111,900 research outputs found

    Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class

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    We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set. To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC)

    Improved Bounds on Quantum Learning Algorithms

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    In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. Hunziker et al. conjectured that for any class C of Boolean functions, the number of quantum black-box queries which are required to exactly identify an unknown function from C is O(logCγ^C)O(\frac{\log |C|}{\sqrt{{\hat{\gamma}}^{C}}}), where γ^C\hat{\gamma}^{C} is a combinatorial parameter of the class C. We essentially resolve this conjecture in the affirmative by giving a quantum algorithm that, for any class C, identifies any unknown function from C using O(logCloglogCγ^C)O(\frac{\log |C| \log \log |C|}{\sqrt{{\hat{\gamma}}^{C}}}) quantum black-box queries. We consider a range of natural problems intermediate between the exact learning problem (in which the learner must obtain all bits of information about the black-box function) and the usual problem of computing a predicate (in which the learner must obtain only one bit of information about the black-box function). We give positive and negative results on when the quantum and classical query complexities of these intermediate problems are polynomially related to each other. Finally, we improve the known lower bounds on the number of quantum examples (as opposed to quantum black-box queries) required for (ϵ,δ)(\epsilon,\delta)-PAC learning any concept class of Vapnik-Chervonenkis dimension d over the domain {0,1}n\{0,1\}^n from Ω(dn)\Omega(\frac{d}{n}) to Ω(1ϵlog1δ+d+dϵ)\Omega(\frac{1}{\epsilon}\log \frac{1}{\delta}+d+\frac{\sqrt{d}}{\epsilon}). This new lower bound comes closer to matching known upper bounds for classical PAC learning.Comment: Minor corrections. 18 pages. To appear in Quantum Information Processing. Requires: algorithm.sty, algorithmic.sty to buil

    Efficient Flow-based Approximation Algorithms for Submodular Hypergraph Partitioning via a Generalized Cut-Matching Game

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    In the past 20 years, increasing complexity in real world data has lead to the study of higher-order data models based on partitioning hypergraphs. However, hypergraph partitioning admits multiple formulations as hyperedges can be cut in multiple ways. Building upon a class of hypergraph partitioning problems introduced by Li & Milenkovic, we study the problem of minimizing ratio-cut objectives over hypergraphs given by a new class of cut functions, monotone submodular cut functions (mscf's), which captures hypergraph expansion and conductance as special cases. We first define the ratio-cut improvement problem, a family of local relaxations of the minimum ratio-cut problem. This problem is a natural extension of the Andersen & Lang cut improvement problem to the hypergraph setting. We demonstrate the existence of efficient algorithms for approximately solving this problem. These algorithms run in almost-linear time for the case of hypergraph expansion, and when the hypergraph rank is at most O(1)O(1). Next, we provide an efficient O(logn)O(\log n)-approximation algorithm for finding the minimum ratio-cut of GG. We generalize the cut-matching game framework of Khandekar et. al. to allow for the cut player to play unbalanced cuts, and matching player to route approximate single-commodity flows. Using this framework, we bootstrap our algorithms for the ratio-cut improvement problem to obtain approximation algorithms for minimum ratio-cut problem for all mscf's. This also yields the first almost-linear time O(logn)O(\log n)-approximation algorithms for hypergraph expansion, and constant hypergraph rank. Finally, we extend a result of Louis & Makarychev to a broader set of objective functions by giving a polynomial time O(logn)O\big(\sqrt{\log n}\big)-approximation algorithm for the minimum ratio-cut problem based on rounding 22\ell_2^2-metric embeddings.Comment: Comments and feedback welcom

