De novo protein structure prediction from amino acid sequence is one of the most challenging problems in computational biology. As one of the extensively explored mathematical models for protein folding, Hydrophobic-Polar (HP) model enables thorough investigation of protein structure formation and evolution. Although HP model discretizes the conformational space and simplifies the folding energy function, it has been proven to be an NP-complete problem. In this paper, we propose a novel protein folding framework FoldingZero, self-folding a de novo protein 2D HP structure from scratch based on deep reinforcement learning. FoldingZero features the coupled approach of a two-head (policy and value heads) deep convolutional neural network (HPNet) and a regularized Upper Confidence Bounds for Trees (R-UCT). It is trained solely by a reinforcement learning algorithm, which improves HPNet and R-UCT iteratively through iterative policy optimization. Without any supervision and domain knowledge, FoldingZero not only achieves comparable results, but also learns the latent folding knowledge to stabilize the structure. Without exponential computation, FoldingZero shows promising potential to be adopted for real-world protein properties prediction.
Approximation algorithms offer rigorous mathematical tools and fold a protein structure within polynomial time. However, it may lead to a weak approximation ratio, resulting in a structure far from the optimal solution. Hart and Istrail proposed an approximation algorithm with ratio 3 8 of the optimal score for the 3D cubic lattice structure @cite_16 . An improved approximation algorithm with 2 5 performance guarantees was further developed by the same authors @cite_1 . For the 2D square lattice, an approximation algorithm @cite_7 can achieve the approximation ratio of 1 3.
@cite_16 @cite_1 @cite_7
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We consider the problem of protein folding in the HP model on the two-dimensional square lattice. This problem is combinatorially equivalent to folding a string of 0's and 1's so that the string forms a self-avoiding walk on the lattice and the number of adjacent pairs of 1's is maximized. We present a linear-time 1 3-approximation algorithm for this problem, improving on the previous best approximation factor of 1 4. The approximation guarantee of this algorithm is based on an upper bound presented by Hart and Istrail [6] and used in all previous papers that address this problem. We show that this upper bound cannot be used to obtain an approximation factor better than 1 2. We prove new results for approximating the graph-TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graph-TSP itself, we improve the approximation ratio to 7=5. For a generalization, the minimum T-tour problem, we obtain the first nontrivial approximation algorithm, with ratio 3=2. This contains the s-t-path graph-TSP as a special case. Our approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4=3. The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios. ABSTRACT This paper considers the protein energy minimization problem for lattice and off-lattice protein folding models that explicitly represent side chains. Lattice models of proteins have proven useful tools for reasoning about protein folding in unrestricted continuous space through analogy. This paper provides the first illustration of how rigorous algorithmic analyses of lattice models can lead to rigorous algorithmic analyses of off-lattice models. We consider two side chain models: a lattice model that generalizes the HP model (Dill, 1985) to explicitly represent side chains on the cubic lattice and a new off-lattice model, the HP Tangent Spheres Side Chain model (HP-TSSC), that generalizes this model further by representing the backbone and side chains of proteins with tangent spheres. We describe algorithms with mathematically guaranteed error bounds for both of these models. In particular, we describe a linear time performance guaranteed approximation algorithm for the HP side chain model that... We prove new results for approximating the graphic TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graphic TSP itself, we improve the approximation ratio to 7 5. For a generalization, the connected- @math -join problem, we obtain the first nontrivial approximation algorithm, with ratio 3 2. This contains the graphic @math - @math -path-TSP as a special case. Our improved approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4 3. The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios. In this paper, we revisit the classical k-median problem. Using the standard LP relaxation for k-median, we give an efficient algorithm to construct a probability distribution on sets of k centers that matches the marginals specified by the optimal LP solution. Analyzing the approximation ratio of our algorithm presents significant technical difficulties: we are able to show an upper bound of 3.25. While this is worse than the current best known 3+e guarantee of [2], because: (1) it leads to 3.25 approximation algorithms for some generalizations of the k-median problem, including the k-facility location problem introduced in [10], (2) our algorithm runs in @math time to achieve 