We present two new deterministic algorithms for the Feedback Vertex Set problem parameterized by the solution size. We begin with a simple algorithm, which runs in O*((2 + ϕ)^k) time, where ϕ< 1.619 is the golden ratio. It already surpasses the previously fastest O*((1+2sqrt(2))^k)-time deterministic algorithm due to Cao et al. [SWAT 2010]. In our developments we follow the approach of Cao et al., however, thanks to a new reduction rule, we obtain not only better dependency on the parameter in the running time, but also a solution with simple analysis and only a single branching rule. Then, we present a modification of the algorithm which, using a more involved set of branching rules, achieves O*(3.592^k) running time.
In the Cluster Vertex Deletion problem the input is a graph $G$ and an integer $k$. The goal is to decide whether there is a set of vertices $S$ of size at most $k$ such that the deletion of the vertices of $S$ from $G$ results a graph in which every connected component is a clique. We give an algorithm for Cluster Vertex Deletion whose running time is $O^*(1.811^k)$.
We present the first deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph $G = (V,E)$, with $|V| = n$ and $|E| =m$, in $o(\sqrt{m}\,)$ time per update. In particular, for minimum vertex cover we provide deterministic data structures for maintaining a $(2+\eps)$ approximation in $O(\log n/\eps^2)$ amortized time per update. For maximum matching, we show how to maintain a $(3+\eps)$ approximation in $O(\min(\sqrt{n}/ε, m^{1/3}/\eps^2))$ {\em amortized} time per update, and a $(4+\eps)$ approximation in $O(m^{1/3}/\eps^2)$ {\…
A directed graph $G$ is called a pumpkin if $G$ is a union of induced paths with a common start vertex $s$ and a common end vertex $t$, and the internal vertices of every two paths are disjoint. We give an algorithm that given a directed graph $G$ and an integer $k$, decides whether a pumpkin can be obtained from $G$ by deleting at most $k$ vertices. The algorithm runs in $O^*(2^k)$ time.
In the k-Path problem, the input is a directed graph $G$ and an integer $k\geq 1$, and the goal is to decide whether there is a simple directed path in $G$ with exactly $k$ vertices. We give a deterministic algorithm for k-Path with time complexity $O^*(2.554^k)$. This improves the previously best deterministic algorithm for this problem of Zehavi [ESA 2015] whose time complexity is $O^*(2.597^k)$. The technique used by our algorithm can also be used to obtain faster deterministic algorithms for k-Tree, r-Dimensional k-Matching, Graph Motif, and Partial Cover.
In the vertex connectivity problem, given an undirected $n$-vertex $m$-edge graph $G$, we need to compute the minimum number of vertices that can disconnect $G$ after removing them. This problem is one of the most well-studied graph problems. From 2019, a new line of work [Nanongkai et al.~STOC'19;SODA'20;STOC'21] has used randomized techniques to break the quadratic-time barrier and, very recently, culminated in an almost-linear time algorithm via the recently announced maxflow algorithm by Chen et al. In contrast, all known deterministic algorithms are much slower. The fastest algorithm [Gab…
In the vertex connectivity survivable network design problem we are given an undirected graph G = (V,E) and connectivity requirement r(u,v) for each pair of vertices u,v. We are also given a cost function on the set of edges. Our goal is to find the minimum cost subset of edges such that for every pair (u,v) of vertices we have r(u,v) vertex disjoint paths in the graph induced by the chosen edges. Recently, Chuzhoy and Khanna presented a randomized algorithm that achieves a factor of O(k^3 log n) for this problem where k is the maximum connectivity requirement. In this paper we derandomize the…
We show that Odd Cycle Transversal and Vertex Multiway Cut admit deterministic polynomial kernels when restricted to planar graphs and parameterized by the solution size. This answers a question of Saurabh. On the way to these results, we provide an efficient sparsification routine in the flavor of the sparsification routine used for the Steiner Tree problem in planar graphs (FOCS 2014). It differs from the previous work because it preserves the existence of low-cost subgraphs that are not necessarily Steiner trees in the original plane graph, but structures that turn into (supergraphs of) Ste…
A cactus is a connected graph that does not contain $K_4 - e$ as a minor. Given a graph $G = (V, E)$ and integer $k \ge 0$, Cactus Vertex Deletion (also known as Diamond Hitting Set) is the problem of deciding whether $G$ has a vertex set of size at most $k$ whose removal leaves a forest of cacti. The current best deterministic parameterized algorithm for this problem was due to Bonnet et al. [WG 2016], which runs in time $26^kn^{O(1)}$, where $n$ is the number of vertices of $G$. In this paper, we design a deterministic algorithm for Cactus Vertex Deletion, which runs in time $17.64^kn^{O(1)}…
We present a deterministic dynamic connectivity data structure for undirected graphs with worst case update time $O\left(\sqrt{\frac{n(\log\log n)^2}{\log n}}\right)$ and constant query time. This improves on the previous best deterministic worst case algorithm of Frederickson (STOC 1983) and Eppstein Galil, Italiano, and Nissenzweig (J. ACM 1997), which had update time $O(\sqrt{n})$. All other algorithms for dynamic connectivity are either randomized (Monte Carlo) or have only amortized performance guarantees.