    Parameterized Algorithms for Graph Partitioning Problems

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    In parameterized complexity, a problem instance (I, k) consists of an input I and an extra parameter k. The parameter k usually a positive integer indicating the size of the solution or the structure of the input. A computational problem is called fixed-parameter tractable (FPT) if there is an algorithm for the problem with time complexity O(f(k).nc ), where f(k) is a function dependent only on the input parameter k, n is the size of the input and c is a constant. The existence of such an algorithm means that the problem is tractable for fixed values of the parameter. In this thesis, we provide parameterized algorithms for the following NP-hard graph partitioning problems: (i) Matching Cut Problem: In an undirected graph, a matching cut is a partition of vertices into two non-empty sets such that the edges across the sets induce a matching. The matching cut problem is the problem of deciding whether a given graph has a matching cut. The Matching Cut problem is expressible in monadic second-order logic (MSOL). The MSOL formulation, together with Courcelle’s theorem implies linear time solvability on graphs with bounded tree-width. However, this approach leads to a running time of f(||ϕ||, t) · n, where ||ϕ|| is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. The dependency of f(||ϕ||, t) on ||ϕ|| can be as bad as a tower of exponentials. In this thesis we give a single exponential algorithm for the Matching Cut problem with tree-width alone as the parameter. The running time of the algorithm is 2O(t) · n. This answers an open question posed by Kratsch and Le [Theoretical Computer Science, 2016]. We also show the fixed parameter tractability of the Matching Cut problem when parameterized by neighborhood diversity or other structural parameters. (ii) H-Free Coloring Problems: In an undirected graph G for a fixed graph H, the H-Free q-Coloring problem asks to color the vertices of the graph G using at most q colors such that none of the color classes contain H as an induced subgraph. That is every color class is H-free. This is a generalization of the classical q-Coloring problem, which is to color the vertices of the graph using at most q colors such that no pair of adjacent vertices are of the same color. The H-Free Chromatic Number is the minimum number of colors required to H-free color the graph. For a fixed q, the H-Free q-Coloring problem is expressible in monadic secondorder logic (MSOL). The MSOL formulation leads to an algorithm with time complexity f(||ϕ||, t) · n, where ||ϕ|| is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. In this thesis we present the following explicit combinatorial algorithms for H-Free Coloring problems: • An O(q O(t r ) · n) time algorithm for the general H-Free q-Coloring problem, where r = |V (H)|. • An O(2t+r log t · n) time algorithm for Kr-Free 2-Coloring problem, where Kr is a complete graph on r vertices. The above implies an O(t O(t r ) · n log t) time algorithm to compute the H-Free Chromatic Number for graphs with tree-width at most t. Therefore H-Free Chromatic Number is FPT with respect to tree-width. We also address a variant of H-Free q-Coloring problem which we call H-(Subgraph)Free q-Coloring problem, which is to color the vertices of the graph such that none of the color classes contain H as a subgraph (need not be induced). We present the following algorithms for H-(Subgraph)Free q-Coloring problems. • An O(q O(t r ) · n) time algorithm for the general H-(Subgraph)Free q-Coloring problem, which leads to an O(t O(t r ) · n log t) time algorithm to compute the H- (Subgraph)Free Chromatic Number for graphs with tree-width at most t. • An O(2O(t 2 ) · n) time algorithm for C4-(Subgraph)Free 2-Coloring, where C4 is a cycle on 4 vertices. • An O(2O(t r−2 ) · n) time algorithm for {Kr\e}-(Subgraph)Free 2-Coloring, where Kr\e is a graph obtained by removing an edge from Kr. • An O(2O((tr2 ) r−2 ) · n) time algorithm for Cr-(Subgraph)Free 2-Coloring problem, where Cr is a cycle of length r. (iii) Happy Coloring Problems: In a vertex-colored graph, an edge is happy if its endpoints have the same color. Similarly, a vertex is happy if all its incident edges are happy. we consider the algorithmic aspects of the following Maximum Happy Edges (k-MHE) problem: given a partially k-colored graph G, find an extended full k-coloring of G such that the number of happy edges are maximized. When we want to maximize the number of happy vertices, the problem is known as Maximum Happy Vertices (k-MHV). We show that both k-MHE and k-MHV admit polynomial-time algorithms for trees. We show that k-MHE admits a kernel of size k + `, where ` is the natural parameter, the number of happy edges. We show the hardness of k-MHE and k-MHV for some special graphs such as split graphs and bipartite graphs. We show that both k-MHE and k-MHV are tractable for graphs with bounded tree-width and graphs with bounded neighborhood diversity. vii In the last part of the thesis we present an algorithm for the Replacement Paths Problem which is defined as follows: Let G (|V (G)| = n and |E(G)| = m) be an undirected graph with positive edge weights. Let PG(s, t) be a shortest s − t path in G. Let l be the number of edges in PG(s, t). The Edge Replacement Path problem is to compute a shortest s − t path in G\{e}, for every edge e in PG(s, t). The Node Replacement Path problem is to compute a shortest s−t path in G\{v}, for every vertex v in PG(s, t). We present an O(TSP T (G) + m + l 2 ) time and O(m + l 2 ) space algorithm for both the problems, where TSP T (G) is the asymptotic time to compute a single source shortest path tree in G. The proposed algorithm is simple and easy to implement

