Three men, each with a sister, must cross a river using a boat that can carry only two people in such a way that a sister is never left in the company of another man if her brother is not present. This very famous pro...
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Three men, each with a sister, must cross a river using a boat that can carry only two people in such a way that a sister is never left in the company of another man if her brother is not present. This very famous problem appeared in the Latin book "Problems to Sharpen the Young," one of the earliest collections of recreational mathematics. This paper considers a generalization of such "river crossing problems" and provides a new formulation that can treat wide variations. The main result is that, if there is no upper bound on the number of transportations (river crossings), a large class of subproblems can be solved in polynomialtime even when the passenger capacity of the boat is arbitrarily large. The authors speculated this fact at FUN 2012. On the other hand, this paper also demonstrates that, if an upper bound on the number of transportations is given, the problem is NP-hard even when the boat capacity is three, although a large class of subproblems can be solved in polynomialtime if the boat capacity is two.
Papadimitriou and Steiglitz constructed ‘traps’ for the symmetric travelling salesman problem (TSP) with n = 8 k cities. The constructed problem instances have exponentially many suboptimal solutions with arbitraril...
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Papadimitriou and Steiglitz constructed ‘traps’ for the symmetric travelling salesman problem (TSP) with n = 8 k cities. The constructed problem instances have exponentially many suboptimal solutions with arbitrarily large weight, which differ from the unique optimal solution in exactly 3 k edges, and hence local search algorithms are ineffective to solve this problem. However, we show that this class of ‘catastrophic’ examples can be solved by linear programming relaxation appended with k subtour elimination constraints. It follows that this class of problem instances of TSP can be optimized in polynomialtime.
In this paper, we study classical single machine scheduling problems with the additional constraint that a set of special jobs must be scheduled at certain positions in the job sequence. In other words, a special job ...
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In this paper, we study classical single machine scheduling problems with the additional constraint that a set of special jobs must be scheduled at certain positions in the job sequence. In other words, a special job must start when a certain number of jobs have finished on the machine. We analyze several classical objective functions for this more general setting with the additional constraint of fixed positioned jobs. We show that for some of them they are still polynomially solvable. Then, we focus on the objective of minimizing the number of tardy jobs. Considering the case of just one special job, this allows for a polynomial -time algorithm.
The haplotype inference problem (HIP) asks to find a set of haplotypes which resolve a given set of genotypes. This problem is important in practical fields such as the investigation of diseases or other types of gene...
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The haplotype inference problem (HIP) asks to find a set of haplotypes which resolve a given set of genotypes. This problem is important in practical fields such as the investigation of diseases or other types of genetic mutations. In order to find the haplotypes which are as close as possible to the real set of haplotypes that comprise the genotypes, two models have been suggested which are by now well-studied: The perfect phylogeny model and the pure parsimony model. All known algorithms up till now for haplotype inference may find haplotypes that are not necessarily plausible, i.e., very rare haplotypes or haplotypes that were never observed in the population. In order to overcome this disadvantage, we study in this paper, a new constrained version of HIP under the above-mentioned models. In this new version, a pool of plausible haplotypes (H) over tilde is given together with the set of genotypes G, and the goal is to find a subset H subset of (H) over tilde that resolves G. For constrained perfect phylogeny haplotyping (CPPH), we provide initial insights and polynomial-time algorithms for some restricted cases of the problem. For constrained parsimony haplotyping (CPH), we show that the problem is fixed parameter tractable when parameterized by the size of the solution set of haplotypes.
We consider the Directed Steiner Network problem, also called the POINT-TO-POINT CONNECTION problem. Given a directed graph G and p pairs {(s(1), t(1)),..., (s(p), t(p))} of nodes in the graph, one has to find the sma...
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We consider the Directed Steiner Network problem, also called the POINT-TO-POINT CONNECTION problem. Given a directed graph G and p pairs {(s(1), t(1)),..., (s(p), t(p))} of nodes in the graph, one has to find the smallest subgraph H of G that contains paths from s(i) to t(i) for all i. The problem is NP-hard for general p, since the DIRECTED STEINER Tree problem is a special case. Until now, the complexity was unknown for constant p >= 3. We prove that the problem is polynomially solvable if p is any constant number, even if nodes and edges in G are weighted and the goal is to minimize the total weight of the subgraph H. In addition, we give an efficient algorithm for the STRONGLY CONNECTED STEINER SUBGRAPH problem for any constant p, where given a directed graph and p nodes in the graph, one has to compute the smallest strongly connected subgraph containing the p nodes.
