We study the Maximum Independent Set of Rectangles (MISR) problem, where we are given a set of axis-parallel rectangles in the plane and the goal is to select a subset of non-overlapping rectangles of maximum cardinality. In a recent breakthrough, Mitchell (FOCS’21) obtained the first constant-factor approximation algorithm for MISR. His algorithm achieves an approximation ratio of 10, and it is based on a dynamic program that intuitively recursively partitions the input plane into special polygons called Corner-Clipped Rectangles (CCRs) , without intersecting certain special horizontal line segments called fences . In this article, we present a \((2+\varepsilon)\) -approximation algorithm for MISR, which is also based on a recursive partitioning scheme. First, we use a partition into a class of axis-parallel polygons with constant complexity each that are more general than CCRs. This allows us to provide an arguably simpler analysis and, at the same time, already improves the approximation ratio to 4. Then, using a more elaborate charging scheme and a recursive partitioning into general axis-parallel polygons with constant complexity, we improve our approximation ratio to \(2+\varepsilon\) . In particular, we construct a recursive partitioning based on more general fences which can be sequences of up to \(O(1/\varepsilon)\) line segments each. This partitioning routine and our other new ideas may be useful for future work toward a PTAS for MISR.
In the \(\textsf{LOCAL}\) model of distributed computing, low-diameter decomposition is a fundamental tool for algorithm design, as it enables a reduction from general graphs to low-diameter graphs where brute-force information gathering can be performed efficiently. Chang and Su [PODC 2022] showed that any high-conductance network excluding a fixed minor contains a high-degree vertex \(v^{\star}\) , allowing the entire graph topology to be gathered at \(v^{\star}\) efficiently in the \(\textsf{CONGEST}\) model via expander routing . Consequently, in such networks, many problems that admit efficient \(\textsf{LOCAL}\) algorithms via low-diameter decomposition can also be solved efficiently in \(\textsf{CONGEST}\) using expander decomposition . In this work, we present improved decomposition and routing algorithms for networks excluding a fixed minor. We define an \((\epsilon,D,T)\) -decomposition of a graph \(G=(V,E)\) as a partition of \( V \) into clusters of diameter at most \( D \) , with at most \(\epsilon|E|\) inter-cluster edges, such that information gathering within each cluster can be completed in \( T \) rounds in parallel. We show that an \((\epsilon,D,T)\) -decomposition with \(\begin{align*} D=O(\epsilon^{-1})\quad\text{and}\quad T=\min\left\{2^{O\left(\log^{2}\frac{1} {\epsilon}\right)}\cdot O(\log\Delta),\ \operatorname{poly}(\epsilon^{-1},\log \Delta)\right\} \nonumber\end{align*}\) can be computed deterministically in \(\begin{align*} O(\epsilon^{-1}\log^{\ast}n)+\min\left\{2^{O\left(\log^{2}\frac{1}{\epsilon} \right)}\cdot O(\log\Delta),\ \operatorname{poly}(\epsilon^{-1},\log\Delta)\right\}\nonumber \end{align*}\) rounds in the \(\textsf{CONGEST}\) model for networks excluding a fixed minor. Our algorithm has a wide range of applications, including the following results in \(\textsf{CONGEST}\) : — A \((1-\epsilon)\) -approximate maximum independent set in networks excluding a fixed minor can be computed deterministically in \(O(\epsilon^{-1}\log^{\ast}n)+\operatorname{poly}(\epsilon^{-1})\) rounds, nearly matching the \(\Omega(\epsilon^{-1}\log^{\ast}n)\) lower bound of Lenzen and Wattenhofer [DISC 2008]. — Property testing of any additive minor-closed property can be performed deterministically in \(O(\log n)\) rounds for constant \(\epsilon\) , or in \(O(\epsilon^{-1}\log n)+\operatorname{poly}(\epsilon^{-1})\) rounds for constant \(\Delta\) , nearly matching the \(\Omega(\epsilon^{-1}\log n)\) lower bound of Levi et al. [PODC 2018].