    Topology Matters: Smoothed Competitiveness of Metrical Task Systems

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    We consider online problems that can be modeled as metrical task systems: An online algorithm resides in a graph of n nodes and may move in this graph at a cost equal to the distance. The algorithm has to service a sequence of tasks that arrive over time; each task specifies for each node a request cost that is incurred if the algorithm services the task in this particular node. The objective is to minimize the total request plus travel cost. Borodin, Linial and Saks gave a deterministic work function algorithm (WFA) for metrical task systems having a tight competitive ratio of 2n-1. We present a smoothed competitive analysis of WFA. Given an adversarial task sequence, we add some random noise to the request costs and analyze the competitive ratio of WFA on the perturbed sequence. We prove upper and matching lower bounds. Our analysis reveals that the smoothed competitive ratio of WFA is much better than its (worst case) competitive ratio and that it depends on several topological parameters of the graph underlying the metric, such as maximum degree, diameter, etc. For example, already for moderate perturbations, the smoothed competitive ratio of WFA is O(log(n)) on a clique and O(sqrt{n}) on a line. We also provide the first average case analysis of WFA. For a large class of probability distributions, we prove that WFA has O(log(D)) expected competitive ratio, where D is the maximum degree of the underlying graph

    Efikasni algoritmi za probleme iz diskretne geometrije

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    The first class of problem we study deals with geometric matchings. Given a set of points in the plane, we study perfect matchings of those points by straight line segments so that the segments do not cross. Bottleneck matching is such a matching that minimizes the length of the longest segment. We are interested in finding a bottleneck matching of points in convex position. In the monochromatic case, where any two points are allowed to be matched, we give an O(n 2 )-time algorithm for finding a bottleneck matching, improving upon previously best known algorithm of O(n 3 ) time complexity. We also study a bichromatic version of this problem, where each point is colored either red or blue, and only points of different color can be matched. We develop a range of tools, for dealing with bichromatic non-crossing matchings of points in convex position. Combining that set of tools with a geometric analysis enable us to solve the problem of finding a bottleneck matching in O(n 2 ) time. We also design an O(n)-time algorithm for the case where the given points lie on a circle. Previously best known results were O(n 3 ) for points in convex position, and O(n log n) for points on a circle. The second class of problems we study deals with dilation of geometric networks. Given a polygon representing a network, and a point p in the same plane, we aim to extend the network by inserting a line segment, called a feed-link, which connects p to the boundary of the polygon. Once a feed link is fixed, the geometric dilation of some point q on the boundary is the ratio between the length of the shortest path from p to q through the extended network, and their Euclidean distance. The utility of a feed-link is inversely proportional to the maximal dilation over all boundary points. We give a linear time algorithm for computing the feed-link with the minimum overall dilation, thus improving upon the previously known algorithm of complexity that is roughly O(n log n).Prva klasa problema koju proučavamo tičee se geometrijskih mečinga. Za dat skup tačaaka u ravni, posmatramo savršene mečinge tih tačaka spajajućii ih  dužima koje   se ne smeju sećui. Bottleneck mečing je takav mečing koji minimizuje dužinu najduže duži. Naš cilj je da nađemo bottleneck mečiing tačaka u konveksnom položaju.Za monohromatski slučaj, u kom je dozvoljeno upariti svaki par tačaka, dajemo algoritam vremenske složenosti O(n 2) za nalaženje bottleneck mečinga. Ovo  je bolje od prethodno najbolji poznatog algoritam, čiija je složenost O(n 3 ). Takođe proučavamo bihromatsku verziju ovog problema, u kojoj je svaka tačka  obojena ili u crveno ili u plavo, i dozvoljeno je upariti samo tačke različite boje. Razvijamo niz alata za rad sa bihromatskim nepresecajućim mečinzima tačaka u konveksnom položaju. Kombinovanje ovih alata sa geometrijskom analizom omogućava nam da rešimo problem nalaženja bottleneck mečinga u O(n 2 ) vremenu. Takođe, konstruišemo algoritam vremenske složenosti O(n) za slučaj kada  sve date tačkke leže na krugu. Prethodno najbolji poznati algoritmi su imali složenosti  O(n 3 ) za tačkeke u konveksnom položaju i O(n log n) za tačke na krugu. Druga klasa problema koju proučaavamo tiče se dilacije u geometrijskim mrežama. Za datu mrežu predstavljenu poligonom, i tačku p u istoj ravni, želimo proširiti mrežu  dodavanjem duži zvane feed-link koja povezuje p sa obodom poligona. Kada je feed- link fiksiran, definišemo geometrijsku dilaciju neke tačke q na obodu kao odnos izme  đu  dužine najkraćeg puta od p do q kroz proširenu mrežu i njihovog Euklidskog rastojanja. Korisnost feed-linka je obrnuto proporcionalna najvećoj dilaciji od svih ta čaka na obodu poligona. Konstruišemo algoritam linearne vremenske složenosti koji nalazi feed-link sa najmanom sveukupnom dilacijom. Ovim postižemo bolji rezultat od prethodno najboljeg poznatog algoritma složenosti približno O(n log n)

