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Parity Matching Parity Matching Martin Loebl1 1 KAM MFF UK, Charles University, Malostranske n. 25, 118 00 Praha 1 Czech republic [email protected] Abstract We study the Parity-Matching problem: Given a graph on n vertices with edge-weights from {0, 1, −1}, and pairs of edges p1, . , pg, find the maximum weight of a perfect matching M of G such that |M ∩ pi| is even for each i = 1, . , g. A straightforward algorithm for the Parity-Matching problem has complexity 2g × poly(n) and we wonder if any deterministic or randomized algorithm does better than that. Our study of the Parity-Matching problem is an attempt to pinpoint the curse of dimensionality of the Max-Cut problem for graphs embedded into orientable surfaces. We show that existence of an algorithm for the Parity-Matching problem beating the 2g bound implies that in the class of graphs where the crossing number is equal to the genus, the complexity of the Max-Cut problem is smaller than the additive determinantal complexity of cuts enumeration. At present, no natural class of graphs with this property is known. A well established way to approach matching problems is to determine, by means of the Isolating lemma [14], whether a coefficient of the generating function of the perfect matchings (with suitable substitutions) is non-zero. We argue that this approach will unlikely lead to an algorithm with complexity better than 2g for the Parity-Matching problem. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases complexity, matching, MaxCut, embedded graphs, enumeration, Pfaffian Digital Object Identifier 10.4230/LIPIcs... 1 Introduction Given a graph G = (V, E), a set of edges M ⊆ E is called perfect matching if the graph (V, M) has degree one at each vertex. Matching theory is perhaps the most developed part of graph theory and it is also closely linked to the study of algorithms and their complexity. In this paper we introduce and study a matching problem called the Parity-Matching problem: Given a graph G on n vertices with edge-weights from {0, 1, −1}, and pairs of edges p1, . , pg, find the maximum weight of a perfect matching M of G such that |M ∩ pi| is even for each i = 1, . , g. ` For a collection of disjoint pairs of edges {pi}i=1 we say that a perfect matching M of G is admissible if |M ∩ pi| is even for each i = 1, . , `. The Parity-Matching problem thus asks for maximum weight of an admissible perfect matching. As far as we know, the problem was not studied before; it somehow resembles the extensively studied Matroid Parity problem (see [12]). Leslie Valiant (personal communication) observed that NP-completness of the Parity-Matching problem follows in a simple way from the hardness of the Multiple choice matching problem, using techiques developed in [17]. NP-completeness is also a straightforward consequence of the results presented here. We provide evidence that the Parity-Matching problem pinpoints the time complexity as a function of the surface genus of the Max-Cut problem for graphs on surfaces. This © Martin Loebl; licensed under Creative Commons License CC-BY Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany XX:2 Parity Matching problem is studied extensively e.g. in statistical physics. Experience with the embedded Max-Cut problems leads us to formulate the Frustration Conjecture: No algorithm can solve the Parity-Matching problem with time complexity asymptotically better than 2g. A straightforward algorithm for the Parity-Matching problem consists in solving 2g maximum weight perfect matching problems and thus has complexity 2g × poly(n). The Frustration Conjecture thus proposes that no algorithm can significantly outperform the described straightforward algorithm. We show that existence of an algorithm beating the 2g bound of the Frustration Conjecture would mean that in the class of graphs where the crossing number is equal to the genus, the complexity of the Max-Cut problem is smaller than the additive determinantal complexity of the cuts enumeration. At present, no natural class of graphs with this property is known. Let G = (V, E) be a graph. A set of edges C ⊆ E is called an edge-cut of G, if there is a V 0 ⊆ V so that C = {e ∈ E : |e ∩ V 0| = 1}. The Max-Cut problem, one of the basic optimization problems, asks for the maximum size of an edge-cut in the input graph G, or, if weights on the edges are given, for the maximum total weight of an edge-cut. Let g ≥ 0. The g-Emb-Max-Cut problem asks for the maximum total weight of an edge-cut in the input graph G given along with its embedding in the 2-dimensional closed surface of genus g. 1.1 Optimization by enumeration The motivation of this paper is a curious phenomenon: There is a strongly polynomial algorithm to solve the 