In this paper, we devise three deterministic algorithms for solving the $m$-set $k$-packing, $m$-dimensional $k$-matching, and $t$-dominating set problems in time $O^*(5.44^{mk})$, $O^*(5.44^{(m-1)k})$ and $O^*(5.44^{t})$, respectively. Although recently there has been remarkable progress on randomized solutions to those problems, our bounds make good improvements on the best known bounds for deterministic solutions to those problems.
We revisit the vertex-failure connectivity oracle problem. This is one of the most basic graph data structure problems under vertex updates, yet its complexity is still not well-understood. We essentially settle the complexity of this problem by showing a new data structure whose space, preprocessing time, update time, and query time are simultaneously optimal up to sub-polynomial factors assuming popular conjectures. Moreover, the data structure is deterministic. More precisely, for any integer $d_{\star}$, the data structure preprocesses a graph $G$ with $n$ vertices and $m$ edges in $\hat{O…
In the Connected Vertex Cover problem we are given an undirected graph G together with an integer k and we are to find a subset of vertices X of size at most k, such that X contains at least one end-point of each edge and moreover X induces a connected subgraph. For this problem we present a deterministic algorithm running in O(2^k n^O(1)) time and polynomial space, improving over previously best O(2.4882^k n^O(1)) deterministic algorithm and O(2^k n^O(1)) randomized algorithm. Furthermore, when usage of exponential space is allowed, we present an O(2^k k(n+m)) time algorithm that solves a mor…
We study dynamic planar graphs with $n$ vertices, subject to edge deletion, edge contraction, edge insertion across a face, and the splitting of a vertex in specified corners. We dynamically maintain a combinatorial embedding of such a planar graph, subject to connectivity and $2$-vertex-connectivity (biconnectivity) queries between pairs of vertices. Whenever a query pair is connected and not biconnected, we find the first and last cutvertex separating them. Additionally, we allow local changes to the embedding by flipping the embedding of a subgraph that is connected by at most two vertices …
The problem of finding an optimal vertex cover in a graph is a classic NP-complete problem, and is a special case of the hitting set question. On the other hand, the hitting set problem, when asked in the context of induced geometric objects, often turns out to be exactly the vertex cover problem on restricted classes of graphs. In this work we explore a particular instance of such a phenomenon. We consider the problem of hitting all axis-parallel slabs induced by a point set P, and show that it is equivalent to the problem of finding a vertex cover on a graph whose edge set is the union of tw…
An out-tree $T$ is an oriented tree with only one vertex of in-degree zero. A vertex $x$ of $T$ is internal if its out-degree is positive. We design randomized and deterministic algorithms for deciding whether an input digraph contains a given out-tree with $k$ vertices. The algorithms are of runtime $O^*(5.704^k)$ and $O^*(5.704^{k(1+o(1))})$, respectively. We apply the deterministic algorithm to obtain a deterministic algorithm of runtime $O^*(c^k)$, where $c$ is a constant, for deciding whether an input digraph contains a spanning out-tree with at least $k$ internal vertices. This answers i…