Cut arcs , or strong bridges , are one of the most fundamental reachability notions in directed graphs. Specifically, in a strongly connected graph \(G=(V,E)\) ( \(|V|=n\) , \(|E|=m\) ), a cut arc is an arc \(e\in E\) for which there exist \(u,v\in V\) , such that all \(u\) - \(v\) walks contain \(e\) . In this paper we generalise this notion to cut paths , that is, walks \(W\) for which there exist \(u,v\in V\) , such that all \(u\) - \(v\) walks contain \(W\) as subwalk. We first prove various properties of cut paths and define their remainder structure , which we use to present a simple \(O(m)\) -time verification algorithm for a cut path. We further show that a graph contains at most \(O(n)\) maximal cut paths of length at most \(O(n)\) each, and present an optimal \(O(n^{2})\) enumeration algorithm for maximal cut paths. We apply cut paths and their remainder structure to improve several reachability problems from bioinformatics, as follows. A walk is called safe if it is a subwalk of every node-covering closed walk of a strongly connected graph. Multi-safety is defined analogously, by considering node-covering sets of closed walks instead. Cut paths provide simple \(O(m)\) -time algorithms verifying if a walk is safe or multi-safe. Further, by simultaneous computation of remainder structures of all subwalks of a cut path in linear time, we can identify all maximal multi-safe walks in \(O(mn)\) time. This improves over the state-of-the-art algorithm running in time \(O(m^{2}+n^{3}\log n)\) .
A complete deterministic finite (semi)automaton (DFA) with a set of states Q is completely reachable if every nonempty subset of Q is the image of the action of some word applied to Q. The concept of completely reachable automata appeared, in particular, in connection with synchronizing automata; the class contains the Cerny automata and covers several distinguished subclasses. The notion was introduced by Bondar and Volkov (2016), who also raised the question about the complexity of deciding whether an automaton is completely reachable. We develop an algorithm solving this problem, which works in O(vertical bar Sigma vertical bar . 2(2)) time and O(vertical bar Sigma vertical bar . n) space, where n = vertical bar Q| is the number of states and vertical bar Sigma vertical bar is the size of the input alphabet. In the second part, we prove a weak Don's conjecture for this class of automata: a nonempty subset of states S subset of Q is reachable with a word of length at most 2n(n - vertical bar S vertical bar) - n . Hn - vertical bar S vertical bar, where H-i is the ith harmonic number. This implies a quadratic upper bound in = on the length of the shortest synchronizing words (reset threshold) for the class of completely reachable automata and generalizes earlier upper bounds derived for its subclasses.
Cut arcs, or strong bridges, are one of the most fundamental reachability notions in directed graphs. Specifically, in a strongly connected graph G = (V, E) (|V | = n, |E| = m), a cut arc is an arc e is an element of E for which there exist u, v is an element of V, such that all u-v walks contain e. In this article, we generalise this notion to cut paths, that is, walks W for which there exist u, v is an element of V, such that all u-v walks contain W as subwalk. We first prove various properties of cut paths and define their remainder structure, which we use to present a simple O (m)-time verification algorithm for a cut path. We further show that a graph contains at most O(n) maximal cut paths of length at most O(n) each, and present an optimal O(n2) enumeration algorithm for maximal cut paths. We apply cut paths and their remainder structure to improve several reachability problems from bioinformatics, as follows. A walk is called safe if it is a subwalk of every node-covering closed walk of a strongly connected graph. Multi-safety is defined analogously, by considering node-covering sets of closed walks instead. Cut paths provide simple O (m)-time algorithms verifying if a walk is safe or multi-safe. Further, by simultaneous computation of remainder structures of all subwalks of a cut path in linear time, we can identify all maximal multi-safe walks in O(mn) time. This improves over the state-of-the-art algorithm running in time O(m2 + n3 log n).
We study a family of matroid optimization problems with a linear constraint (MOL). In these problems, we seek a subset of elements that optimizes (i.e., maximizes or minimizes) a linear objective function simultaneously subject to (i) a matroid independent set, or a matroid basis constraint and (ii) additional linear constraint. A notable member in this family is BUDGETED MATROmINDEPENDENT SET (BM), which can be viewed as classic 0/1-KNAPSACK with a matroid constraint. While special cases of BM, such as KNAPSACK WITH CARDINALITY CONSTRAINT and MULTIPLE-CHOICE KNAPSACK, admit a fully polynomial-time approximation scheme (Fully PTAS), the best-known result for BM on a general matroid is an Efficient PTAS. Prior to this work, the existence of a Fully PTAS for BM, and more generally, for any problem in the family of MOL problems, has been open. In this article, we answer this question negatively by showing that none of the (non-trivial) problems in this family admits a Fully PTAS. This resolves the complexity status of several well-studied problems. Our main result is obtained by showing first that EXACT WEIGHT MATROID BASIS (EMB) does not admit a pseudo-polynomial time algorithm. We then obtain unconditional hardness results for the family of MOL problems in the oracle model (even if randomization is allowed) and show that the same results hold when the matroids are encoded as part of the input, assuming P # NP.
