We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition. As a further generalisation, we obtain the same conclusion only assuming an extendability property and the existence of a robust fractional clique decomposition.
Another recent result, due to Ferber, Hod, Krivelevich and Sudakov @cite_22 gives a short probabilistic construction of almost Steiner systems', in which every @math -subset is covered by either one or two @math -subsets.
@cite_22
1524753800
Given a graph @math and a subset @math of terminals, a of @math is a tree that spans @math . In the vertex-weighted Steiner tree (VST) problem, each vertex is assigned a non-negative weight, and the goal is to compute a minimum weight Steiner tree of @math . We study a natural generalization of the VST problem motivated by multi-level graph construction, the (V-GSST), which can be stated as follows: given a graph @math and terminals @math , where each terminal @math requires a facility of a minimum grade of service @math , compute a Steiner tree @math by installing facilities on a subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in @math with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multi-level variants such as this one can be useful in network design problems where vertices may require facilities of varying priority. While similar problems have been studied in the edge-weighted case, they have not been studied as well in the more general vertex-weighted case. We first describe a simple heuristic for the V-GSST problem whose approximation ratio depends on @math , the number of grades of service. We then generalize the greedy algorithm of [Klein & Ravi, 1995] to show that the V-GSST problem admits a @math -approximation, where @math is the set of terminals requiring some facility. This result is surprising, as it shows that the (seemingly harder) multi-grade problem can be approximated as well as the VST problem, and that the approximation ratio does not depend on the number of grades of service. Given a set @math of m unit disks and a set @math of n points in the plane, the discrete unit disk cover problem is to select a minimum cardinality subset @math to cover @math . This problem is NP-hard [14] and the best previous practical solution is a 38-approximation algorithm by [5]. We first consider the line-separable discrete unit disk cover problem (the set of disk centers can be separated from the set of points by a line) for which we present an O(n(log n + m))-time algorithm that finds an exact solution. Combining our line-separable algorithm with techniques from the algorithm of [5] results in an O(m2n4) time 22-approximate solution to the discrete unit disk cover problem. In the hidden clique problem, one needs to find the maximum clique in an @math -vertex graph that has a clique of size @math but is otherwise random. An algorithm of Alon, Krivelevich and Sudakov that is based on spectral techniques is known to solve this problem (with high probability over the random choice of input graph) when @math for a sufficiently large constant @math . In this manuscript we present a new algorithm for finding hidden cliques. It too provably works when @math for a sufficiently large constant @math . However, our algorithm has the advantage of being much simpler (no use of spectral techniques), running faster (linear time), and experiments show that the leading constant @math is smaller than in the spectral approach. We also present linear time algorithms that experimentally find even smaller hidden cliques, though it remains open whether any of these algorithms finds hidden cliques of size @math . An instance of the generalized partial cover problem consists of a ground set U and a family of subsets @math . Each element e∈U is associated with a profit p(e), whereas each subset @math has a cost c(S). The objective is to find a minimum cost subcollection @math such that the combined profit of the elements covered by @math is at least P, a specified profit bound. In the prize-collecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e∈U uncovered, we incur a penalty of π(e). The goal is to identify a subcollection @math that minimizes the cost of @math plus the penalties of uncovered elements. Although problem-specific connections between the partial cover and the prize-collecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first non-trivial approximability results. Let @math be a ground set of @math elements, and let @math be a family of subsets of @math , @math , with a positive cost @math associated with each @math . Consider the following online version of the set cover problem, described as a game between an algorithm and an adversary. An adversary gives elements to the algorithm from @math one by one. Once a new element is given, the algorithm has to cover it by some set of @math containing it. We assume that the elements of @math and the members of @math are known in advance to the algorithm; however, the set @math of elements given by the adversary is not known in advance to the algorithm. (In general, @math may be a strict subset of @math .) The objective is to minimize the total cost of the sets chosen by the algorithm. Let @math denote the family of sets in @math that the algorithm chooses. At the end of the game the adversary also produces (offline) a family of sets @math that covers @math . The performance of the algorithm is the ratio between the cost of @math and the cost of @math . The maximum ratio, taken over all input sequences, is the competitive ratio of the algorithm. We present an @math competitive deterministic algorithm for the problem and establish a nearly matching @math lower bound for all interesting values of @math and @math . The techniques used are motivated by similar techniques developed in computational