3.25(1+δ)-approximation compared to the O(n8) time required by the local search algorithm of [2] to guarantee a 3.25 approximation, and (3) our approach has the potential to beat the decade old bound of 3+e for k-median. We also give a 34-approximation for the knapsack median problem, which greatly improves the approximation constant in [13]. Using the same technique, we also give a 9-approximation for matroid median problem introduced in [11], improving on their 16-approximation. We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s,t -tour out of those constructed from a family @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math for such an s,t -tour, which is the first improvement after the 5 3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13 8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8 5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x* 2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) and its complement, which induces a decomposition of x*. This decomposition can be refined and then efficiently used to complete x* 2 without using linear programming or particular properties of T, but by adding to each cut deficient for x* 2 an individually tailored explicitly given vector, inherent in x*. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s,t -tour than 3 2 times the incidentally common optima of the problem and of its fractional relaxation. The best approximation algorithm for Max Cut in graphs of maximum degree 3 uses semidefinite programming, has approximation ratio 0.9326, and its running time is @Q(n^3^.^5logn); but the best combinatorial algorithms have approximation ratio 4 5 only, achieved in O(n^2) time [J.A. Bondy, S.C. Locke, J. Graph Theory 10 (1986) 477-504; E. Halperin, et al, J. Algorithms 53 (2004) 169-185]. Here we present an improved combinatorial approximation, which is a 5 6-approximation algorithm that runs in O(n^2) time, perhaps improvable even to O(n). Our main tool is a new type of vertex decomposition for graphs of maximum degree 3. A local-ratio theorem for approximating the weighted vertex cover problem is presented. It consists of reducing the weights of vertices in certain subgraphs and has the effect of local-approximation. Putting together the Nemhauser-Trotter local optimization algorithm and the local-ratio theorem yields several new approximation techniques which improve known results from time complexity, simplicity and performance-ratio point of view. The main approximation algorithm guarantees a ratio of where K is the smallest integer s.t. † This is an improvement over the currently known ratios, especially for a “practical” number of vertices (e.g. for graphs which have less than 2400, 60000, 10 12 vertices the ratio is bounded by 1.75, 1.8, 1.9 respectively). We obtain a 1.5-approximation algorithm for the metric uncapacitated facility location problem (UFL), which improves on the previously best known 1.52-approximation algorithm by Mahdian, Ye and Zhang. Note, that the approximability lower bound by Guha and Khuller is 1.463. An algorithm is a ( @math , @math )-approximation algorithm if the solution it produces has total cost at most @math , where @math and @math are the facility and the connection cost of an optimal solution. Our new algorithm, which is a modification of the @math -approximation algorithm of Chudak and Shmoys, is a (1.6774,1.3738)-approximation algorithm for the UFL problem and is the first one that touches the approximability limit curve @math established by Jain, Mahdian and Saberi. As a consequence, we obtain the first optimal approximation algorithm for instances dominated by connection costs. When combined with a (1.11,1.7764)-approximation algorithm proposed by , and later analyzed by , we obtain the overall approximation guarantee of 1.5 for the metric UFL problem. We also describe how to use our algorithm to improve the approximation ratio for the 3-level version of UFL. We consider the problem of finding dense subgraphs with specified upper or lower bounds on the number of vertices. We introduce two optimization problems: the densest at-least-k-subgraph problem (dalks), which is to find an induced subgraph of highest average degree among all subgraphs with at least k vertices, and the densest at-most-k-subgraph problem (damks), which is defined similarly. These problems are relaxed versions of the well-known densest k-subgraph problem (dks), which is to find the densest subgraph with exactly k vertices. Our main result is that dalks can be approximated efficiently, even for web-scale graphs. We give a (1 3)-approximation algorithm for dalks that is based on the core decomposition of a graph, and that runs in time O(m + n), where n is the number of nodes and m is the number of edges. In contrast, we show that damks is nearly as hard to approximate as the densest k-subgraph problem, for which no good approximation algorithm is known. In particular, we show that if there exists a polynomial time approximation algorithm for damks with approximation ratio γ, then there is a polynomial time approximation algorithm for dks with approximation ratio γ 2 8. In the experimental section, we test the algorithm for dalks on large publicly available web graphs. We observe that, in