arXiv subjects

Radu Curticapean

Publications and source records attributed to Radu Curticapean.

28 records · Page 2Linked to original sources

arXiv2017

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 $Θ(\sqrt 2^k)$ 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 $Ω(1.97^k)$ for any prime $p\neq 2$ and even $Ω(2.15^k)$ for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time $O^*((6-ε)^{\mathsf{pw}})$ for any $ε>0$ 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.

cs.DS↗

arXiv2017

Block Interpolation: A Framework for Tight Exponential-Time Counting Complexity

We devise a framework for proving tight lower bounds under the counting exponential-time hypothesis #ETH introduced by Dell et al. (ACM Transactions on Algorithms, 2014). Our framework allows us to convert classical #P-hardness results for counting problems into tight lower bounds under #ETH, thus ruling out algorithms with running time $2^{o(n)}$ on graphs with $n$ vertices and $O(n)$ edges. As exemplary applications of this framework, we obtain tight lower bounds under #ETH for the evaluation of the zero-one permanent, the matching polynomial, and the Tutte polynomial on all non-easy points except for one line. This remaining line was settled very recently by Brand et al. (IPEC 2016).

cs.CC↗

arXiv2017

Homomorphisms Are a Good Basis for Counting Small Subgraphs

We introduce graph motif parameters, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lovász show that many interesting quantities have this form, including, for fixed graphs $H$, the number of $H$-copies (induced or not) in an input graph $G$, and the number of homomorphisms from $H$ to $G$. Using the framework of graph motif parameters, we obtain faster algorithms for counting subgraph copies of fixed graphs $H$ in host graphs $G$: For graphs $H$ on $k$ edges, we show how to count subgraph copies of $H$ in time $k^{O(k)}\cdot n^{0.174k + o(k)}$ by a surprisingly simple algorithm. This improves upon previously known running times, such as $O(n^{0.91k + c})$ time for $k$-edge matchings or $O(n^{0.46k + c})$ time for $k$-cycles. Furthermore, we prove a general complexity dichotomy for evaluating graph motif parameters: Given a class $\mathcal C$ of such parameters, we consider the problem of evaluating $f\in \mathcal C$ on input graphs $G$, parameterized by the number of induced subgraphs that $f$ depends upon. For every recursively enumerable class $\mathcal C$, we prove the above problem to be either FPT or #W[1]-hard, with an explicit dichotomy criterion. This allows us to recover known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms in a uniform and simplified way, together with improved lower bounds. Finally, we extend graph motif parameters to colored subgraphs and prove a complexity trichotomy: For vertex-colored graphs $H$ and $G$, where $H$ is from a fixed class $\mathcal H$, we want to count color-preserving $H$-copies in $G$. We show that this problem is either polynomial-time solvable or FPT or #W[1]-hard, and that the FPT cases indeed need FPT time under reasonable assumptions.

cs.DS↗

arXiv2017

Finding Detours is Fixed-parameter Tractable

We consider the following natural "above guarantee" parameterization of the classical Longest Path problem: For given vertices s and t of a graph G, and an integer k, the problem Longest Detour asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k. Furthermore, we study the related problem Exact Detour that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k, and a deterministic algorithm with running time about 6.745^k, showing that this problem is FPT as well. Our algorithms for Exact Detour apply to both undirected and directed graphs.

cs.DS↗

arXiv2016

Counting matchings with k unmatched vertices in planar 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 statistical physics, and it is a special case of counting perfect matchings in k-apex graphs (graphs that can be turned planar by removing at most k vertices). Starting from a recent #W[1]-hardness proof for counting perfect matchings on k-apex graphs, we obtain that counting matchings with k unmatched vertices in planar graphs is #W[1]-hard. In contrast, given a plane graph G with s distinguished faces, there is an $O(2^s \cdot n^3)$ time algorithm for counting those matchings with k unmatched vertices such that all unmatched vertices lie on the distinguished faces. This implies an $f(k,s)\cdot n^{O(1)}$ time algorithm for counting perfect matchings in k-apex graphs whose apex neighborhood is covered by s faces.

cs.CC↗

arXiv2015

Parity Separation: A Scientifically Proven Method for Permanent Weight Loss

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 from classical #P-hardness proofs. We derive the following applications: 1. An alternative #P-completeness proof for counting unweighted perfect matchings. 2. C=P-completeness for deciding whether two given unweighted graphs have the same number of perfect matchings. To the best of our knowledge, this is the first C=P-completeness result for the "equality-testing version" of any natural counting problem that is not already #P-hard under parsimonious reductions. 3. An alternative tight lower bound for counting unweighted perfect matchings under the counting exponential-time hypothesis #ETH. Our technique is based upon matchgates and the Holant framework. To make our #P-hardness proof self-contained, we also apply matchgates for an alternative #P-hardness proof of PerfMatch on graphs with edge-weights -1 and 1.