learning theory for online prediction (e.g., the WINNOW algorithm) together with a novel way of converting a fractional solution into a deterministic online algorithm. Given a collection @math of weighted subsets of a ground set @math , the set cover problem is to find a minimum weight subset of @math which covers all elements of @math . We study a strategic game defined upon this classical optimization problem. Every element of @math is a player which chooses one set of @math where it appears. Following a public tax function, every player is charged a fraction of the weight of the set that it has selected. Our motivation is to design a tax function having the following features: it can be implemented in a distributed manner, existence of an equilibrium is guaranteed and the social cost for these equilibria is minimized. We study the problem of constructing universal Steiner trees for undirected graphs. Given a graph @math and a root node @math , we seek a single spanning tree @math of minimum stretch , where the stretch of @math is defined to be the maximum ratio, over all terminal sets @math , of the cost of the minimal sub-tree @math of @math that connects @math to @math to the cost of an optimal Steiner tree connecting @math to @math in @math . Universal Steiner trees (USTs) are important for data aggregation problems where computing the Steiner tree from scratch for every input instance of terminals is costly, as for example in low energy sensor network applications. We provide a polynomial time UST construction for general graphs with @math -stretch. We also give a polynomial time polylogarithmic-stretch construction for minor-free graphs. One basic building block of our algorithms is a hierarchy of graph partitions, each of which guarantees small strong diameter for each cluster and bounded neighbourhood intersections for each node. We show close connections between the problems of constructing USTs and building such graph partitions. Our construction of partition hierarchies for general graphs is based on an iterative cluster merging procedure, while the one for minor-free graphs is based on a separator theorem for such graphs and the solution to a cluster aggregation problem that may be of independent interest even for general graphs. To our knowledge, this is the first sub polynomial-stretch ( @math for any @math ) UST construction for general graphs, and the first polylogarithmic-stretch UST construction for minor-free graphs. We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most @math , the algorithm achieves a performance ratio of @math for large @math , which improves the previously known ratio of @math obtained by Halld orsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For k-uniform hypergraphs with n vertices, we achieve a ratio of @math for large n, and for k-uniform hypergraphs with maximum degree at most @math the algorithm achieves a ratio of @math for large @math . These results considerably improve the previous best ratio of @math for bounded degree k-uniform hypergraphs, and @math for general k-uniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldorsson. The NP-hard Subset Interconnection Design problem, also known as Minimum Topic-Connected Overlay, is motivated by numerous applications including the design of scalable overlay networks and vacuum systems. It has as input a finite set @math and a collection of subsets @math , and asks for a minimum-cardinality edge set @math such that for the graph @math all induced subgraphs @math are connected. We study Subset Interconnection Design in the context of polynomial-time data reduction rules that preserve the possibility of constructing optimal solutions. Our contribution is threefold: First, we show the incorrectness of earlier polynomial-time data reduction rules. Second, we show linear-time solvability in case of a constant number @math of subsets, implying fixed-parameter tractability for the parameter @math . Third, we provide a fixed-parameter tractability result for small subset sizes and tree-like output graphs. To achieve our results, we elaborate o... In this paper we study the problem of finding maximally sized subsets of binary strings (codes) of equal length that are immune to a given number @math of repetitions, in the sense that no two strings in the code can give rise to the same string after @math repetitions. We propose explicit number theoretic constructions of such subsets. In the case of @math repetition, the proposed construction is asymptotically optimal. For @math , the proposed construction is within a constant factor of the best known upper bound on the cardinality of a set of strings immune to @math repetitions. Inspired by these constructions, we then develop a prefixing method for correcting any prescribed number @math of repetition errors in an arbitrary binary linear block code. The proposed method constructs for each string in the given code a carefully chosen prefix such that the resulting strings are all of the same length and such that despite up to any @math repetitions in the concatenation of the prefix and the codeword, the original codeword can be recovered. In this construction, the prefix length is made to scale logarithmically with the length of strings in the original code. As a result, the guaranteed immunity to repetition errors is achieved while the added redundancy is asymptotically negligible. We prove two results regarding cycles in properly edge-colored graphs. First, we make a small improvement to the recent breakthrough work of Alon, Pokrovskiy and Sudakov who showed that every properly edge-colored complete graph @math on @math vertices has a rainbow cycle on at least @math vertices, by showing that @math has a rainbow cycle on at least @math vertices. Second, by