The problem of $d$-Path Vertex Cover, $d$-PVC lies in determining a subset $F$ of vertices of a given graph $G=(V,E)$ such that $G \setminus F$ does not contain a path on $d$ vertices. The paths we aim to cover need not to be induced. It is known that the $d$-PVC problem is NP-complete for any $d \ge 2$. When parameterized by the size of the solution $k$, 5-PVC has direct trivial algorithm with $\mathcal{O}(5^kn^{\mathcal{O}(1)})$ running time and, since $d$-PVC is a special case of $d$-Hitting Set, an algorithm running in $\mathcal{O}(4.0755^kn^{\mathcal{O}(1)})$ time is known. In this paper …
Deterministic online matching under short augmenting paths and restricted vertex reassignments
Department of Mathematical Engineering, Universidad de Chile（智利大学数学工程系）；Center for Mathematical Modeling CNRS-IRL 2807, Universidad de Chile（智利大学数学建模中心）
We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011], Solomon [FOCS 2016] gave a randomized dynamic algorithm for this problem that has an approximation ratio of $2$ and an amortised update time of $O(1)$ with high probability. This algorithm requires the assump…
We consider the problem of maintaining an (approximately) minimum vertex cover in an $n$-node graph $G = (V, E)$ that is getting updated dynamically via a sequence of edge insertions/deletions. We show how to maintain a $(2+ε)$-approximate minimum vertex cover, "deterministically", in this setting in $O(1/ε^2)$ amortized update time. Prior to our work, the best known deterministic algorithm for maintaining a $(2+ε)$-approximate minimum vertex cover was due to Bhattacharya, Henzinger and Italiano [SODA 2015]. Their algorithm has an update time of $O(\log n/ε^2)$. Recently, Bhattacharya, Chakrab…
We present a deterministic distributed $2$-approximation algorithm for the Minimum Weight Vertex Cover problem in the CONGEST model whose round complexity is $O(\log n \log Δ/ \log^2 \log Δ)$. This improves over the currently best known deterministic 2-approximation implied by [KVY94]. Our solution generalizes the $(2+ε)$-approximation algorithm of [BCS17], improving the dependency on $ε^{-1}$ from linear to logarithmic. In addition, for every $ε=(\log Δ)^{-c}$, where $c\geq 1$ is a constant, our algorithm computes a $(2+ε)$-approximation in $O(\log Δ/ \log \log Δ)$~rounds (which is asymptotic…
Faster Directed Hopsets, Distance Preservers, and Deterministic Shortcut Sets
CWI, Amsterdam, The Netherlands（荷兰阿姆斯特丹CWI）；Vrije Universiteit Amsterdam, The Netherlands（荷兰阿姆斯特丹自由大学）；TTIC, Chicago, United States（美国芝加哥陶尔森理论计算机科学研究所）
A graph $G$ is called a \emph{block graph} if each maximal $2$-connected component of $G$ is a clique. In this paper we study the Block Graph Vertex Deletion from the perspective of fixed parameter tractable (FPT) and kernelization algorithm. In particular, an input to Block Graph Vertex Deletion consists of a graph $G$ and a positive integer $k$ and the objective to check whether there exists a subset $S \subseteq V(G)$ of size at most $k$ such that the graph induced on $V(G)\setminus S$ is a block graph. In this paper we give an FPT algorithm with running time $4^kn^{O(1)}$ and a polynomial …
The best known unconditional deterministic complexity bound for computing the prime factorization of an integer N is O(M_int(N^(1/4) log N)), where M_int(k) denotes the cost of multiplying k-bit integers. This result is due to Bostan--Gaudry--Schost, following the Pollard--Strassen approach. We show that this bound can be improved by a factor of (log log N)^(1/2).