cs.CC↗

arXiv2015

Parameterizing the Permanent: Genus, Apices, Minors, Evaluation mod 2^k

We identify and study relevant structural parameters for the problem PerfMatch of counting perfect matchings in a given input graph $G$. These generalize the well-known tractable planar case, and they include the genus of $G$, its apex number (the minimum number of vertices whose removal renders $G$ planar), and its Hadwiger number (the size of a largest clique minor). To study these parameters, we first introduce the notion of combined matchgates, a general technique that bridges parameterized counting problems and the theory of so-called Holants and matchgates: Using combined matchgates, we can simulate certain non-existing gadgets $F$ as linear combinations of $t=O(1)$ existing gadgets. If a graph $G$ features $k$ occurrences of $F$, we can then reduce $G$ to $t^k$ graphs that feature only existing gadgets, thus enabling parameterized reductions. As applications of this technique, we simplify known $4^g n^{O(1)}$ time algorithms for PerfMatch on graphs of genus $g$. Orthogonally to this, we show #W[1]-hardness of the permanent on $k$-apex graphs, implying its #W[1]-hardness under the Hadwiger number. Additionally, we rule out $n^{o(k/\log k)}$ time algorithms under the counting exponential-time hypothesis #ETH. Finally, we use combined matchgates to prove parity-W[1]-hardness of evaluating the permanent modulo $2^k$, complementing an $O(n^{4k-3})$ time algorithm by Valiant and answering an open question of Björklund. We also obtain a lower bound of $n^{Ω(k/\log k)}$ under the parity version of the exponential-time hypothesis.

cs.CC↗

arXiv2014

Complexity of counting subgraphs: only the boundedness of the vertex-cover number counts

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]-hard parameterized by the size of $H$ and hence not polynomial-time solvable and not even fixed-parameter tractable, unless FPT = #W[1]. As a first step of the proof, we show that counting $k$-matchings in bipartite graphs is #W[1]-hard. Recently, Curticapean [ICALP 2013] proved the #W[1]-hardness of counting $k$-matchings in general graphs; our result strengthens this statement to bipartite graphs with a considerably simpler proof and even shows that, assuming the Exponential Time Hypothesis (ETH), there is no $f(k)n^{o(k/\log k)}$ time algorithm for counting $k$-matchings in bipartite graphs for any computable function $f(k)$. As a consequence, we obtain an independent and somewhat simpler proof of the classical result of Flum and Grohe [SICOMP 2004] stating that counting paths of length $k$ is #W[1]-hard, as well as a similar almost-tight ETH-based lower bound on the exponent.

cs.CC↗

arXiv2014

Counting perfect matchings in graphs that exclude a single-crossing minor

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 independent recent result (Straub et al. 2014) for graphs excluding $K_5$, it is one of the first nontrivial algorithms to not inherently rely on Pfaffian orientations.

cs.DS↗

arXiv2013

Counting Triangulations and other Crossing-free Structures via Onion Layers

Let $P$ be a set of $n$ points in the plane. A crossing-free structure on $P$ is a plane graph with vertex set $P$. Examples of crossing-free structures include triangulations of $P$, spanning cycles of $P$, also known as polygonalizations of $P$, among others. In this paper we develop a general technique for computing the number of crossing-free structures of an input set $P$. We apply the technique to obtain algorithms for computing the number of triangulations, matchings, and spanning cycles of $P$. The running time of our algorithms is upper bounded by $n^{O(k)}$, where $k$ is the number of onion layers of $P$. In particular, for $k = O(1)$ our algorithms run in polynomial time. In addition, we show that our algorithm for counting triangulations is never slower than $O^{*}(3.1414^{n})$, even when $k = Θ(n)$. Given that there are several well-studied configurations of points with at least $Ω(3.464^{n})$ triangulations, and some even with $Ω(8^{n})$ triangulations, our algorithm asymptotically outperforms any enumeration algorithm for such instances. In fact, it is widely believed that any set of $n$ points must have at least $Ω(3.464^{n})$ triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the "Restricted-Triangulation-Counting-Problem", which we prove to be $W[2]$-hard in the parameter $k$. This implies a "no free lunch" result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.

cs.CG↗