addition to producing near-optimal solutions for dalks, the algorithm also produces near-optimal solutions for dks for nearly all values of k. Abstract In this paper we study the Steiner tree problem in graphs for the case when vertices as well as edges have weights associated with them. A greedy approximation algorithm based on “spider decompositions” was developed by Klein and Ravi for this problem. This algorithm provides a worst case approximation ratio of 2 ln k , where k is the number of terminals. However, the best known lower bound on the approximation ratio is (1− o (1)) ln k , assuming that NP ⊈ DTIME [ n O (log log n ) ], by a reduction from set cover. We show that for the unweighted case we can obtain an approximation factor of ln k . For the weighted case we develop a new decomposition theorem and generalize the notion of “spiders” to “branch-spiders” that are used to design a new algorithm with a worst case approximation factor of 1.5 ln k . We then generalize the method to yield an approximation factor of (1.35+ e ) ln k , for any constant e >0. These algorithms, although polynomial, are not very practical due to their high running time, since we need to repeatedly find many minimum weight matchings in each iteration. We also develop a simple greedy algorithm that is practical and has a worst case approximation factor of 1.6103 ln k . The techniques developed for this algorithm imply a method of approximating node weighted network design problems defined by 0–1 proper functions as well. These new ideas also lead to improved approximation guarantees for the problem of finding a minimum node weighted connected dominating set. The previous best approximation guarantee for this problem was 3 ln n by Guha and Khuller. By a direct application of the methods developed in this paper we are able to develop an algorithm with an approximation factor of (1.35+ e ) ln n for any fixed e >0. We present a deterministic (1+√5 2)-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on @math vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5 3-approximation algorithm for this problem, and this asymptotically tight bound in fact had been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of 1+√5 2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP. We analyze approximation algorithms for several variants of the traveling salesman problem with multiple objective functions. First, we consider the symmetric TSP (STSP) with γ-triangle inequality. For this problem, we present a deterministic polynomial-time algorithm that achieves an approximation ratio of @math and a randomized approximation algorithm that achieves a ratio of @math . In particular, we obtain a 2+e approximation for multi-criteria metric STSP. Then we show that multi-criteria cycle cover problems admit fully polynomial-time randomized approximation schemes. Based on these schemes, we present randomized approximation algorithms for STSP with γ-triangle inequality (ratio @math ), asymmetric TSP (ATSP) with γ-triangle inequality (ratio @math ), STSP with weights one and two (ratio 4 3) and ATSP with weights one and two (ratio 3 2). Submodularity is one of the most important properties of combinatorial optimization, and k-submodularity is a generalization of submodularity. Maximization of a k-submodular function is NP-hard, and approximation algorithms have been studied. Most of algorithms use randomization and achieve the approximation ratio as the expected value. For unconstrained submodular maximization, [Buchbinder and Feldman 2016] gave a derandomization scheme, and showed that randomness is not necessary for obtaining optimal results. In this paper, we extend their scheme for unconstrained k-submodular maximization and give two deterministic algorithms for k-submodular maximization. The first is for monotone k-submodular functions. We derandomize (k (2k-1) )-approximation algorithm [Iwata, Tanigawa, and Yoshida 2016], and achieve the same approximation ratio. The second is for nonmonotone functions with (k 2 ). It achieves (k (3k-2) )-approximation, and it is better than a simple greedy algorithm [Ward and Živný 2016] which achieves 1 3-approximation for (k 3 ). Given a set Q of squares with positive profits, the square packing problem is to select and pack a subset of squares of maximum profit into a rectangular bin R. We present a polynomial time approximation scheme for this problem, that for any value Ɛ > 0 finds and packs a subset Q′ ⊆ Q of profit at least (1 - Ɛ)OPT, where OPT is the profit of an optimum solution. This settles the approximability of the problem and improves on the previously best approximation ratio of 5 4 +Ɛ achieved by Harren's algorithm. We consider the problem of designing a revenue-maximizing auction for a single item, when the values of the bidders are drawn from a correlated distribution. We observe that there exists an algorithm that finds the optimal randomized mechanism that runs in time polynomial in the size of the support. We leverage this result to show that in the oracle model introduced by Ronen and Saberi [FOCS'02], there exists a polynomial time truthful in expectation mechanism that provides a (1.5+e)-approximation to the revenue achievable by an optimal truthful-in-expectation