A tournament is a directed graph T such that every pair of vertices are connected by an arc. A feedback vertex set is a set S of vertices in T such that T - S is acyclic. In this article we consider the Feedback Vertex Set problem in tournaments. Here input is a tournament T and integer k, and the task is to determine whether T has a feedback vertex set of size at most k. We give a new algorithm for Feedback Vertex Set in Tournaments. The running time of our algorithm is upper bounded by O(1.618^k + n^{O(1)}) and by O(1.46^n). Thus our algorithm simultaneously improves over the fastest known p…
We give an $O(k^3 n \log n \min(k,\log^2 n) \log^2(nC))$-time algorithm for computing maximum integer flows in planar graphs with integer arc {\em and vertex} capacities bounded by $C$, and $k$ sources and sinks. This improves by a factor of $\max(k^2,k\log^2 n)$ over the fastest algorithm previously known for this problem [Wang, SODA 2019]. The speedup is obtained by two independent ideas. First we replace an iterative procedure of Wang that uses $O(k)$ invocations of an $O(k^3 n \log^3 n)$-time algorithm for maximum flow algorithm in a planar graph with $k$ apices [Borradaile et al., FOCS 20…
We present a simple deterministic distributed algorithm that computes a $(Δ+1)$-vertex coloring in $O(\log^2 Δ\cdot \log n)$ rounds. The algorithm can be implemented with $O(\log n)$-bit messages. The algorithm can also be extended to the more general $(degree+1)$-list coloring problem. Obtaining a polylogarithmic-time deterministic algorithm for $(Δ+1)$-vertex coloring had remained a central open question in the area of distributed graph algorithms since the 1980s, until a recent network decomposition algorithm of Rozhoň and Ghaffari [STOC'20]. The current state of the art is based on an impr…
We consider the fundamental problems of determining the rooted and global edge and vertex connectivities (and computing the corresponding cuts) in directed graphs. For rooted (and hence also global) edge connectivity with small integer capacities we give a new randomized Monte Carlo algorithm that runs in time $\tilde{O}(n^2)$. For rooted edge connectivity this is the first algorithm to improve on the $Ω(n^3)$ time bound in the dense-graph high-connectivity regime. Our result relies on a simple combination of sampling coupled with sparsification that appears new, and could lead to further trad…
We study deterministic algorithms for computing graph cuts, with focus on two fundamental problems: balanced sparse cut and $k$-vertex connectivity for small $k$ ($k=O(\polylog n)$). Both problems can be solved in near-linear time with randomized algorithms, but their previous deterministic counterparts take at least quadratic time. In this paper, we break this bound for both problems. Interestingly, achieving this for one problem crucially relies on doing so for the other. In particular, via a divide-and-conquer argument, a variant of the cut-matching game by [Khandekar et al.`07], and the lo…
We consider the Modular Subset Sum problem: given a multiset $X$ of integers from $\mathbb{Z}_m$ and a target integer $t$, decide if there exists a subset of $X$ with a sum equal to $t \pmod{m}$. Recent independent works by Cardinal and Iacono (SOSA'21), and Axiotis et al. (SOSA'21) provided simple and near-linear algorithms for this problem. Cardinal and Iacono gave a randomized algorithm that runs in $O(m \log m)$ time, while Axiotis et al. gave a deterministic algorithm that runs in $O(m \text{ polylog } m)$ time. Both results work by reduction to a text problem, which is solved using a dyn…
Owing to their versatility, graph structures admit representations of intricate relationships between the separate entities comprising the data. We formalise the notion of connection between two vertex sets in terms of edge and vertex features by introducing graph-walking programs. We give two algorithms for mining of deterministic graph-walking programs that yield programs in the order of increasing length. These programs characterise linear long-distance relationships between the given two vertex sets in the context of the whole graph.
