Greedy algorithm

About: Greedy algorithm is a research topic. Over the lifetime, 15347 publications have been published within this topic receiving 393945 citations.

Sublinear algorithms for (Δ + 1) vertex coloring

Abstract: Any graph with maximum degree Δ admits a proper vertex coloring with Δ + 1 colors that can be found via a simple sequential greedy algorithm in linear time and space. But can one find such a coloring via a sublinear algorithm?We answer this fundamental question in the affirmative for several canonical classes of sublinear algorithms including graph streaming, sublinear time, and massively parallel computation (MPC) algorithms. In particular, we design:• A single-pass semi-streaming algorithm in dynamic streams using O(n) space. The only known semi-streaming algorithm prior to our work was a folklore O(log n)-pass algorithm obtained by simulating classical distributed algorithms in the streaming model.• A sublinear-time algorithm in the standard query model that allows neighbor queries and pair queries using [MATH HERE] time. We further show that any algorithm that outputs a valid coloring with sufficiently large constant probability requires [MATH HERE] time. No non-trivial sublinear time algorithms were known prior to our work.• A parallel algorithm in the massively parallel computation (MPC) model using O(n) memory per machine and O(1) MPC rounds. Our number of rounds significantly improves upon the recent O(log log Δ · log* (n))-round algorithm of Parter [ICALP 2018].At the core of our results is a remarkably simple meta-algorithm for the (Δ + 1) coloring problem: Sample O(log n) colors for each vertex independently and uniformly at random from the Δ + 1 colors; find a proper coloring of the graph using only the sampled colors of each vertex. As our main result, we prove that the sampled set of colors with high probability contains a proper coloring of the input graph. The sublinear algorithms are then obtained by designing efficient algorithms for finding a proper coloring of the graph from the sampled colors in each model.We note that all our upper bound results for (Δ + 1) coloring are either optimal or close to best possible in each model studied. We also establish new lower bounds that rule out the possibility of achieving similar results in these models for the closely related problems of maximal independent set and maximal matching. Collectively, our results highlight a sharp contrast between the complexity of (Δ+1) coloring vs maximal independent set and maximal matching in various models of sublinear computation even though all three problems are solvable by a simple greedy algorithm in the classical setting.

The fixed charge facility location problem with coverage restrictions

Abstract: This paper develops a fixed charge facility location model with coverage restrictions, minimizing cost while maintaining an appropriate level of service, in identifying facility locations Further, it discusses the insights that can be gained using the model Two Lagrangian relaxation based heuristics are presented and tested Both heuristics use a greedy adding algorithm to calculate upper bounds and subgradient optimization to calculate lower bounds While both procedures are capable of generating good solutions, one is computationally superior

Stowage planning in maritime container transportation

Abstract: We consider a stowage-planning problem of arranging containers on a container ship in the maritime transportation system. Since containers are accessible only from the top of the stack, temporary unloading and reloading of containers, called shifting, is unavoidable if a container required to be unloaded at the current port is stacked under containers to be unloaded at later ports on the route of the ship. The objective of the stowage planning problem is to minimize the time required for shifting and crane movements on a tour of a container ship while maintaining the stability of the ship. For the problem, we develop a heuristic solution method in which the problem is divided into two subproblems, one for assigning container groups into the holds and one for determining a loading pattern of containers assigned to each hold. The former subproblem is solved by a greedy heuristic based on the transportation simplex method, while the latter is solved by a tree search method. These two subproblems are solved iteratively using information obtained from solutions of each other. To see the performance of the suggested algorithm, computational tests are performed on problem instances generated based on information obtained from an ocean container liner. Results show that the suggested algorithm works better than existing algorithms.

Minimization and Reliability Analyses of Attack Graphs

Abstract: : An attack graph is a succinct representation of all paths through a system that end in a state where an intruder has successfully achieved his goal. Today Red Teams determine the vulnerability of networked systems by drawing gigantic attack graphs by hand. Constructing attack graphs by hand is tedious, error-prone, and impractical for large systems. By viewing an attack as a violation of a safety property, we can use model checking to produce attack graphs automatically: a successful path from the intruder's viewpoint is a counterexample produced by the model checker. In this paper we present an algorithm for generating attack graphs using model checking. Security analysts use attack graphs for detection, defense, and forensics. In this paper we present a minimization technique that allows analysts to decide which minimal set of security measures would guarantee the safety of the system. We provide a formal characterization of this problem: we prove that it is polynomially equivalent to the minimum hitting set problem and we present a greedy algorithm with provable bounds. We also present a reliability technique that allows analysts to perform a simple cost-benefit analysis depending on the likelihoods of attacks. By interpreting attack graphs as Markov Decision Processes we can use a standard MDP value iteration algorithm to compute the probabilities of intruder success for each attack the graph. We illustrate our work in the context of a small example that includes models of a firewall and an intrusion detection system.