0-Emb-Max-Cut problem (planar graphs) based on a reduction to the weighted perfect matching problem. For g ≥ 1 the situation is different: There is a weakly polynomial algorithm by Galluccio and Loebl ([4]; see also [5], [6]) to solve the problem for g ≥ 1. The algorithm is called Algorithm 1 in this paper and it is sketched in the appendix. Algorithm 1 was implemented several times and applied in extensive statistical physics calculations (see [13] ). Recently other related algorithms based on the Valiant’s theory [16] of holomorphic algorithms appeared (see [1], [3]). All presently known approaches are in principle of enumeration nature, even for the toroidal square grids. The seminal observation was formulated by Kasteleyn [9] and proved by Galluccio, Loebl [4] and independently by Tesler [15]: The generating function of perfect matchings of a graph of genus g can be written as a linear combination of 4g Pfaffians. Cimasoni and Reshetikhin [2] provided a beautiful interpretation of the formula which then became known as the Arf invariant formula. The Arf invariant formula produces, together with the value of the g-Emb-Max-Cut also the number of edge-cuts of maximum total weight, and in fact the whole generating function of edge-cuts of the given graph. 1.2 Additive determinantal complexity As mentioned above, Algorithm 1 of Galluccio and Loebl consists in calculating 22g Pfaffians. A recent result of Loebl and Masbaum [11] indicates that this might be optimum for the cuts enumeration. It is shown by Loebl and Masbaum in [11] that, if we want to enumerate the edge-cuts of each possible size of an input graph G of genus g, then in a strongly restricted setting called additive determinantal complexity the number of the Pfaffian calculations cannot be smaller than 22g. M. Loebl XX:3 1.3 Main Contribution In this paper we show that for a graph G with n vertices and embedded to the plane with g crossings, finding a maximum size of an edge-cut in G is polynomial time reducible to determining a maximum weight perfect matching M in a graph G0 which is admisible for a set of 2g pairs of G0 edges. The Frustration Conjecture is our attempt to pinpoint a puzzle: Is there an algorithm for solving the Max-Cut problem in a natural class of embedded graphs, whose complexity beats the additive determinantal complexity of the cuts enumeration. At present no such algorithm for a natural class of graphs is known. A well-established way to design a randomized algorithm for a matching problem is to determine whether some particular coefficient of a multivariable generating function of the perfect matchings (with suitable substitutions) is non-zero. In section 2 we argue that such algorithms will unlikely contradict the Frustration Conjecture. 1.4 The Exponential Time Hypothesis The Exponential Time Hypothesis (ETH) is an unproven computational hardness assumption that was formulated by Impagliazzo and Paturi [7]. ETH states that 3-SAT cannot be solved in subexponential time in the worst case. This was strenghthened by Impagliazzo, Paturi and Zane [8] to the Strong Exponential Time Hypothesis (SETH): For all d < 1 there is a k such that k-SAT cannot be solved in O(2dn) time. The ETH and SETH have a very natural role: they are used to argue that known algorithms are probably optimal. A lower bound for the time complexity of the perfect matchings enumeration in the class of graphs of genus g, assuming an enumeration variant of ETH, was recently obtained by Curticapean and Xia in [3]. Acknowledgement. This project initially started as a joint work with Marcos Kiwi. I would like to thank to Marcos for many helpful discussions. 2 The Coefficient Problem A well-established way to approach matching problems is to determine whether a coefficient of the generating function of the perfect matchings (with suitable substitutions) is non-zero. In this section we analyse this approach for the Parity-Matching problem and argue that unlikely it disproves the Frustration Conjecture. Given a graph G = (V, E), let P(G) denote the set of all perfect matchings of G. We assume that a variable xe is associated with each edge e, we write x = (xe)e∈E(G), and define over Z[x] the generating polynomial for perfect matchings, denoted PG, as follows: X Y PG(x) = xe . M∈P(G) e∈M 2.1 Pfaffians Assume the vertices of G are numbered from 1 to n. If D is an orientation of G, we denote by A(G, D) the skew-symmetric adjacency matrix of D defined as follows: The diagonal entries of A(G, D) are zero, and the off-diagonal entries are X A(G, D)ij = ±xe , XX:4 Parity Matching where the sum is over all edges e connecting vertices i and j, and the sign in front of xe is 1 if e is oriented from i to j in the orientation D, and −1 otherwise.
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