modifying the argument of Hatami and Shor which gives a lower bound for the length of a partial transversal in a Latin Square, we prove that every properly colored complete graph has a Hamilton cycle in which at least @math different colors appear. For large @math , this is an improvement of the previous best known lower bound of @math of Andersen. We show how to approximate a data matrix @math with a much smaller sketch @math that can be used to solve a general class of constrained k-rank approximation problems to within @math error. Importantly, this class of problems includes @math -means clustering and unconstrained low rank approximation (i.e. principal component analysis). By reducing data points to just @math dimensions, our methods generically accelerate any exact, approximate, or heuristic algorithm for these ubiquitous problems. For @math -means dimensionality reduction, we provide @math relative error results for many common sketching techniques, including random row projection, column selection, and approximate SVD. For approximate principal component analysis, we give a simple alternative to known algorithms that has applications in the streaming setting. Additionally, we extend recent work on column-based matrix reconstruction, giving column subsets that not only cover' a good subspace for @math , but can be used directly to compute this subspace. Finally, for @math -means clustering, we show how to achieve a @math approximation by Johnson-Lindenstrauss projecting data points to just @math dimensions. This gives the first result that leverages the specific structure of @math -means to achieve dimension independent of input size and sublinear in @math . We propose approximation algorithms under game-theoretic considerations. We indroduce and study the general covering problem which is a natural generalization of the well-studied max-n -cover problem. In the general covering problem, we are given a universal set of weighted elements E and n collections of subsets of the elements. The task is to choose one subset from each collection such that the total weight of their union is as large as possible. In our game-theoretic setting, the choice in each collection is made by an independent player. For covering an element, the players receive a payoff defined by a non-increasing utility sharing function. This function defines the fraction that each covering player receives from the weight of the elements. We show how to construct a utility sharing function such that every Nash Equilibrium approximates the optimal solution by a factor of @math . We also prove that any sequence of unilateral improving steps is polynomially bounded. This gives rise to a polynomial-time local search approximation algorithm whose approximation ratio is best possible. We give efficient algorithms for volume sampling, i.e., for picking @math -subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). In other words, we can efficiently sample @math -subsets of @math with probabilities proportional to the corresponding @math by @math principal minors of any given @math by @math positive semi definite matrix. This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section @math of KV , also implicit in BDM, DRVW ). Our first algorithm for volume sampling @math -subsets of rows from an @math -by- @math matrix runs in @math arithmetic operations (where @math is the exponent of matrix multiplication) and a second variant of it for @math -approximate volume sampling runs in @math arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small @math . Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation: (1) Given @math , in @math arithmetic operations we can find @math of its rows such that projecting onto their span gives a @math -approximation to the matrix of rank @math closest to @math under the Frobenius norm. This improves the @math -approximation of Boutsidis, Drineas and Mahoney BDM and matches the lower bound shown in DRVW . The method of conditional expectations gives a algorithm with the same complexity. The running time can be improved to @math at the cost of losing an extra @math in the approximation factor. (2) The same rows and projection as in the previous point give a @math -approximation to the matrix of rank @math closest to @math under the spectral norm. In this paper, we show an almost matching lower bound of @math , even for @math . This paper makes two contributions to the areas of anti-concentration and non-asymptotic random matrix theory. First, we study the counting problem in inverse Littlewood-Offord theory for general random variables: for random variables @math which are i.i.d. copies of a random variable @math (satisfying some mild hypotheses), how many integer vectors @math in a prescribed box have large @math ? Building on recent work of Ferber, Jain, Luh, and Samotij (who treated the case when @math is a Rademacher random variable), we provide significantly better bounds for this problem than those obtained using the inverse Littlewood-Offord theorems of Tao and Vu, and Nguyen and Vu. Next, we study the non-asymptotic behavior of the least singular value @math of a random @math square matrix @math with i.i.d. entries, which are only assumed to be centered with variance @math . As an application of our counting theorem, and utilizing and developing a recent work by the author, we show that for all @math , @math , where @math is an absolute constant, thereby providing a considerably simpler proof of an approximate version of an essentially optimal result due to Rebrova and Tikhomirov. We study a natural extension of the Maximum Weight Independent Set Problem (MWIS), one of the most studied optimization problems in Graph algorithms. We are given a graph @math , a weight function @math , a budget