    Efikasni algoritmi za probleme iz diskretne geometrije

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    The first class of problem we study deals with geometric matchings. Given a set of points in the plane, we study perfect matchings of those points by straight line segments so that the segments do not cross. Bottleneck matching is such a matching that minimizes the length of the longest segment. We are interested in finding a bottleneck matching of points in convex position. In the monochromatic case, where any two points are allowed to be matched, we give an O(n 2 )-time algorithm for finding a bottleneck matching, improving upon previously best known algorithm of O(n 3 ) time complexity. We also study a bichromatic version of this problem, where each point is colored either red or blue, and only points of different color can be matched. We develop a range of tools, for dealing with bichromatic non-crossing matchings of points in convex position. Combining that set of tools with a geometric analysis enable us to solve the problem of finding a bottleneck matching in O(n 2 ) time. We also design an O(n)-time algorithm for the case where the given points lie on a circle. Previously best known results were O(n 3 ) for points in convex position, and O(n log n) for points on a circle. The second class of problems we study deals with dilation of geometric networks. Given a polygon representing a network, and a point p in the same plane, we aim to extend the network by inserting a line segment, called a feed-link, which connects p to the boundary of the polygon. Once a feed link is fixed, the geometric dilation of some point q on the boundary is the ratio between the length of the shortest path from p to q through the extended network, and their Euclidean distance. The utility of a feed-link is inversely proportional to the maximal dilation over all boundary points. We give a linear time algorithm for computing the feed-link with the minimum overall dilation, thus improving upon the previously known algorithm of complexity that is roughly O(n log n).Prva klasa problema koju proučavamo tičee se geometrijskih mečinga. Za dat skup tačaaka u ravni, posmatramo savršene mečinge tih tačaka spajajućii ih  dužima koje   se ne smeju sećui. Bottleneck mečing je takav mečing koji minimizuje dužinu najduže duži. Naš cilj je da nađemo bottleneck mečiing tačaka u konveksnom položaju.Za monohromatski slučaj, u kom je dozvoljeno upariti svaki par tačaka, dajemo algoritam vremenske složenosti O(n 2) za nalaženje bottleneck mečinga. Ovo  je bolje od prethodno najbolji poznatog algoritam, čiija je složenost O(n 3 ). Takođe proučavamo bihromatsku verziju ovog problema, u kojoj je svaka tačka  obojena ili u crveno ili u plavo, i dozvoljeno je upariti samo tačke različite boje. Razvijamo niz alata za rad sa bihromatskim nepresecajućim mečinzima tačaka u konveksnom položaju. Kombinovanje ovih alata sa geometrijskom analizom omogućava nam da rešimo problem nalaženja bottleneck mečinga