mechanism, and a polynomial time deterministic truthful mechanism that guarantees 5 3 approximation to the revenue achievable by an optimal deterministic truthful mechanism. We show that the 5 3-approximation mechanism provides the same approximation ratio also with respect to the optimal truthful-in-expectation mechanism. This shows that the performance gap between truthful-in-expectation and deterministic mechanisms is relatively small. En route, we solve an open question of Mehta and Vazirani [EC'04]. Finally, we extend some of our results to the multi-item case, and show how to compute the optimal truthful-in-expectation mechanisms for bidders with more complex valuations. We present improved combinatorial approximation algorithms for the uncapacitated facility location and k-median problems. Two central ideas in most of our results are cost scaling and greedy improvement. We present a simple greedy local search algorithm which achieves an approximation ratio of 2.414+ spl epsiv in O spl tilde (n sup 2 spl epsiv ) time. This also yields a bicriteria approximation tradeoff of (1+ spl gamma , 1+2 spl gamma ) for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible. Combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to K. Jain and V. Vazirani (1999), we get an approximation ratio of 1.853 in O spl tilde (n sup 3 ) time. This is already very close to the approximation guarantee of the best known algorithm which is LP-based. Further combined with the best known LP-based algorithm for facility location, we get a very slight improvement in the approximation factor for facility location, achieving 1.728. We present improved approximation algorithms for capacitated facility location and a variant. We also present a 4-approximation for the k-median problem, using similar ideas, building on the 6-approximation of Jain and Vazirani. The algorithm runs in O spl tilde (n sup 3 ) time. The maximin share guarantee is, in the context of allocating indivisible goods to a set of agents, a recent fairness criterion. A solution achieving a constant approximation of this guarantee always exists and can be computed in polynomial time. We extend the problem to the case where the goods collectively received by the agents satisfy a matroidal constraint. Polynomial approximation algorithms for this generalization are provided: a 1 2-approximation for any number of agents, a ((1- ) )-approximation for two agents, and a ((8 9- ) )-approximation for three agents. Apart from the extension to matroids, the ((8 9- ) )-approximation for three agents improves on a ((7 8- ) )-approximation by (ICALP 2015). The smallest k-ECSS problem is, given a graph along with an integer k, find a spanning subgraph that is k-edge connected and contains the fewest possible number of edges. We examine a natural approximation algorithm based on rounding an LP solution. A tight bound on the approximation ratio is 1 + 3-k for undirected graphs with k > 1 odd, 1 + 2-k for undirected graphs with k even, and 1 + 2-k for directed graphs with k arbitrary. Using iterated rounding improves the first upper bound to 1 + 2-k. On the hardness side we show that for some absolute constant c > 0, for any integer k ≥ 2 (k ≥ 1), a polynomial-time algorithm approximating the smallest k-ECSS on undirected (directed) multigraphs to within ratio 1 + c-k would imply P = NP. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009 k-means++ [5] seeding procedure is a simple sampling based algorithm that is used to quickly find k centers which may then be used to start the Lloyd’s method. There has been some progress recently on understanding this sampling algorithm. [10] showed that if the data satisfies the separation condition that ( k-1 (P) (P) c ) (Δ i (P) is the optimal cost w.r.t. i centers, c > 1 is a constant, and P is the point set), then the sampling algorithm gives an O(1)-approximation for the k-means problem with probability that is exponentially small in k. Here, the distance measure is the squared Euclidean distance. Ackermann and Blomer [2] showed the same result when the distance measure is any μ-similar Bregman divergence. Arthur and Vassilvitskii [5] showed that the k-means++ seeding gives an O(logk) approximation in expectation for the k-means problem. They also give an instance where k-means++ seeding gives Ω(logk) approximation in expectation. However, it was unresolved whether the seeding procedure gives an O(1) approximation with probability ( ( 1 poly(k) ) ), even when the data satisfies the above-mentioned separation condition. Brunsch and Roglin [8] addressed this question and gave an instances on which k-means++ achieves an approximation ratio of (2 3 − e) ·logk only with exponentially small probability. However, the instances that they give satisfy ( k-1 (P) (P) = 1+o(1) ). In this work, we show that the sampling algorithm gives an O(1) approximation with probability ( ( 1 k ) ) for any k-means problem instance where the point set satisfy separation condition ( k-1 (P) (P) 1 + ), for some fixed constant γ. Our results hold for any distance measure that is a metric in an approximate sense. For point sets that do not satisfy the above separation condition, we show O(1) approximation with probability Ω(2− 2k ).