A connection tree of a graph G for a terminal set W is a tree subgraph T of G such that leaves(T) subset of W subset of V(T). A non-terminal vertex is called linker if its degree in T is exactly 2, and it is called ro...
详细信息
A connection tree of a graph G for a terminal set W is a tree subgraph T of G such that leaves(T) subset of W subset of V(T). A non-terminal vertex is called linker if its degree in T is exactly 2, and it is called router if its degree in T is at least 3. The Terminal connection problem (TCP) asks whether G admits a connection tree for W with at most l linkers and at most r routers, while the Steiner tree problem asks whether G admits a connection tree for W with at most k non-terminal vertices. We prove that, if r >= 1 is fixed, then TCP is polynomial-time solvable when restricted to split graphs. This result separates the complexity of TCP from the complexity of Steiner tree, which is known to be NP-complete on split graphs. Additionally, we prove that TCP is NP-complete on strongly chordal graphs, even if r >= 0 is fixed, whereas Steiner tree is known to be polynomial-time solvable. We also prove that, when parameterized by clique-width, TCP is W[1]-hard, whereas STeiner tree is known to be in FPT. On the other hand, agreeing with the complexity of Steiner tree, we prove that TCP is linear-time solvable when restricted to cographs (i.e. graphs of clique-width 2). Finally, we prove that, even if either l >= 0 or r >= 0 is fixed, TCP remains NP-complete on graphs of maximum degree 3.
We formally treat cryptographic constructions based on the hardness of deciding ideal membership in multivariate polynomial rings. Of particular interest to us is a class of schemes known as "Polly Cracker."...
详细信息
We formally treat cryptographic constructions based on the hardness of deciding ideal membership in multivariate polynomial rings. Of particular interest to us is a class of schemes known as "Polly Cracker." We start by formalising and studying the relation between the ideal membership problem and the problem of computing a Grobner basis. We show both positive and negative results. On the negative side, we define a symmetric Polly Cracker encryption scheme and prove that this scheme only achieves bounded security under the hardness of the ideal membership problem. Furthermore, we show that a large class of algebraic transformations cannot convert this scheme to a fully secure Polly Cracker-style scheme. On the positive side, we formalise noisy variants of the ideal-theoretic problems. These problems can be seen as natural generalisations of the learning with errors () and the approximate GCD problems over polynomial rings. After formalising and justifying the hardness of the noisy assumptions, we show that noisy encoding of messages results in a fully -secure and somewhat homomorphic encryption scheme. Together with a standard symmetric-to-asymmetric transformation for additively homomorphic schemes, we provide a positive answer to the long-standing open problem of constructing a secure Polly Cracker-style cryptosystem reducible to the hardness of solving a random system of equations. Indeed, our results go beyond this and also provide a new family of somewhat homomorphic encryption schemes based on generalised hard problems. Our results also imply that Regev's -based public-key encryption scheme is (somewhat) multiplicatively homomorphic for appropriate choices of parameters.
A graph is 2K2-partitionable if its vertex set can be partitioned into four nonempty parts A, B, C, D such that each vertex of A is adjacent to each vertex of B, and each vertex of C is adjacent to each vertex of D. D...
详细信息
We study the concept of an H-partition of the vertex set of a graph G, which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the st...
详细信息
We study the concept of an H-partition of the vertex set of a graph G, which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the structure of a model graph H, with the exception of two cases, one that has already been classified as polynomial, and the other one remains unclassified. In the context of more general vertex-partition problems, the problems addressed in this paper have these properties: non-list, 4-part, external constraints only ( no internal constraints), each part non-empty. We describe tools that yield for each problem considered in this paper a simple and low complexity polynomial-time algorithm.
A skew partition is a partition of the vertex set of a graph into four nonempty parts A, B, C, D such that there are all possible edges between A and B, and no edges between C and D. A stable skew partition is a skew ...
详细信息
A skew partition is a partition of the vertex set of a graph into four nonempty parts A, B, C, D such that there are all possible edges between A and B, and no edges between C and D. A stable skew partition is a skew partition where A induces a stable set of the graph. We show that determining if a graph permits a stable skew partition is NP-complete. We discuss limits of such reductions by adding cardinality constraints. (C) 2004 Elsevier B.V. All rights reserved.
The goal of this paper is to point out the differences between jitter ( the perturbations in sampling points reading) and the measurement errors. In some cases jitter may have significantly smaller influence on the ra...
详细信息
The goal of this paper is to point out the differences between jitter ( the perturbations in sampling points reading) and the measurement errors. In some cases jitter may have significantly smaller influence on the radius of information than the measurement error. The class of Lipschitz functions is considered and two problems, integration and approximation, are studied.
Let G = (V,E') be a simple graph. The NON-PLANAR DELETION problem consists in finding a smallest subset E' subset of E such that H=(V,E\E') is a planar graph. The SPLITTING NUMBER problem consists in findi...
详细信息
Let G = (V,E') be a simple graph. The NON-PLANAR DELETION problem consists in finding a smallest subset E' subset of E such that H=(V,E\E') is a planar graph. The SPLITTING NUMBER problem consists in finding the smallest integer k greater than or equal to 0, such that a planar graph H can be defined from G by k vertex splitting operations. We establish the Max SNP-hardness of SPLITTING NUMBER and NON-PLANAR DELETION problems for cubic graphs. (C) 2003 Elsevier B.V. All rights reserved.
Let G = (V,E') be a simple graph. The NON-PLANAR DELETION problem consists in finding a smallest subset E' subset of E such that H=(V,E\E') is a planar graph. The SPLITTING NUMBER problem consists in findi...
详细信息
Let G = (V,E') be a simple graph. The NON-PLANAR DELETION problem consists in finding a smallest subset E' subset of E such that H=(V,E\E') is a planar graph. The SPLITTING NUMBER problem consists in finding the smallest integer k greater than or equal to 0, such that a planar graph H can be defined from G by k vertex splitting operations. We establish the Max SNP-hardness of SPLITTING NUMBER and NON-PLANAR DELETION problems for cubic graphs. (C) 2003 Elsevier B.V. All rights reserved.
暂无评论