We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time $2^{O(\sqrt{k})}(n+m)$. Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time $2^{o(\sqrt{k})}(n+m)^{O(1)}$ [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the $2^{O(\sqrt{k})}(n+m)^{O(1)}$-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a $2k$-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time $2^{O(\sqrt{k}\log k)}(n+m)$. This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width $O(\sqrt{k})$.
We study exact algorithms for Euclidean TSP in $\mathbb{R}^d$. In the early 1990s algorithms with $n^{O(\sqrt{n})}$ running time were presented for the planar case, and some years later an algorithm with $n^{O(n^{1-1/d})}$ running time was presented for any $d\geq 2$. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on Euclidean TSP, except for a lower bound stating that the problem admits no $2^{O(n^{1-1/d-ε})}$ algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of Euclidean TSP by giving…
In this paper, we study problems of connecting classes of points via noncrossing structures. Given a set of colored terminal points, we want to find a graph for each color that connects all terminals of its color with the restriction that no two graphs cross each other. We consider these problems both on the Euclidean plane and in planar graphs. On the algorithmic side, we give a Gap-ETH-tight EPTAS for the two-colored traveling salesman problem as well as for the red-blue-green separation problem (in which we want to separate terminals of three colors with two noncrossing polygons of minimum …
De Berg et al. in [SICOMP 2020] gave an algorithmic framework for subexponential algorithms on geometric graphs with tight (up to ETH) running times. This framework is based on dynamic programming on graphs of weighted treewidth resulting in algorithms that use super-polynomial space. We introduce the notion of weighted treedepth and use it to refine the framework of de Berg et al. for obtaining polynomial space (with tight running times) on geometric graphs. As a result, we prove that for any fixed dimension $d \ge 2$ on intersection graphs of similarly-sized fat objects many well-known graph…
We give an algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to intersection graphs of similarly-sized fat objects, yielding algorithms with running time $2^{O(n^{1-1/d})}$ for any fixed dimension $d \geq 2$ for many well known graph problems, including Independent Set, $r$-Dominating Set for constant $r$, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersectio…
We show that there is no $2^{o(k^2)} n^{O(1)}$ time algorithm for Independent Set on $n$-vertex graphs with rank-width $k$, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the $2^{O(k^2)} n^{O(1)}$ time algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010] and it answers the open question of Bergougnoux and Kanté [SIAM J. Discret. Math., 2021]. We also show that the known $2^{O(k^2)} n^{O(1)}$ time algorithms for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set parameterized by rank-width $k$ are optimal assuming ETH…
The Planar Steiner Tree problem is one of the most fundamental NP-complete problems as it models many network design problems. Recall that an instance of this problem consists of a graph with edge weights, and a subset of vertices (often called terminals); the goal is to find a subtree of the graph of minimum total weight that connects all terminals. A seminal paper by Erickson et al. [Math. Oper. Res., 1987] considers instances where the underlying graph is planar and all terminals can be covered by the boundary of $k$ faces. Erickson et al. show that the problem can be solved by an algorithm…
We prove that the Hadwiger number of an $n$-vertex graph $G$ (the maximum size of a clique minor in $G$) cannot be computed in time $n^{o(n)}$, unless the Exponential Time Hypothesis (ETH) fails. This resolves a well-known open question in the area of exact exponential algorithms. The technique developed for resolving the Hadwiger number problem has a wider applicability. We use it to rule out the existence of $n^{o(n)}$-time algorithms (up to ETH) for a large class of computational problems concerning edge contractions in graphs.
