Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs

We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the problems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the logspace complexity class \SPL. Since \SPL\ is contained in the logspace counting classes \oplus\L (in fact in \modk\ for all $k\geq 2$), \CeqL, and \PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs is in \FL^{\SPL}. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a logspace computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function.Comment: 23 pages, 13 figure

Immunity and Simplicity for Exact Counting and Other Counting Classes

Ko [RAIRO 24, 1990] and Bruschi [TCS 102, 1992] showed that in some relativized world, PSPACE (in fact, ParityP) contains a set that is immune to the polynomial hierarchy (PH). In this paper, we study and settle the question of (relativized) separations with immunity for PH and the counting classes PP, C_{=}P, and ParityP in all possible pairwise combinations. Our main result is that there is an oracle A relative to which C_{=}P contains a set that is immune to BPP^{ParityP}. In particular, this C_{=}P^A set is immune to PH^{A} and ParityP^{A}. Strengthening results of Tor\'{a}n [J.ACM 38, 1991] and Green [IPL 37, 1991], we also show that, in suitable relativizations, NP contains a C_{=}P-immune set, and ParityP contains a PP^{PH}-immune set. This implies the existence of a C_{=}P^{B}-simple set for some oracle B, which extends results of Balc\'{a}zar et al. [SIAM J.Comp. 14, 1985; RAIRO 22, 1988] and provides the first example of a simple set in a class not known to be contained in PH. Our proof technique requires a circuit lower bound for ``exact counting'' that is derived from Razborov's [Mat. Zametki 41, 1987] lower bound for majority.Comment: 20 page

Optimizing the double description method for normal surface enumeration

Many key algorithms in 3-manifold topology involve the enumeration of normal surfaces, which is based upon the double description method for finding the vertices of a convex polytope. Typically we are only interested in a small subset of these vertices, thus opening the way for substantial optimization. Here we give an account of the vertex enumeration problem as it applies to normal surfaces, and present new optimizations that yield strong improvements in both running time and memory consumption. The resulting algorithms are tested using the freely available software package Regina.Comment: 27 pages, 12 figures; v2: Removed the 3^n bound from Section 3.3, fixed the projective equation in Lemma 4.4, clarified "most triangulations" in the introduction to section 5; v3: replace -ise with -ize for Mathematics of Computation (note that this changes the title of the paper

A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

For even $k$, the matchings connectivity matrix $\mathbf{M}_k$ encodes which pairs of perfect matchings on $k$ vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_2$ is $\Theta(\sqrt 2^k)$ and used this to give an $O^*((2+\sqrt{2})^{\mathsf{pw}})$ time algorithm for counting Hamiltonian cycles modulo $2$ on graphs of pathwidth $\mathsf{pw}$. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within $\mathbf{M}_k$, which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar pattern propagation when only a black-box lower bound on the asymptotic rank of $\mathbf{M}_k$ is given; no stronger structural insights such as the existence of large permutation submatrices in $\mathbf{M}_k$ are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes $p$) parameterized by pathwidth. To apply this technique, we prove that the rank of $\mathbf{M}_k$ over the rationals is $4^k / \mathrm{poly}(k)$. We also show that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_p$ is $\Omega(1.97^k)$ for any prime $p\neq 2$ and even $\Omega(2.15^k)$ for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time $O^*((6-\epsilon)^{\mathsf{pw}})$ for any $\epsilon>0$ unless SETH fails. This bound is tight due to a $O^*(6^{\mathsf{pw}})$ time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes $p\neq 2$ in time $O^*(3.97^\mathsf{pw})$, indicating that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in SODA 201

Dynamic programming for graphs on surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2O(k·log k). Our approach combines tools from topological graph theory and analytic combinatorics.Postprint (updated version