Subexponential-time Algorithm Research Articles

True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs

We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set \(\mathcal{D}\) of \(n\) unit disks inducing a unit-disk graph \(G_{\mathcal{D}}\) and a number \(p\in[n]\) , one can partition \(\mathcal{D}\) into \(p\) subsets \(\mathcal{D}_{1},\dots,\mathcal{D}_{p}\) such that for every \(i\in[p]\) and every \(\mathcal{D}^{\prime}\subseteq\mathcal{D}_{i}\) , the graph obtained from \(G_{\mathcal{D}}\) by contracting all edges between the vertices in \(\mathcal{D}_{i}\backslash\mathcal{D}^{\prime}\) admits a tree decomposition in which each bag consists of \(O(p+|\mathcal{D}^{\prime}|)\) cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved recently by Marx et al. [SODA ’22] and Bandyapadhyay et al. [SODA ’22]. By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem for unit-disk graphs, resolving an open question in the work by Panolan et al. [SODA ’19]. On the algorithmic side, we obtain a new algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in \(2^{O(\sqrt{k}\log k)}\cdot n^{O(1)}\) time, where \(k\) denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA ’22] which runs in \(2^{O(k^{27/28})}\cdot n^{O(1)}\) time. We also show that the problem cannot be solved in \(2^{o(\sqrt{k})}\cdot n^{O(1)}\) time assuming the Exponential Time Hypothesis, which implies that our algorithm is almost optimal.

Approximate and Randomized Algorithms for Computing a Second Hamiltonian Cycle

In this paper we consider the following problem: Given a Hamiltonian graph G, and a Hamiltonian cycle C of G, can we compute a second Hamiltonian cycle C′≠C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{\\prime } \ e C$$\\end{document} of G, and if yes, how quickly? If the input graph G satisfies certain conditions (e.g. if every vertex of G is odd, or if the minimum degree is large enough), it is known that such a second Hamiltonian cycle always exists. Despite substantial efforts, no subexponential-time algorithm is known for this problem. In this paper we relax the problem of computing a second Hamiltonian cycle in two ways. First, we consider approximating the length of a second longest cycle on n-vertex graphs with minimum degree δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} and maximum degree Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}. We provide a linear-time algorithm for computing a cycle C′≠C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{\\prime } \ e C$$\\end{document} of length at least n-4α(n+2α)+8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-4\\alpha (\\sqrt{n}+2\\alpha )+8$$\\end{document}, where α=Δ-2δ-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha = \\frac{\\Delta -2}{\\delta -2}$$\\end{document}. This results provides a constructive proof of a recent result by Girão, Kittipassorn, and Narayanan in the regime of Δδ=o(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{\\Delta }{\\delta } = o(\\sqrt{n})$$\\end{document}. Our second relaxation of the problem is probabilistic. We propose a randomized algorithm which computes a second Hamiltonian cycle with high probability, given that the input graph G has a large enough minimum degree. More specifically, we prove that for every 0<p≤0.02\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<p\\le 0.02$$\\end{document}, if the minimum degree of G is at least 8plog8n+4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{8}{p} \\log \\sqrt{8}n + 4$$\\end{document}, then a second Hamiltonian cycle can be computed with probability at least 1-1n50p4+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 - \\frac{1}{n}\\left( \\frac{50}{p^4} + 1 \\right) $$\\end{document} in poly(n)·24pn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$poly(n) \\cdot 2^{4pn}$$\\end{document} time. This result implies that, when the minimum degree δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} is sufficiently large, we can compute with high probability a second Hamiltonian cycle faster than any known deterministic algorithm. In particular, when δ=ω(logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta = \\omega (\\log n)$$\\end{document}, our probabilistic algorithm works in 2o(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{o(n)}$$\\end{document} time.

Max Weight Independent Set in Graphs with No Long Claws: An Analog of the Gyárfás’ Path Argument

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw S t,t,t as an induced subgraph and provide a subexponential-time algorithm with improved running time \(2^{\mathcal {O}(\sqrt {nt}\log n)}\) and a quasipolynomial-time approximation scheme with improved running time \(2^{\mathcal {O}(\varepsilon ^{-1}t \log ^{5} n)}\) . The Gyárfás’ path argument, a powerful tool that is the main building block for many algorithms in P t -free graphs, ensures that given an n -vertex P t -free graph, in polynomial time we can find a set P of at most t -1 vertices such that every connected component of G-N[P] has at most n /2 vertices. Our main technical contribution is an analog of this result for S t,t,t -free graphs: given an n -vertex S t,t,t -free graph, in polynomial time we can find a set P of \(\mathcal {O}(t \log n)\) vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of G-N[P] such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most n /2 vertices.

Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits

An Algebraic Circuit for a multivariate polynomial P is a computational model for constructing the polynomial P using only additions and multiplications. It is a syntactic model of computation, as opposed to the Boolean Circuit model, and hence lower bounds for this model are widely expected to be easier to prove than lower bounds for Boolean circuits. Despite this, we do not have superpolynomial lower bounds against general algebraic circuits of depth 3 (except over constant-sized finite fields) and depth 4 (over any field other than F 2 ), while constant-depth Boolean circuit lower bounds have been known since the early 1980s. In this paper, we prove the first superpolynomial lower bounds against algebraic circuits of all constant depths over all fields of characteristic 0. We also observe that our super-polynomial lower bound for constant-depth circuits implies the first deterministic sub-exponential time algorithm for solving the Polynomial Identity Testing (PIT) problem for all small-depth circuits using the known connection between algebraic hardness and randomness.

FPT approximation and subexponential algorithms for covering few or many edges

We study the α-Fixed Cardinality Graph Partitioning (α-FCGP) problem, the generic local graph partitioning problem introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given a graph G, two numbers k,p and 0≤α≤1, the question is whether there is a set S⊆V of size k with a specified coverage function covα(S) at least p (or at most p for the minimization version). The coverage function covα(⋅) counts edges with exactly one endpoint in S with weight α and edges with both endpoints in S with weight 1−α. α-FCGP generalizes a number of fundamental graph problems such as Densestk-Subgraph, Maxk-Vertex Cover, and Max(k,n−k)-Cut.A natural question in the study of α-FCGP is whether the algorithmic results known for its special cases, like Maxk-Vertex Cover, could be extended to more general settings. One of the simple but powerful methods for obtaining parameterized approximation [Manurangsi, SOSA 2019] and subexponential algorithms [Fomin et al. IPL 2011] for Maxk-Vertex Cover is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greedy vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for α>0 and subexponential-time algorithms for the problem on apex-minor free graphs for maximization with α>1/3 and minimization with α<1/3.4

Grid induced minor theorem for graphs of small degree

A graph H is an induced minor of a graph G if H can be obtained from G by vertex deletions and edge contractions. We show that there is a function f(k,d)=O(k10+2d5) so that if a graph has treewidth at least f(k,d) and maximum degree at most d, then it contains a k×k-grid as an induced minor. This proves the conjecture of Aboulker, Adler, Kim, Sintiari, and Trotignon (2021) [1] that any graph with large treewidth and bounded maximum degree contains a large wall or the line graph of a large wall as an induced subgraph. It also implies that for any fixed planar graph H, there is a subexponential time algorithm for maximum weight independent set on H-induced-minor-free graphs.

Subexponential-Time Algorithms for Sparse PCA

We study the computational cost of recovering a unit-norm sparse principal component $$x \in \mathbb {R}^n$$ planted in a random matrix, in either the Wigner or Wishart spiked model (observing either $$W + \lambda xx^\top $$ with W drawn from the Gaussian orthogonal ensemble, or N independent samples from $$\mathcal {N}(0, I_n + \beta xx^\top )$$ , respectively). Prior work has shown that when the signal-to-noise ratio ( $$\lambda $$ or $$\beta \sqrt{N/n}$$ , respectively) is a small constant and the fraction of nonzero entries in the planted vector is $$\Vert x\Vert _0 / n = \rho $$ , it is possible to recover x in polynomial time if $$\rho \lesssim 1/\sqrt{n}$$ . While it is possible to recover x in exponential time under the weaker condition $$\rho \ll 1$$ , it is believed that polynomial-time recovery is impossible unless $$\rho \lesssim 1/\sqrt{n}$$ . We investigate the precise amount of time required for recovery in the “possible but hard” regime $$1/\sqrt{n} \ll \rho \ll 1$$ by exploring the power of subexponential-time algorithms, i.e., algorithms running in time $$\exp (n^\delta )$$ for some constant $$\delta \in (0,1)$$ . For any $$1/\sqrt{n} \ll \rho \ll 1$$ , we give a recovery algorithm with runtime roughly $$\exp (\rho ^2 n)$$ , demonstrating a smooth tradeoff between sparsity and runtime. Our family of algorithms interpolates smoothly between two existing algorithms: the polynomial-time diagonal thresholding algorithm and the $$\exp (\rho n)$$ -time exhaustive search algorithm. Furthermore, by analyzing the low-degree likelihood ratio, we give rigorous evidence suggesting that the tradeoff achieved by our algorithms is optimal.

