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Computers And Intractability A Guide To The Theory Of Np Completeness

We present a fast algorithm for the subset convolution problem:given functions f and g defined on the lattice of subsets of ann-element set n, compute their subset convolution f*g, defined for S⊆ N by [ (f * g)(S) = [T ⊆ S] f(T) g(S/T),,]where addition and multiplication is carried out in an arbitrary ring. Via Mobius transform and inversion, our algorithm evaluates the subset convolution in O(n2 2n) additions and multiplications, substanti y improving upon the straightforward O(3n) algorithm. Specifically, if the input functions have aninteger range [-M,-M+1,...,M], their subset convolution over the ordinary sum--product ring can be computed in O(2n log M) time; the notation O suppresses polylogarithmic factors.Furthermore, using a standard embedding technique we can compute the subset convolution over the max--sum or min--sum semiring in O(2n M) time.To demonstrate the applicability of fast subset convolution, wepresent the first O(2k n2 + n m) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the O(3kn + 2k n2 + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent O(2n)-time algorithms for covering and partitioning problems (Bjorklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).

Fourier meets Möbius: fast subset convolution

In this paper we analyze the complexity of algorithms for two problems that arise in automatic test path generation for programs: the problem of building a path through a specified set of program statements and the problem of building a path which satisfies impossible-pairs restrictions on statement pairs. These problems are both reduced to graph traversal problems. We give an efficient algorithm for the first, and show that the second is NP-complete.

On Two Problems in the Generation of Program Test Paths ; CU-CS-081-75

Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration 
 $$\varDelta $$
 , referred to as $$\varDelta $$
 -restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the “restless variant” of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the distance to disjoint path of the underlying graph, which implies W[1]-hardness for many other parameters like feedback vertex number and pathwidth. A natural question is thus whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

/pdf/finding-temporal-paths-under-waiting-time-constraints-1e5m85kxng.pdf

Finding Temporal Paths Under Waiting Time Constraints

Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration $\Delta$, referred to as $\Delta$-restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the "restless variant" of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the distance to disjoint path of the underlying graph, which implies W[1]-hardness for many other parameters like feedback vertex number and pathwidth. A natural question is thus whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

The Computational Complexity of Finding Temporal Paths under Waiting Time Constraints.

Participatory budgeting systems allow city residents to jointly decide on projects they wish to fund using public money, by letting residents vote on such projects. While participatory budgeting is gaining popularity, existing aggregation methods do not take into account the natural possibility of project interactions, such as substitution and complementarity effects. Here we take a step towards fixing this issue: First, we augment the standard model of participatory budgeting by introducing a partition over the projects and model the type and extent of project interactions within each part using certain functions. We study the computational complexity of finding bundles that maximize voter utility, as defined with respect to such functions. Motivated by the desire to incorporate project interactions in real-world participatory budgeting systems, we identify certain cases that admit efficient aggregation in the presence of such project interactions.

/pdf/participatory-budgeting-with-project-interactions-2j1tab9ksj.pdf

Participatory Budgeting with Project Interactions

On minimizing vertex bisection using a memetic algorithm

In the mixed dominating set (mds) problem, we are given an n-vertex graph G and a positive integer k, and the objective is to decide whether there exists a set $S \subseteq V(G) \cup E(G)$ of cardinality at most k such that every element $x \in (V(G) \cup E(G)) \setminus S$ is either adjacent to or incident with an element of S. We show that mds can be solved in time ${7.465^k n^{\mathcal {O}(1)}} $ on general graphs, and in time $2^{\mathcal {O}(\sqrt{k})} n^{\mathcal {O}(1)}$ on planar graphs. We complement this result by showing that mds does not admit an algorithm with running time $2^{o(k)} n^{\mathcal {O}(1)}$ unless the Exponential Time Hypothesis (ETH) fails, and that it does not admit a polynomial kernel unless coNP $ \subseteq \mathsf{NP / poly}$. In addition, we provide an algorithm which, given a graph G together with a tree decomposition of width $\mathsf{tw}$, solves mds in time $6^{\mathsf{tw}} n^{\mathcal {O}(1)}$. We finally show that unless the Set Cover Conjecture (SeCoCo) fails, mds does not admit an algorithm with running time $\mathcal {O}((2-\epsilon )^{\mathsf{tw}(G)} n^{\mathcal {O}(1)})$ for any $\epsilon >0$, where $\mathsf{tw}(G)$ is the tree-width of G.

Mixed Dominating Set: A Parameterized Perspective

Vertex Bisection Minimization problem (VBMP) consists of partitioning a vertex set V of graph G = (V, E) into two sets B and B′ where ∣B∣ = such that vertex width (VW) is minimized where vertex width is defined as the number of vertices in B which are adjacent to at least one vertex in B′. It is an NP-complete problem in general. VBMP has applications in fault tolerance and is related to the complexity of sending messages to processors in interconnection networks via vertex disjoint paths. In this paper, we have proposed a new integer linear programming (ILP) and quadratically constrained quadratic programming (QCQP) formulation for VBMP. Both of them require number of variables and constraints lesser than existing ILPs and QCQP. We have also implemented ILP and obtained optimal results for various classes of graphs. The result of the experiments with the benchmark graphs shows that the proposed model outperforms the state of the art. Moreover, proposed model obtains optimal result for all the benchmark graphs.

A new Integer Linear Programming and Quadratically Constrained Quadratic Programming Formulation for Vertex Bisection Minimization Problem

Vertex Bisection Minimization problem (VBMP) consists of partitioning the vertex set V of a graph G = (V, E) into two sets B and B′ where $\left| B \right| \, = \left\lfloor {|V|/ 2} \right\rfloor$ such that its vertex width (VW) is minimized. Vertex width is defined as the number of vertices in B which are adjacent to at least one vertex in B′. It is an NP-complete problem in general but polynomially solvable for trees and hypercubes. VBMP has applications in fault tolerance and is related to the complexity of sending messages to processors in interconnection networks via vertex disjoint paths. In this paper, we propose a branch and bound algorithm for VBMP which uses a greedy heuristic to determine upper bound for the vertex width. We have devised a strategy to obtain lower bounds on the vertex width of partial solutions. A tree pruning procedure which reduces the size of search tree is also incorporated into the algorithm. This algorithm has been experimented on selected benchmark graphs. Results indicate that except for five of the selected graphs, the algorithm is able to, run through the search tree very fast.

Pallavi Jain

Papers

Participatory Budgeting with Project Interactions

On minimizing vertex bisection using a memetic algorithm

Mixed Dominating Set: A Parameterized Perspective

A new Integer Linear Programming and Quadratically Constrained Quadratic Programming Formulation for Vertex Bisection Minimization Problem

Branch and Bound Algorithm for Vertex Bisection Minimization Problem