u O(n 2 ) vremenu. Takođe, konstruišemo algoritam vremenske složenosti O(n) za slučaj kada  sve date tačkke leže na krugu. Prethodno najbolji poznati algoritmi su imali složenosti  O(n 3 ) za tačkeke u konveksnom položaju i O(n log n) za tačke na krugu. Druga klasa problema koju proučaavamo tiče se dilacije u geometrijskim mrežama. Za datu mrežu predstavljenu poligonom, i tačku p u istoj ravni, želimo proširiti mrežu  dodavanjem duži zvane feed-link koja povezuje p sa obodom poligona. Kada je feed- link fiksiran, definišemo geometrijsku dilaciju neke tačke q na obodu kao odnos izme  đu  dužine najkraćeg puta od p do q kroz proširenu mrežu i njihovog Euklidskog rastojanja. Korisnost feed-linka je obrnuto proporcionalna najvećoj dilaciji od svih ta čaka na obodu poligona. Konstruišemo algoritam linearne vremenske složenosti koji nalazi feed-link sa najmanom sveukupnom dilacijom. Ovim postižemo bolji rezultat od prethodno najboljeg poznatog algoritma složenosti približno O(n log n)

    Algorithms and lower bounds for testing properties of structured distributions

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    In this doctoral thesis we consider various property testing problems for structured distributions. A distribution is said to be structured if it belongs to a certain class which can be simply described in approximation terms. Such distributions often arise in practice, e.g. log-concave distributions, easily approximated by polynomials (see [Bir87a]), often appear in econometric research. For structured distributions, testing a property often requires far less samples than for general unrestricted distributions. In this thesis we prove that this is indeed the case for several distance-related properties. Namely, we give explicit sub-linear time algorithms for L1 and L2 distance testing between two structured distributions for the cases when either one or both of them are available as a “black box”. We also prove that the given algorithms have the best possible asymptotic complexity by proving matching lower bounds in the form of explicit problem instances (albeit constructed using randomized techniques) demanding at least a specified amount of data to be tested successfully. As the main numerical result, we prove that testing that total variation distance to an explicitly given distribution is at least e requires O(√k/e²) samples, where k is an approximation parameter, dependent on the class of distribution being tested and independent of the support size. Testing that the total variation distance between two “black box” distributions is at least e requires O(k⁴/⁵e⁶/⁵). In some cases, when k ~ n, this result may be worse than using an unrestricted testing algorithm (which requires O( n²/3/e² ) samples where n is the domain size). To address this issue, we develop a third algorithm, which requires O(k²/³e⁴/³ log⁴/³(n/k) log log(n/k)) and serves as a bridge between the cases of small and large domain sizes
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