On Subgraph Complementation to H-free Graphs

For a class $$\mathcal {G}$$ of graphs, the problem Subgraph Complement to $$\mathcal {G}$$ asks whether one can find a subset S of vertices of the input graph G such that complementing the subgraph induced by S in G results in a graph in $$\mathcal {G}$$ . We investigate the complexity of the problem when $$\mathcal {G}$$ is H-free for H being a complete graph, a star, a path, or a cycle. We obtain the following results: Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time $$2^{o(\mid V(G)\mid )}$$ ), assuming the Exponential Time Hypothesis. We show that the complexity results on a graph class $$\mathcal {G}$$ is also true for the class $$\overline{\mathcal {G}}$$ of the complement graphs of $$\mathcal {G}$$ . Therefore, each of the above results mentioned for the H-free class of graphs is also valid for the $$\overline{H}$$ -free class of graphs. It is noteworthy that our results generalize two main results, namely, Subgraph Complement to triangle-free graphs and Subgraph Complement to d-degenerate graphs, and resolves one open question due to Fomin et al. (Algorithmica, 2020).

Defective Coloring on Classes of Perfect Graphs

In Defective Coloring we are given a graph $G$ and two integers $\chi_d$, $\Delta^*$ and are asked if we can $\chi_d$-color $G$ so that the maximum degree induced by any color class is at most $\Delta^*$. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters $\chi_d$, $\Delta^*$ is set to the smallest possible fixed value that does not trivialize the problem ($\chi_d = 2$ or $\Delta^* = 1$). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either $\chi_d$ or $\Delta^*$ is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both $\chi_d$ and $\Delta^*$ are unbounded.

Quantum algorithms and approximating polynomials for composed functions with shared inputs

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Letfbe anm-bit Boolean function and consider ann-bit functionFobtained by applyingfto conjunctions of possibly overlapping subsets ofnvariables. Iffhas quantum query complexityQ(f), we give an algorithm for evaluatingFusingO~(Q(f)⋅n)quantum queries. This improves on the bound ofO(Q(f)⋅n)that follows by treating each conjunction independently, and our bound is tight for worst-case choices off. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree off.By recursively applying our composition theorems, we obtain a nearly optimalO~(n1−2−d)upper bound on the quantum query complexity and approximate degree of linear-size depth-dAC0circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC0circuits.As an additional consequence, we show that AC0∘⊕circuits of depthd+1require sizeΩ~(n1/(1−2−d))≥ω(n1+2−d)to compute the Inner Product function even on average. The previous best size lower bound wasΩ(n1+4−(d+1))and only held in the worst case (Cheraghchi et al., JCSS 2018).

The exponential-time hypothesis and the relative complexity of optimization and logical reasoning problems

Obtaining lower bounds for NP-hard problems has for a long time been an active area of research. Algebraic techniques introduced by Jonsson et al. (2017) [4] show that the fine-grained time complexity of the parameterized ▪ problem correlates to the lattice of strong partial clones. With this ordering they isolated a relation R such that ▪ can be solved at least as fast as any other NP-hard ▪ problem. In this paper we extend this method and show that such languages also exist for the surjective SAT problem, the max ones problem, the propositional abduction problem, and the Boolean valued constraint satisfaction problem over finite-valued constraint languages. These languages may be interesting when investigating the borderline between polynomial time, subexponential time and exponential-time algorithms since they in a precise sense can be regarded as NP-hard problems with minimum time complexity. Indeed, with the help of these languages we relate all of the above problems to the exponential time hypothesis (ETH) in several different ways.

