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Computers And Intractability A Guide To The Theory Of Np Completeness

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Combinatorial Optimization Networks And Matroids

Thank you very much for reading computers and intractability a guide to the theory of np completeness. As you may know, people have look hundreds times for their favorite novels like this computers and intractability a guide to the theory of np completeness, but end up in infectious downloads. Rather than enjoying a good book with a cup of tea in the afternoon, instead they juggled with some harmful virus inside their desktop computer.

In many problems we want to reason about the ranking of items. For example, in information retrieval, when aggregating several search results, we may have ties and consequently rank orders. (e.g. [2, 3]). As a second example, we may wish to construct an overall ranking of tennis player based on pairwise comparisons between players. One principled method for constructing a ranking is the Kemeny distance [5] as this is the unique scheme that is neutral, consistent, and Condorcet. Unfortunately, determining this ranking is NP-hard, and remains so when we permit ties in the input or output [4]. As a third example, tasks in a scheduling problem may run in parallel, resulting in a ranking. In a ranking, unlike a permutation, we can have ties. Thus, 12225 is a ranking whilst 12345 is a permutation. To reason about permutations, we have efficient and effective global constraints. Regin [7] proposed an Oðn4Þ GAC propagator for permutations. For BC, there is an even faster Oðn log nÞ propagator [6]. Every constraint toolkit now provides propagators for permutation constraints. Surprisingly, ranking constraints are not yet supported. In [1], we tackle this weakness by proposing a global ranking constraint. We show that simple decompositions of this constraint hurt pruning. We then show that GAC can be achieved in polynomial time and we propose an Oðn3 log nÞ algorithm for achieving RC as well as an efficient quadratic algorithm offering a better tradeoff.

Principles and Practice of Constraint Programming

In the context of DPLL(T), theory propagation is the process of dynamically selecting consequences of a conjunction of constraints from a given set of candidate constraints. We present improvements to a fast theory propagation procedure for difference constraints of the form x - y < c. These improvements are demonstrated experimentally.

Fast and flexible difference constraint propagation for DPLL(T)

Finding read-once resolution refutations in systems of 2CNF clauses

In this paper, we discuss the design and analysis of polynomial time algorithms for two problems associated with a linearly infeasible system of Unit Two Variable Per Inequality (UTVPI) constraints, viz., (a) the read-once refutation (ROR) problem, and (b) the literal-once refutation (LOR) problem. Recall that a UTVPI constraint is a linear relationship of the form: $$a_{i}\cdot x_{i}+a_{j} \cdot x_{j} \le b_{ij}$$
 , where $$a_{i},a_{j} \in \{0,1,-1\}$$
 . A conjunction of such constraints is called a UTVPI constraint system (UCS) and can be represented in matrix form as: $$\mathbf{A \cdot x \le b}$$
 . These constraints find applications in a host of domains including but not limited to operations research and program verification. For the linear system $$\mathbf{A\cdot x \le b}$$
 , a refutation is a collection of m variables $$\mathbf{y}=[y_{1},y_{2},\ldots , y_{m}]^{T} \in \mathfrak {R}^{m}_{+}$$
 , such that $$\mathbf{y\cdot A =0}$$
 , $$\mathbf{y \cdot b } < 0$$
 . Such a refutation is guaranteed to exist for any infeasible linear program, as per Farkas’ lemma. The refutation is said to be read-once, if each $$y_{i} \in \{0,1\}$$
 . Read-once refutations are incomplete in that their existence is not guaranteed for infeasible linear programs, in general. Indeed they are not complete, even for UCSs. Hence, the question of whether an arbitrary UCS has an ROR is both interesting and non-trivial. In this paper, we reduce this problem to the problem of computing a minimum weight perfect matching (MWPM) in an undirected graph. This transformation results in an algorithm that runs in in time polynomial in the size of the input UCS. Additionally, we design a polynomial time algorithm (also via a reduction to the MWPM problem) for a variant of read-once resolution called literal-once resolution. The advantage of reducing our problems to the MWPM problem is that we can leverage recent advances in algorithm design for the MWPM problem towards solving the ROR and LOR problems in UCSs. Finally, we show that another variant of read-once refutation problem called the non-literal read-once refutation (NLROR) problem is NP-complete in UCSs.

A Polynomial Time Algorithm for Read-Once Certification of Linear Infeasibility in UTVPI Constraints

In this paper, we study two closely related problems on bipartite graphs, viz, the partial vertex cover problem and the budgeted maximum coverage problem Both these problems arise in a number of 

Partial Vertex Cover and Budgeted Maximum Coverage in Bipartite Graphs

http://www.csl.sri.com/users/gehani/papers/CN-2013.Risk.pdf

Analytical models for risk-based intrusion response

This paper is concerned with the design and analysis of improved algorithms for determining the optimal length resolution refutation (OLRR) of a system of difference constraints over an integral domain. The problem of finding short explanations for unsatisfiable Difference Constraint Systems (DCS) finds applications in a number of design domains including program verification, proof theory, real-time scheduling, and operations research. These explanations have also been called "certificates" and "refutations" in the literature. This problem was first studied in Subramani (J Autom Reason 43(2):121---137, 2009), wherein the first polynomial time algorithm was proposed. In this paper, we propose two new strongly polynomial algorithms which improve on the existing time bound. Our first algorithm, which we call the edge progression approach, runs in O(n2 · k + m · n · k) time, while our second algorithm, which we call the edge relaxation approach, runs in O(m · n · k) time, where m is the number of constraints in the DCS, n is the number of program variables, and k denotes the length of the shortest refutation. We conducted an extensive empirical analysis of the three OLRR algorithms discussed in this paper. Our experiments indicate that in the case of sparse graphs, the new algorithms discussed in this paper are superior to the algorithm in Subramani (J Autom Reason 43(2):121---137, 2009). Likewise, in the case of dense graphs, the approach in Subramani (J Autom Reason 43(2):121---137, 2009) is superior to the algorithms described in this paper. One surprising observation is the superiority of the edge relaxation algorithm over the edge progression algorithm in all cases, although both algorithms have the same asymptotic time complexity.

Improved algorithms for optimal length resolution refutation in difference constraint systems

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