We present improved learning-augmented algorithms for the multi-option ski rental problem. Learning-augmented algorithms take machine learning (ML) predictions as an added part of the input and incorporate these predictions in solving the given problem. Due to their unique strength that combines the power of ML predictions with provable performance guarantees, they have been extensively studied in the context of online optimization problems. While the multi-option ski rental problem provides a natural generalization of the classical rent-or-buy variant, only deterministic algorithms for this problem were previously known, with or without learning augmentation. In this article, we first present that a very simple modification to a previously known algorithm suffices to give an improved deterministic learning-augmented algorithm. In fact, we prove that this algorithm has the best-possible performance of a deterministic algorithm by giving a matching lower bound. Then we present the first randomized learning-augmented algorithm, which surpasses the lower bound of deterministic algorithms; this learning-augmented algorithm is based on a new best-possible randomized competitive algorithm. These results are complemented by lower bounds for randomized competitive/learning-augmented algorithms.
The prophet secretary problem is a combination of the prophet inequality and the secretary problem, where elements are drawn from known independent distributions and arrive in uniformly random order. In this work, we design 1) a 0.688-competitive algorithm, that breaks the 0.675 barrier of blind strategies (Correa, Saona, Ziliotto, 2021), and 2) a 0.641-competitive algorithm for the prophet secretary matching problem, that breaks the 1-1/e≈ 0.632 barrier for the first time. Our second result also applies to the query-commit model of weighted stochastic matching and improves the state-of-the-art ratio (Derakhshan and Farhadi, 2023).
The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP-hard graph problems, like maximum-weight independent set, can be solved in time \(n^{\mathcal{O}(k)}\) if the input \(n\)-vertex graph is given together with a tree decomposition of independence number \(k\). Yolov, in SODA 2018, gave an algorithm that, given an \(n\)-vertex graph \(G\) and an integer \(k\), in time \(n^{\mathcal{O}(k^{3})}\) either constructs a tree decomposition of \(G\) whose independence number is \(\mathcal{O}(k^{3})\) or correctly reports that the tree-independence number of \(G\) is larger than \(k\). In this article, we first give an algorithm for computing the tree-independence number with a better approximation ratio and running time and then prove that our algorithm is, in some sense, the best one can hope for. More precisely, our algorithm runs in time \(2^{\mathcal{O}(k^{2})}n^{\mathcal{O}(k)}\) and either outputs a tree decomposition of \(G\) with independence number at most \(8k\) or determines that the tree-independence number of \(G\) is larger than \(k\). This implies \(2^{\mathcal{O}(k^{2})}n^{\mathcal{O}(k)}\)-time algorithms for various problems, like maximum-weight independent set, parameterized by the tree-independence number \(k\) without needing the decomposition as an input. Assuming Gap-ETH, an \(n^{\Omega(k)}\) factor in the running time is unavoidable for any approximation algorithm for the tree-independence number. Our second result is that the exact computation of the tree-independence number is para-NP-hard: We show that for every constant \(k\geq 4\) it is NP-complete to decide whether a given graph has the tree-independence number at most \(k\).
We study the parameterized complexity of a generalization of the coordinated motion planning problem on graphs, where the goal is to route a specified subset of a given set of k robots to their destinations with the aim of minimizing the total energy (i.e., the total length traveled). We develop novel techniques to push beyond previously-established results that were restricted to solid grids. We design a fixed-parameter additive approximation algorithm for this problem parameterized by k alone. This result, which is of independent interest, allows us to prove the following two results pertaining to well-studied coordinated motion planning problems: (1) A fixed-parameter algorithm, parameterized by k, for routing a single robot to its destination while avoiding the other robots, which is related to the famous Rush-Hour Puzzle; and (2) a fixed-parameter algorithm, parameterized by k plus the treewidth of the input graph, for the standard Coordinated Motion Planning (CMP) problem in which we need to route all the k robots to their destinations. The latter of these results implies, among others, the fixed-parameter tractability of CMP parameterized by k on graphs of bounded outerplanarity, which include bounded-height subgrids. We complement the above results with a lower bound which rules out the fixed-parameter tractability for CMP when parameterized by the total energy. This contrasts the recently-obtained tractability of the problem on solid grids under the same parameterization. As our final result, we strengthen the aforementioned fixed-parameter tractability to hold not only on solid grids but all graphs of bounded local treewidth – a class including, among others, all graphs of bounded genus.