We prove that unless Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. Combined with the reduction of Cygan, Pachocki, and Socała, our result rules out (subject to ETH) a possibility of $|V(G)|^{o(|V(G)|)}$-time algorithm deciding if graph $H$ is a subgraph of $G$. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental prob…
A binary matrix has the consecutive ones property (C1P) if it is possible to order the columns so that all 1s are consecutive in every row. In [McConnell, SODA 2004 768-777] the notion of incompatibility graph of a binary matrix was introduced and it was shown that odd cycles of this graph provide a certificate that a matrix does not have the consecutive ones property. A bound of (k+2) was claimed for the smallest odd cycle of a non-C1P matrix with k columns. In this note we show that this result can be obtained simply and directly via Tucker patterns, and that the correct bound is (k+2) when …
We prove that unless the Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. We also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of $|V(H)|^{o(|V(H)|)}$-time algorithm deciding if graph $G$ is a subgraph of $H$. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known comple…
Fine-grained reductions have established equivalences between many core problems with $\tilde{O}(n^3)$-time algorithms on $n$-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also have $\tilde{O}(mn)$-time algorithms on $m$-edge $n$-node weighted graphs, and such algorithms have wider applicability. Are these $mn$ bounds optimal when $m \ll n^2$? Starting from the hypothesis that the minimum weight $(2\ell+1)$-Clique problem in edge weighted graphs requires $n^{2\ell+1-o(1)}$ time, we prov…
The fastest algorithms for edge coloring run in time $2^m n^{O(1)}$, where $m$ and $n$ are the number of edges and vertices of the input graph, respectively. For dense graphs, this bound becomes $2^{Θ(n^2)}$. This is a somewhat unique situation, since most of the studied graph problems admit algorithms running in time $2^{O(n\log n)}$. It is a notorious open problem to either show an algorithm for edge coloring running in time $2^{o(n^2)}$ or to refute it, assuming Exponential Time Hypothesis (ETH) or other well established assumption. We notice that the same question can be asked for list edg…
In the multicoloring problem, also known as ($a$:$b$)-coloring or $b$-fold coloring, we are given a graph G and a set of $a$ colors, and the task is to assign a subset of $b$ colors to each vertex of G so that adjacent vertices receive disjoint color subsets. This natural generalization of the classic coloring problem (the $b=1$ case) is equivalent to finding a homomorphism to the Kneser graph $KG_{a,b}$, and gives relaxations approaching the fractional chromatic number. We study the complexity of determining whether a graph has an ($a$:$b$)-coloring. Our main result is that this problem does …
We study the Maximum Weight Matching (MWM) problem for general graphs through the max-product Belief Propagation (BP) and related Linear Programming (LP). The BP approach provides distributed heuristics for finding the Maximum A Posteriori (MAP) assignment in a joint probability distribution represented by a Graphical Model (GM) and respective LPs can be considered as continuous relaxations of the discrete MAP problem. It was recently shown that a BP algorithm converges to the correct MWM assignment under a simple GM formulation of MAP/MWM as long as the corresponding LP relaxation is tight. F…
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…
In $(k,r)$-Center we are given a (possibly edge-weighted) graph and are asked to select at most $k$ vertices (centers), so that all other vertices are at distance at most $r$ from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically: - For any $r\ge 1$, we show an algorithm that solves the problem in $O^*((3r+1)^{\textrm{cw}})$ time, where $\textrm{cw}$ is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm's performance. As a coroll…
We sharpen run-time analysis for algorithms under the partial rejection sampling framework. Our method yields improved bounds for: the cluster-popping algorithm for approximating all-terminal network reliability; the cycle-popping algorithm for sampling rooted spanning trees; the sink-popping algorithm for sampling sink-free orientations. In all three applications, our bounds are not only tight in order, but also optimal in constants.
An upper dominating set is a minimal dominating set in a graph. In the \textsc{Upper Dominating Set} problem, the goal is to find an upper dominating set of maximum size. We study the complexity of parameterized algorithms for \textsc{Upper Dominating Set}, as well as its sub-exponential approximation. First, we prove that, under ETH, \textsc{$k$-Upper Dominating Set} cannot be solved in time $O(n^{o(k)})$ (improving on $O(n^{o(\sqrt{k})})$), and in the same time we show under the same complexity assumption that for any constant ratio $r$ and any $\varepsilon > 0$, there is no $r$-approximatio…
In this work we start the investigation of tight complexity bounds for connectivity problems parameterized by cutwidth assuming the Strong Exponential-Time Hypothesis (SETH). Van Geffen et al. posed this question for odd cycle transversal and feedback vertex set. We answer it for these two and four further problems, namely connected vertex cover, connected domintaing set, steiner tree, and connected odd cycle transversal. For the latter two problems it sufficed to prove lower bounds that match the running time inherited from parameterization by treewidth; for the others we provide faster algor…
The Cut & Count technique and the rank-based approach have lead to single-exponential FPT algorithms parameterized by treewidth, that is, running in time $2^{O(tw)}n^{O(1)}$, for Feedback Vertex Set and connected versions of the classical graph problems (such as Vertex Cover and Dominating Set). We show that Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Restricted Edge-Subset Feedback Edge Set, Node Multiway Cut, and Multiway Cut are unlikely to have such running times. More precisely, we match algorithms running in time $2^{O(tw \log tw)}n^{O(1)}$ with tight lower bounds under the…
(see paper for full abstract) Given a vertex-weighted directed graph $G=(V,E)$ and a set $T=\{t_1, t_2, \ldots t_k\}$ of $k$ terminals, the objective of the SCSS problem is to find a vertex set $H\subseteq V$ of minimum weight such that $G[H]$ contains a $t_{i}\rightarrow t_j$ path for each $i\neq j$. The problem is NP-hard, but Feldman and Ruhl [FOCS '99; SICOMP '06] gave a novel $n^{O(k)}$ algorithm for the SCSS problem, where $n$ is the number of vertices in the graph and $k$ is the number of terminals. We explore how much easier the problem becomes on planar directed graphs: - Our main alg…
