Greedy MAXCUT Algorithms and their Information Content

MAXCUT defines a classical NP-hard problem for graph partitioning and it serves as a typical case of the symmetric non-monotone Unconstrained Submodular Maximization (USM) problem. Applications of MAXCUT are abundant in machine learning, computer vision and statistical physics. Greedy algorithms to approximately solve MAXCUT rely on greedy vertex labelling or on an edge contraction strategy. These algorithms have been studied by measuring their approximation ratios in the worst case setting but very little is known to characterize their robustness to noise contaminations of the input data in the average case. Adapting the framework of Approximation Set Coding, we present a method to exactly measure the cardinality of the algorithmic approximation sets of five greedy MAXCUT algorithms. Their information contents are explored for graph instances generated by two different noise models: the edge reversal model and Gaussian edge weights model. The results provide insights into the robustness of different greedy heuristics and techniques for MAXCUT, which can be used for algorithm design of general USM problems.Comment: This is a longer version of the paper published in 2015 IEEE Information Theory Workshop (ITW

Approximation Bounds For Minimum Degree Matching

We consider the MINGREEDY strategy for Maximum Cardinality Matching. MINGREEDY repeatedly selects an edge incident with a node of minimum degree. For graphs of degree at most $\Delta$ we show that MINGREEDY achieves approximation ratio at least $\frac{\Delta-1}{2\Delta-3}$ in the worst case and that this performance is optimal among adaptive priority algorithms in the vertex model, which include many prominent greedy matching heuristics. Even when considering expected approximation ratios of randomized greedy strategies, no better worst case bounds are known for graphs of small degrees.Comment: % CHANGELOG % rev 1 2014-12-02 % - Show that the class APV contains many prominent greedy matching algorithms. % - Adapt inapproximability bound for APV-algorithms to a priori knowledge on |V|. % rev 2 2015-10-31 % - improve performance guarantee of MINGREEDY to be tigh

Collapsing Superstring Conjecture

In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits 2 11/23-approximation in polynomial time (Mucha, SODA\u2713). While this algorithm and its analysis are technically involved, the 30 years old Greedy Conjecture claims that the trivial and efficient Greedy Algorithm gives a 2-approximation for SCS. We develop a graph-theoretic framework for studying approximation algorithms for SCS. The framework is reminiscent of the classical 2-approximation for Traveling Salesman: take two copies of an optimal solution, apply a trivial edge-collapsing procedure, and get an approximate solution. In this framework, we observe two surprising properties of SCS solutions, and we conjecture that they hold for all input instances. The first conjecture, that we call Collapsing Superstring conjecture, claims that there is an elementary way to transform any solution repeated twice into the same graph G. This conjecture would give an elementary 2-approximate algorithm for SCS. The second conjecture claims that not only the resulting graph G is the same for all solutions, but that G can be computed by an elementary greedy procedure called Greedy Hierarchical Algorithm. While the second conjecture clearly implies the first one, perhaps surprisingly we prove their equivalence. We support these equivalent conjectures by giving a proof for the special case where all input strings have length at most 3 (which until recently had been the only case where the Greedy Conjecture was proven). We also tested our conjectures on millions of instances of SCS. We prove that the standard Greedy Conjecture implies Greedy Hierarchical Conjecture, while the latter is sufficient for an efficient greedy 2-approximate approximation of SCS. Except for its (conjectured) good approximation ratio, the Greedy Hierarchical Algorithm provably finds a 3.5-approximation, and finds exact solutions for the special cases where we know polynomial time (not greedy) exact algorithms: (1) when the input strings form a spectrum of a string (2) when all input strings have length at most 2

Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection

We study the problem of selecting a subset of k random variables from a large set, in order to obtain the best linear prediction of another variable of interest. This problem can be viewed in the context of both feature selection and sparse approximation. We analyze the performance of widely used greedy heuristics, using insights from the maximization of submodular functions and spectral analysis. We introduce the submodularity ratio as a key quantity to help understand why greedy algorithms perform well even when the variables are highly correlated. Using our techniques, we obtain the strongest known approximation guarantees for this problem, both in terms of the submodularity ratio and the smallest k-sparse eigenvalue of the covariance matrix. We further demonstrate the wide applicability of our techniques by analyzing greedy algorithms for the dictionary selection problem, and significantly improve the previously known guarantees. Our theoretical analysis is complemented by experiments on real-world and synthetic data sets; the experiments show that the submodularity ratio is a stronger predictor of the performance of greedy algorithms than other spectral parameters

Different Approximation Algorithms for Channel Scheduling in Wireless Networks

We introduce a new two-side approximation method for the channel scheduling problem, which controls the accuracy of approximation in two sides by a pair of parameters . We present a series of simple and practical-for-implementation greedy algorithms which give constant factor approximation in both sides. First, we propose four approximation algorithms for the weighted channel allocation problem: 1. a greedy algorithm for the multichannel with fixed interference radius scheduling problem is proposed and an one side -IS-approximation is obtained; 2. a greedy -approximation algorithm for single channel with fixed interference radius scheduling problem is presented; 3. we improve the existing algorithm for the multichannel scheduling and show an time -approximation algorithm; 4. we speed up the polynomial time approximation scheme for single-channel scheduling through merging two algorithms and show a -approximation algorithm. Next, we study two polynomial time constant factor greedy approximation algorithms for the unweighted channel allocation with variate interference radius. A greedy -approximation algorithm for the multichannel scheduling problem and an -approximation algorithm for single-channel scheduling problem are developed. At last, we do some experiments to verify the effectiveness of our proposed methods