We devise a data structure that can answer shortest path queries for two query points in a polygonal domain $P$ on $n$ vertices. For any $\varepsilon > 0$, the space complexity of the data structure is $O(n^{10+\varepsilon })$ and queries can be answered in $O(\log n)$ time. Alternatively, we can achieve a space complexity of $O(n^{9+\varepsilon })$ by relaxing the query time to $O(\log^2 n)$. This is the first improvement upon a conference paper by Chiang and Mitchell from 1999. They present a data structure with $O(n^{11})$ space complexity and $O(\log n)$ query time. Our main result can be extended to include a space-time trade-off. Specifically, we devise data structures with $O(n^{9+\varepsilon}/\hspace{1pt} \ell^{4 + O(\varepsilon )})$ space complexity and $O(\ell \log^2 n )$ query time, for any integer $1 \leq \ell \leq n$. Furthermore, we present improved data structures with $O(\log n)$ query time for the special case where we restrict one (or both) of the query points to lie on the boundary of $P$. When one of the query points is restricted to lie on the boundary, and the other query point is unrestricted, the space complexity becomes $O(n^{6+\varepsilon})$. When both query points are on the boundary, the space complexity is decreased further to $O(n^{4+\varepsilon })$, thereby improving an earlier result of Bae and Okamoto.
We describe a new dependent-rounding algorithmic framework for bipartite graphs. Given a fractional assignment G (R) of values to edges of a graph G = (* U +, E), the algorithms return an integral solution -(R) such that each right-node E E + has at most one neighboring edge 5 with - f = 1, and the variables -e also satisfy broad nonpositive-correlation properties. In particular, for any edges 41, 42 sharing a left-node D E * , the variables -e1, -e2 have strong negative correlation, i.e. the expectation of -e1-e2 is significantly below Ge1Ge2. This algorithm is based on generating negatively correlated Exponential random variables and using them for a rounding method inspired by a contention-resolution scheme of Im and Shadloo [2020]. Our algorithm gives stronger and much more flexible negative correlation properties. Dependent rounding schemes with negative correlation properties have been used for approximation algorithms for job-scheduling on unrelated machines to minimize weighted completion times [Bansal et al., 2021; Im and Li, 2023; Im and Shadloo, 2020]. Using our new dependent-rounding algorithm, among other improvements, we obtain a 1.398-approximation for this problem. This significantly improves over the prior 1.45-approximation ratio of Im and Li [2023].
According to the classic Chvátal’s Lemma from 1977, a graph \(G\) of minimum degree \(\delta(G)\) contains every tree on \(\delta(G)+1\) vertices. Our main result is the following algorithmic “extension” of Chvátal’s Lemma: For any \(n\)-vertex graph \(G\), an integer \(k\), and a tree \(T\) on at most \(\delta(G)+k\) vertices, deciding whether \(G\) contains a subgraph isomorphic to \(T\) can be done in time \(f(k)\cdot n^{\mathcal{O}(1)}\) for some function \(f\) of \(k\) only. The proof is based on an intricate interplay between extremal graph theory and parameterized algorithms.
We design the first node-differentially private algorithm for approximating the number of connected components in a graph. Given a database representing an n-vertex graph G and a privacy parameter ε, our algorithm runs in polynomial time and, with probability 1-o(1), has additive error Õ(Δ^*łnłn nε ), where Δ^* is the smallest possible maximum degree of a spanning forest of G. Node-differentially private algorithms are known only for a small number of database analysis tasks. A major obstacle for designing such an algorithm for the number of connected components is that this graph statistic is not robust to adding one node with arbitrary connections (a change that node-differential privacy is designed to hide): every graph is a neighbor of a connected graph. We overcome this by designing a family of efficiently computable Lipschitz extensions of the number of connected components or, equivalently, the size of a spanning forest. The construction of the extensions, which is at the core of our algorithm, is based on the forest polytope of G. We prove several combinatorial facts about spanning forests, in particular, that a graph with no induced Δ-stars has a spanning forest of degree at most Δ. With this fact, we show that our Lipschitz extensions for the number of connected components equal the true value of the function for the largest possible monotone families of graphs. More generally, on all monotone sets of graphs, the l∞ error of our Lipschitz extensions is nearly optimal.
In the longest plane spanning tree problem, we are given a finite planar point set \(\mathcal{P}\), and our task is to find a plane (i.e., noncrossing) spanning tree for \(\mathcal{P}\) with maximum total Euclidean edge length. Despite more than two decades of research, it remains open whether this problem is NP-hard. Thus, previous results have focused on polynomial-time algorithms that produce plane trees whose total edge length approximates \(\mathrm{OPT}\), the maximum possible length. The approximate trees in these algorithms all have small unweighted diameter, typically two to four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms. We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree with diameter at most four and total edge length at least \(0.546\cdot\mathrm{OPT}\). This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter \(d\geq 3\), we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most \(d\) (compared to a longest plane tree without constraints).