function @math , and a positive integer @math . The weight (resp. budget) of a subset of vertices is the sum of weights (resp. budgets) of the vertices in the subset. A @math -budgeted independent set in @math is a subset of vertices, such that no pair of vertices in that subset are adjacent, and the budget of the subset is at most @math . The goal is to find a @math -budgeted independent set in @math such that its weight is maximum among all the @math -budgeted independent sets in @math . We refer to this problem as MWBIS. Being a generalization of MWIS, MWBIS also has several applications in Scheduling, Wireless networks and so on. Due to the hardness results implied from MWIS, we study the MWBIS problem in several special classes of graphs. We design exact algorithms for trees, forests, cycle graphs, and interval graphs. In unweighted case we design an approximation algorithm for @math -claw free graphs whose approximation ratio ( @math ) is competitive with the approximation ratio ( @math ) of MWIS (unweighted). Furthermore, we extend Baker's technique Baker83 to get a PTAS for MWBIS in planar graphs. This paper considers the Steiner tree problem in the model of two-stage stochastic optimization with recourse. This model, the focus of much recent research [11, 16, 8, 18], tries to capture the fact that many infrastructure planning problems have to be solved in the presence of uncertainty, and that we have make decisions knowing merely market forecasts (and not the precise set of demands); by the time the actual demands arrive, the costs may be higher due to inflation. In the context of the Stochastic Steiner Tree problem on a graph G = (V,E), the model can be paraphrased thus: on Monday, we are given a probability distribution π on subsets of vertices, and can build some subset EM of edges. On Tuesday, a set of terminals D materializes (drawn from the same distribution π). We now have to buy edges ET so that the set EM ∪ ET forms a Steiner tree on D. The goal is to minimize the expected cost of the solution. We give the first constant-factor approximation algorithm for this problem. To the best of our knowledge, this is the first O(1)-approximation for the stochastic version of a non sub-additive problem. In fact, algorithms for the unrooted stochastic Steiner tree problem we consider are powerful enough to solve the Multicommodity Rent-or-Buy problem, itself a topic of recent interest [3, 7, 15]. A partial Steiner system Sp(t, k, n) is a collection of k-subsets (i.e. subsets of size k) of n element set such that every t-subset is contained in at most one k-subset. To avoid trivial cases, we assume 2 ⩾ t < k < n. It is easy to see that the size of a partial Steiner system Sp(t, k, n) is at most (tn) (kt). Confirming a conjecture of Erdos and Hanani [5], Rodl [19] proved that for fixed positive integers t and k, there is a partial Steiner system Sp(t, k, n) of size at least (1 − o(1)) ( nt)( kt) where o(1) goes to 0 as n goes to infinity. Grable [7] found an explicit bound of the o(1) term above and it was improved by Kostochka and Rodl [15] to n−1 ((kt)+o(1)−1). All proofs used Rodl's nibble method. We also use the method to improve the bound to O(( log nn) k − t(( kt)−1) for 2 ⩾ t < k — 2. (For t = k — 1, slightly better bounds were already known. See [2].) We believe that the nibble method gives no better bound up to a logarithmic factor. A problem arising in the design of vacuum systems and having applications to some natural problems of interconnection design is described as follows. (1) Given a set X and subsets @math of @math , satisfying @math , find a graph G with vertex set X and the minimum number of edges such that for any i, the subgraph induced by @math has a connected component containing @math .Two other problems related to this one are the following ones. (2) Given a set X and subsets @math such that @math , find a graph G with vertex set X and the minimum number of edges such that for any i the subgraph @math induced by @math in G is connected. (3) Given a set X and subsets @math such that @math , find a graph G with vertex set X, find a graph G with vertex set X and the minimum number of edges such that for any subset I of @math , the subgraph induced by @math is co... A Fixed-Parameter Tractable ( ) @math -approximation algorithm for a minimization (resp. maximization) parameterized problem @math is an FPT algorithm that, given an instance @math computes a solution of cost at most @math (resp. @math ) if a solution of cost at most (resp. at least) @math exists; otherwise the output can be arbitrary. For well-known intractable problems such as the W[1]-hard Clique and W[2]-hard Set Cover problems, the natural question is whether we can get any -approximation. It is widely believed that both Clique and Set-Cover admit no FPT @math -approximation algorithm, for any increasing function @math . Assuming standard conjectures such as the Exponential Time Hypothesis (ETH) eth-paturi and the Projection Games Conjecture (PGC) r3 , we make the first progress towards proving this conjecture by showing that 1. Under the ETH and PGC, there exist constants @math such that the Set Cover problem does not admit an FPT approximation algorithm with ratio @math in @math time, where @math is the size of the universe and @math is the number of sets. 2. Unless @math , for every @math there exists a constant @math such that Clique has no FPT cost approximation with ratio @math in @math time, where @math is the number of vertices in the graph. In the second part of the paper we consider various W[1]-hard problems such as , , Directed Steiner Network and . For all these problem we give polynomial time @math -approximation algorithms for some small function @math (the largest approximation ratio we give is @math ).