We give new deterministic bounds for fully-dynamic graph connectivity. Our data structure supports updates (edge insertions/deletions) in $O(\log^2n/\log\log n)$ amortized time and connectivity queries in $O(\log n/\log\log n)$ worst-case time, where $n$ is the number of vertices of the graph. This improves the deterministic data structures of Holm, de Lichtenberg, and Thorup (STOC 1998, J.ACM 2001) and Thorup (STOC 2000) which both have $O(\log^2n)$ amortized update time and $O(\log n/\log\log n)$ worst-case query time. Our model of computation is the same as that of Thorup, i.e., a pointer m…
In this paper we consider the Target Set Selection problem. The problem naturally arises in many fields like economy, sociology, medicine. In the Target Set Selection problem one is given a graph $G$ with a function $\operatorname{thr}: V(G) \to \mathbb{N} \cup \{0\}$ and integers $k, \ell$. The goal of the problem is to activate at most $k$ vertices initially so that at the end of the activation process there is at least $\ell$ activated vertices. The activation process occurs in the following way: (i) once activated, a vertex stays activated forever; (ii) vertex $v$ becomes activated if at l…
Consider the following "local" cut-detection problem in a directed graph: We are given a starting vertex $s$ and need to detect whether there is a cut with at most $k$ edges crossing the cut such that the side of the cut containing $s$ has at most $Δ$ interior edges. If we are given query access to the input graph, then this problem can in principle be solved in sublinear time without reading the whole graph and with query complexity depending on $k$ and $Δ$. We design an elegant randomized procedure that solves a slack variant of this problem with $O(k^2 Δ)$ queries, improving in particular a…
We consider coloring problems in the distributed message-passing setting. The previously-known deterministic algorithms for edge-coloring employed at least (2Delta - 1) colors, even though any graph admits an edge-coloring with Delta + 1 colors [V64]. Moreover, the previously-known deterministic algorithms that employed at most O(Delta) colors required superlogarithmic time [B15,BE10,BE11,FHK15]. In the current paper we devise deterministic edge-coloring algorithms that employ only Delta + o(Delta) colors, for a very wide family of graphs. Specifically, as long as the arboricity is a = O(Delta…
In this work, we introduce bidirectional collision detection --- a new algorithmic tool that applies to the collision problems that arise in many isomorphism problems. For the group isomorphism problem, we show that bidirectional collision detection yields a deterministic n^((1 / 2) log n + O(1)) time algorithm whereas previously the n^(log n + O(1)) generator-enumeration algorithm was the best result for several decades. For the hard special case of solvable groups, we combine bidirectional collision detection with methods from the author's previous work to obtain a deterministic square-root …
Listing triangles is a fundamental graph problem with many applications, and large graphs require fast algorithms. Vertex ordering allows the orientation of edges from lower to higher vertex indices, and state-of-the-art triangle listing algorithms use this to accelerate their execution and to bound their time complexity. Yet, only basic orderings have been tested. In this paper, we show that studying the precise cost of algorithms instead of their bounded complexity leads to faster solutions. We introduce cost functions that link ordering properties with the running time of a given algorithm.…
Real-world networks are prone to breakdowns. Typically in the underlying graph $G$, besides the insertion or deletion of edges, the set of active vertices changes overtime. A vertex might work actively, or it might fail, and gets isolated temporarily. The active vertices are grouped as a set $S$. $S$ is subjected to updates, i.e., a failed vertex restarts, or an active vertex fails, and gets deleted from $S$. Dynamic subgraph connectivity answers the queries on connectivity between any two active vertices in the subgraph of $G$ induced by $S$. The problem is solved by a dynamic data structure,…
Li and Panigrahi, in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time $o(mn)$. They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of $s$-$t$ minimum cut computations. In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connect…
Recent empirical evaluations of exact algorithms for Feedback Vertex Set have demonstrated the efficiency of a highest-degree branching algorithm with a degree-based pruning heuristic. In this paper, we prove that this empirically fast algorithm runs in $O(3.460^k n)$ time, where $k$ is the solution size. This improves the previous best $O(3.619^k n)$-time deterministic algorithm obtained by Kociumaka and Pilipczuk.
Given a set $W = \{w_1,\ldots, w_n\}$ of non-negative integer weights and an integer $C$, the #Knapsack problem asks to count the number of distinct subsets of $W$ whose total weight is at most $C$. In the more general integer version of the problem, the subsets are multisets. That is, we are also given a set $ \{u_1,\ldots, u_n\}$ and we are allowed to take up to $u_i$ items of weight $w_i$. We present a deterministic FPTAS for #Knapsack running in $O(n^{2.5}\varepsilon^{-1.5}\log(n \varepsilon^{-1})\log (n \varepsilon))$ time. The previous best deterministic algorithm [FOCS 2011] runs in $O(…
We present trade-offs in the incremental and fully dynamic settings to maintian a proper coloring. For any fully dynamic $2$-coloring algorithm, the maximum of the update time, number of recolorings, and query time is $Ω(\log n)$. We present a deterministic fully dynamic $2$-coloring algorithm with $O(\log^2 n)$ amortized update time, $O(\log n)$ amortized query time, and one recoloring in the worst case. For any incremental $2$-coloring algorithm which explicitly maintains the color of every vertex after each update, the amortized update time and the amortized number of recolorings is $Ω(\log…