In a classical scheduling problem, we are given a set of $n$ jobs of unit length along with precedence constraints, and the goal is to find a schedule of these jobs on $m$ identical machines that minimizes the makespan. Using the standard 3-field notation, it is known as $Pm|\text{prec}, p_j=1|C_{\max}$. Settling the complexity of $Pm|\text{prec}, p_j=1|C_{\max}$ even for $m=3$ machines is the last open problem from the book of Garey and Johnson [GJ79] for which both upper and lower bounds on the worst-case running times of exact algorithms solving them remain essentially unchanged since the publication of [GJ79]. We present an algorithm for this problem that runs in $(1+\frac{n}{m})^{\mathcal{O}(\sqrt{nm})}$ time. This algorithm is subexponential when $m = o(n)$. In the regime of $m=\Theta(n)$ we show an algorithm that runs in$\mathcal{O}(1.997^n)$ time. Before our work, even for $m=3$ machines there were no algorithms known that run in $\mathcal{O}((2-\varepsilon)^n)$ time for some $\varepsilon > 0$.
We study the problem of recognizing the cluster structure of a graph in the framework of property testing in the bounded degree model. Given a parameter ε, a d-bounded degree graph is defined to be (k, φ)-clusterable, if it can be partitioned into no more than k parts, such that the (inner) conductance of the induced subgraph on each part is at least φ and the (outer) conductance of each part is at most cd,kε4φ2, where cd,k depends only on d,k. Our main result is a sublinear algorithm with the running time ~O(√n ⋅ poly(φ,k,1/ε)) that takes as input a graph with maximum degree bounded by d, parameters k, φ, ε, and with probability at least 2/3, accepts the graph if it is (k,φ)-clusterable and rejects the graph if it is ε-far from (k, φ*)-clusterable for φ* = c'd,kφ2 ε4}/log n, where c'd,k depends only on d,k. By the lower bound of Ω(√n) on the number of queries needed for testing graph expansion, which corresponds to k=1 in our problem, our algorithm is asymptotically optimal up to polylogarithmic factors.
We study the reverse shortest path problem on disk graphs in the plane. In this problem we consider the proximity graph of a set of $n$ disks in the plane of arbitrary radii: In this graph two disks are connected if the distance between them is at most some threshold parameter $r$. The case of intersection graphs is a special case with $r=0$. We give an algorithm that, given a target length $k$, computes the smallest value of $r$ for which there is a path of length at most $k$ between some given pair of disks in the proximity graph. Our algorithm runs in $O^*(n^{5/4})$ randomized expected time, which improves to $O^*(n^{6/5})$ for unit disk graphs, where all the disks have the same radius. Our technique is robust and can be applied to many variants of the problem. One significant variant is the case of weighted proximity graphs, where edges are assigned real weights equal to the distance between the disks or between their centers, and $k$ is replaced by a target weight $w$; that is, we seek a path whose length is at most $w$. In other variants, we want to optimize a parameter different from $r$, such as a scale factor of the radii of the disks. The main technique for the decision version of the problem (determining whether the graph with a given $r$ has the desired property) is based on efficient implementations of BFS (for the unweighted case) and of Dijkstra's algorithm (for the weighted case), using efficient data structures for maintaining the bichromatic closest pair for certain bicliques and several distance functions. The optimization problem is then solved by combining the resulting decision procedure with enhanced variants of the interval shrinking and bifurcation technique of [4].
We prove that a number of computational problems that ask for the largest sparse induced subgraph satisfying some property definable in CMSO2 logic, most notably Feedback Vertex Set, are polynomial-time solvable in the class of $P_6$-free graphs. This generalizes the work of Grzesik, Klimo\v{s}ová, Pilipczuk, and Pilipczuk on the Maximum Weight Independent Set problem in $P_6$-free graphs~[SODA 2019, TALG 2022], and of Abrishami, Chudnovsky, Pilipczuk, Rz\k{a}\.zewski, and Seymour on problems in $P_5$-free graphs~[SODA~2021]. The key step is a new generalization of the framework of potential maximal cliques. We show that instead of listing a large family of potential maximal cliques, it is sufficient to only list their carvers: vertex sets that contain the same vertices from the sought solution and have similar separation properties.
