotomy for a class of counting problems expressible as bipartite 3-regular Holant problems. For every problem of the form $\operatorname{Holant}\left(f\mid =_3 \right)$, where $f$ is any integer-valued ternary symmetric constraint function on Boolean variables, we prove that it is either P-time computable or #P-hard, depending on an explicit criterion of $f$. The constraint function can take both positive and negative values, allowing for cancellations. The dichotomy extends easily to rational valued functions of the same type. In addition, we discover a new phenomenon…
For a class $\mathcal{H}$ of graphs, #Sub$(\mathcal{H})$ is the counting problem that, given a graph $H\in \mathcal{H}$ and an arbitrary graph $G$, asks for the number of subgraphs of $G$ isomorphic to $H$. It is known that if $\mathcal{H}$ has bounded vertex-cover number (equivalently, the size of the maximum matching in $\mathcal{H}$ is bounded), then #Sub$(\mathcal{H})$ is polynomial-time solvable. We complement this result with a corresponding lower bound: if $\mathcal{H}$ is any recursively enumerable class of graphs with unbounded vertex-cover number, then #Sub$(\mathcal{H})$ is #W[1]-ha…
Valiant introduced some 25 years ago an algebraic model of computation along with the complexity classes VP and VNP, which can be viewed as analogues of the classical classes P and NP. They are defined using non-uniform sequences of arithmetic circuits and provides a framework to study the complexity for sequences of polynomials. Prominent examples of difficult (that is, VNP-complete) problems in this model includes the permanent and hamiltonian polynomials. While the permanent and hamiltonian polynomials in general are difficult to evaluate, there have been research on which special cases of …