{mathcal {U}}-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs—a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a richer class of graphs. In particular, mixed unit interval graphs may contain a claw as an induced subgraph, as opposed to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyacı et al. (Inf Process Lett 121:29–33, 2017. https://doi.org/10.1016/j.ipl.2017.01.007). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

Acyclic orders, partition schemes and CSPs: Unified hardness proofs and improved algorithms

Many computational problems arising in, for instance, artificial intelligence can be realized as infinite-domain constraint satisfaction problems (CSPs) based on partition schemes: a set of pairwise disjoint binary relations (containing the equality relation) whose union spans the underlying domain and which is closed under converse. We first consider partition schemes that contain an acyclic order and where the constraint language contains all unions of the basic relations; such CSPs are frequently occurring in e.g. temporal and spatial reasoning. We identify properties of such orders which, when combined, are sufficient to establish NP-hardness of the CSP and strong lower bounds under the exponential-time hypothesis, even for degree-bounded problems. This result explains, in a uniform way, many existing hardness results from the literature, and shows that it is impossible to obtain subexponential time algorithms unless the exponential-time hypothesis fails. However, some of these problems (including several important temporal problems), despite likely not being solvable in subexponential time, admit non-trivial improved exponential-time algorithm, and we present a novel improved algorithm for RCC-8 and related formalisms.

A note on the fine-grained complexity of MIS on regular graphs

We show that there is no subexponential time algorithm for computing the exact solution of the maximum independent set problem in d-regular graphs, unless ETH fails. We expand our method to show that it helps to provide lowerbounds for other covering problems such as vertex cover and clique. We utilize the construction to show the NP-hardness of MIS on 5-regular planar graphs, closing the exact complexity status of the problem on regular planar graphs.

Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

We study algorithmic properties of the graph class {textsc {Chordal}}{-ke}, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from {textsc {Chordal}}{-ke}. More precisely, we identify a large class of optimization problems on {textsc {Chordal}}{-ke} solvable in time 2^{{mathcal{O}}(sqrt{k}log k)}cdot n^{{mathcal{O}}(1)}. Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on {textsc {Chordal}}{-ke} when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of {textsc {Chordal}}{-ke} graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on {textsc {Chordal}}{-ke} graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on {textsc {Chordal}}{-ke} graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of {textsc {Chordal}}{-ke}, namely, {textsc {Interval}}{-ke} and {textsc {Split}}{-ke} graphs.

On the Minimum Consistent Subset Problem

Let $P$ be a set of $n$ colored points in the plane. Introduced by Hart (1968), a consistent subset of $P$, is a set $S\subseteq P$ such that for every point $p$ in $P\setminus S$, the closest point of $p$ in $S$ has the same color as $p$. The consistent subset problem is to find a consistent subset of $P$ with minimum cardinality. This problem is known to be NP-complete even for two-colored point sets. Since the initial presentation of this problem, aside from the hardness results, there has not been a significant progress from the algorithmic point of view. In this paper we present the following algorithmic results: 1. The first subexponential-time algorithm for the consistent subset problem. 2. An $O(n\log n)$-time algorithm that finds a consistent subset of size two in two-colored point sets (if such a subset exists). Towards our proof of this running time we present a deterministic $O(n \log n)$-time algorithm for computing a variant of the compact Voronoi diagram; this improves the previously claimed expected running time. 3. An $O(n\log^2 n)$-time algorithm that finds a minimum consistent subset in two-colored point sets where one color class contains exactly one point; this improves the previous best known $O(n^2)$ running time which is due to Wilfong (SoCG 1991). 4. An $O(n)$-time algorithm for the consistent subset problem on collinear points; this improves the previous best known $O(n^2)$ running time. 5. A non-trivial $O(n^6)$-time dynamic programming algorithm for the consistent subset problem on points arranged on two parallel lines. To obtain these results, we combine tools from planar separators, additively-weighted Voronoi diagrams with respect to convex distance functions, point location in farthest-point Voronoi diagrams, range trees, paraboloid lifting, minimum covering of a circle with arcs, and several geometric transformations.