Network coding substantially increases network throughput. But since it involves mixing of information inside the network, a single corrupted packet generated by a malicious node can end up contaminating all the infor...
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Network coding substantially increases network throughput. But since it involves mixing of information inside the network, a single corrupted packet generated by a malicious node can end up contaminating all the information reaching a destination, preventing decoding. This paper introduces distributed polynomial-time rate-optimal network codes that work in the presence of Byzantine nodes. We present algorithms that target adversaries with different attacking capabilities. When the adversary can eavesdrop on all links and jam z(O) links, our first algorithm achieves a rate of C - 2z(O), where C is the network capacity. In contrast,, when the adversary has limited eavesdropping capabilities, we provide algorithms that achieve the higher rate of C - z(O). Our algorithms attain the optimal rate given the strength of the adversary. They are information-theoretically secure. They operate in a distributed manner, assume no knowledge of the topology, and can be designed and implemented in polynomialtime. Furthermore, only the source and destination need to be modified;nonmalicious nodes inside the network are oblivious to the presence of adversaries and implement a classical distributed network code. Finally, our algorithms work over wired and wireless networks.
The problem of computing minimum distortion embeddings of a given graph into a line (path) was introduced in 2004 and has quickly attracted significant attention with subsequent results appearing at recent STOC and SO...
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The problem of computing minimum distortion embeddings of a given graph into a line (path) was introduced in 2004 and has quickly attracted significant attention with subsequent results appearing at recent STOC and SODA conferences. So far, all such results concern approximation algorithms or exponential-time exact algorithms. We give the first polynomial-time algorithms for computing minimum distortion embeddings of graphs into a path when the input graphs belong to specific graph classes. In particular, we solve this problem in polynomialtime for bipartite permutation graphs and threshold graphs. For both graph classes, the distortion can be arbitrarily large. The graphs that we consider are unweighted. (C) 2011 Elsevier B.V. All rights reserved.
We study the complexity of telling whether a set of bit-vectors represents the set of all satisfying truth assignments of a Boolean expression of a certain type. We show that the problem is coNP-complete when the expr...
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We study the complexity of telling whether a set of bit-vectors represents the set of all satisfying truth assignments of a Boolean expression of a certain type. We show that the problem is coNP-complete when the expression is required to be in conjunctive normal form with three literals per clause (3CNF). We also prove a dichotomy theorem analogous to the classical one by Schaefer, stating that, unless P=NP, the problem can be solved in polynomialtime if and only if the clauses allowed are all Horn, or all anti-Horn, or all 2CNF, or all equivalent to equations modulo two.
A new interior point method for the solution of the linear programming problem is presented. It is shown that the method admits a polynomialtime bound. The method is based on the use of the trajectory of the problem,...
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A new interior point method for the solution of the linear programming problem is presented. It is shown that the method admits a polynomialtime bound. The method is based on the use of the trajectory of the problem, which makes it conceptually very simple. It has the advantage above related methods that it requires no problem transformation (either affine or projective) and that the feasible region may be unbounded. More importantly, the method generates at each stage solutions of both the primal and the dual problem. This implies that, contrary to the simplex method, the quality of the present solution is known at each stage. The paper also contains a practical (i.e., deepstep) version of the algorithm.
The spectrum omega(G) of a finite group G is the set of orders of elements of G. We present a polynomial-time algorithm that, given a finite set M of positive integers, outputs either an empty set or a finite simple g...
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The spectrum omega(G) of a finite group G is the set of orders of elements of G. We present a polynomial-time algorithm that, given a finite set M of positive integers, outputs either an empty set or a finite simple group G. In the former case, there is no finite simple group H with M = omega(H), while in the latter case, M subset of omega(G) and M not equal omega(H) for all finite simple groups H with omega(H) not equal omega(G). (C) 2019 Elsevier Inc. All rights reserved.
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