Matching Cut in Graphs with Large Minimum Degree

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be {mathsf {NP}}-complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is {mathsf {NP}}-complete on graphs with minimum degree two. In this paper, we show that, for any given constant c>1, Matching Cut is {mathsf {NP}}-complete in the class of graphs with minimum degree c and this restriction of Matching Cut has no subexponential-time algorithm in the number of vertices unless the Exponential-Time Hypothesis fails. We also show that, for any given constant epsilon >0, Matching Cut remains {mathsf {NP}}-complete in the class of n-vertex (bipartite) graphs with unbounded minimum degree delta >n^{1-epsilon }. We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree delta ge 3 in O^*(lambda ^n) time, where lambda is the positive root of the polynomial x^{delta +1}-x^{delta }-1. Despite the hardness results, this is a very fast exact exponential-time algorithm for Matching Cut on graphs with large minimum degree; for instance, the running time is O^*(1.0099^n) on graphs with minimum degree delta ge 469. Complementing our hardness results, we show that, for any two fixed constants 1< c <4 and c^{prime }ge 0, Matching Cut is solvable in polynomial time for graphs with large minimum degree delta ge frac{1}{c}n-c^{prime }.

Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Let {mathcal {C}} and {mathcal {D}} be hereditary graph classes. Consider the following problem: given a graph Gin {mathcal {D}}, find a largest, in terms of the number of vertices, induced subgraph of G that belongs to {mathcal {C}}. We prove that it can be solved in 2^{o(n)} time, where n is the number of vertices of G, if the following conditions are satisfied:the graphs in {mathcal {C}} are sparse, i.e., they have linearly many edges in terms of the number of vertices;the graphs in {mathcal {D}} admit balanced separators of size governed by their density, e.g., {mathcal {O}}(varDelta ) or {mathcal {O}}(sqrt{m}), where varDelta and m denote the maximum degree and the number of edges, respectively; andthe considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph. This leads, for example, to the following corollaries for specific classes {mathcal {C}} and {mathcal {D}}:a largest induced forest in a P_t-free graph can be found in 2^{tilde{{mathcal {O}}}(n^{2/3})} time, for every fixed t; anda largest induced planar graph in a string graph can be found in 2^{tilde{{mathcal {O}}}(n^{2/3})} time.

Relation collection using Pollard special-q sieving to solve integer factorization and discrete logarithm problem

The strength of many security protocols lies on the computational intractability of the integer factorization and discrete logarithm problems. Currently, the best-known techniques employed are number field sieve (NFS) family of algorithms. They come under the class of sub-exponential time algorithms. This class of algorithms comprises of multiple steps. The relation collection (sieving step) is one of the computationally costly and highly memory-dependent phase of these algorithms. This paper discusses various ways to improve the efficiency of the relation collection phase by using parallelization techniques. Experiments have been carried out by using function field sieve, which is one of the NFS family algorithms, to show the computation efficiency of parallelization techniques along with the suitable sieving techniques and the key parameters. The result of our basic implementation is compared with the parallelized version of it. The result analysis depicts that the relation collection phase can be improved by using parallelization techniques up to fourfold.

The Complexity of Tree Partitioning

Given a tree T on n vertices, and $$k, b, s_1, \ldots , s_b \in \mathbb {N}$$ , the Tree Partitioning problem asks if at most k edges can be removed from T so that the resulting components can be grouped into b groups such that the number of vertices in group i is $$s_i$$ , for $$i =1, \ldots , b$$ . The case where $$s_1=\cdots =s_b =n/b$$ , referred to as the Balanced Tree Partitioning problem, was shown to be $${\mathcal {NP}}$$ -complete for trees of maximum degree at most 5, and the complexity of the problem for trees of maximum degree 4 and 3 was posed as an open question. The parameterized complexity of Balanced Tree Partitioning was also posed as an open question in another work. In this paper, we answer both open questions negatively. We show that Balanced Tree Partitioning (and hence, Tree Partitioning) is $${\mathcal {NP}}$$ -complete for trees of maximum degree 3, thus closing the door on the complexity of Balanced Tree Partitioning, as the simple case when T is a path is in $${\mathcal {P}}$$ . In terms of the parameterized complexity of the problems, we show that both Balanced Tree Partitioning and Tree Partitioning are W[1]-complete parameterized by k. Using a compact representation of the solution space for an instance of the problem, we present a dynamic programming algorithm for the weighted version of Tree Partitioning (and hence for that of Balanced Tree Partitioning) that runs in subexponential-time $$2^{O(\sqrt{n})}$$ , adding a natural problem to the list of problems that can be solved in subexponential time. Finally, we extend this subexponential-time algorithm to the Weighted Graph Partitioning problem on graphs of treewidth $$o(n/\lg {n})$$ , and we also show an application of this subexponential-time algorithm for approximating the Weighted Graph Partitioning problem.