313 research outputs found
Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT
AbstractThe maximum 2-satisfiability problem (MAX-2-SAT) is: given a Boolean formula in 2-CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-2-SAT is MAX-SNP-complete. Recently, this problem received much attention in the contexts of (polynomial-time) approximation algorithms and (exponential-time) exact algorithms. In this paper, we present an exact algorithm solving MAX-2-SAT in time poly(L)·2K/5, where K is the number of clauses and L is their total length. In fact, the running time is only poly(L)·2K2/5, where K2 is the number of clauses containing two literals. This bound implies the bound poly(L)·2L/10. Our results significantly improve previous bounds: poly(L)·2K/2.88 (J. Algorithms 36 (2000) 62–88) and poly(L)·2K/3.44 (implicit in Bansal and Raman (Proceedings of the 10th Annual Conference on Algorithms and Computation, ISAAC’99, Lecture Notes in Computer Science, Vol. 1741, Springer, Berlin, 1999, pp. 247–258.))As an application, we derive upper bounds for the (MAX-SNP-complete) maximum cut problem (MAX-CUT), showing that it can be solved in time poly(M)·2M/3, where M is the number of edges in the graph. This is of special interest for graphs with low vertex degree
Lower Bounds on the Oracle Complexity of Nonsmooth Convex Optimization via Information Theory
We present an information-theoretic approach to lower bound the oracle
complexity of nonsmooth black box convex optimization, unifying previous lower
bounding techniques by identifying a combinatorial problem, namely string
guessing, as a single source of hardness. As a measure of complexity we use
distributional oracle complexity, which subsumes randomized oracle complexity
as well as worst-case oracle complexity. We obtain strong lower bounds on
distributional oracle complexity for the box , as well as for the
-ball for (for both low-scale and large-scale regimes),
matching worst-case upper bounds, and hence we close the gap between
distributional complexity, and in particular, randomized complexity, and
worst-case complexity. Furthermore, the bounds remain essentially the same for
high-probability and bounded-error oracle complexity, and even for combination
of the two, i.e., bounded-error high-probability oracle complexity. This
considerably extends the applicability of known bounds
A Linear Algebra Approach for Detecting Binomiality of Steady State Ideals of Reversible Chemical Reaction Networks
Motivated by problems from Chemical Reaction Network Theory, we investigate
whether steady state ideals of reversible reaction networks are generated by
binomials. We take an algebraic approach considering, besides concentrations of
species, also rate constants as indeterminates. This leads us to the concept of
unconditional binomiality, meaning binomiality for all values of the rate
constants. This concept is different from conditional binomiality that applies
when rate constant values or relations among rate constants are given. We start
by representing the generators of a steady state ideal as sums of binomials,
which yields a corresponding coefficient matrix. On these grounds we propose an
efficient algorithm for detecting unconditional binomiality. That algorithm
uses exclusively elementary column and row operations on the coefficient
matrix. We prove asymptotic worst case upper bounds on the time complexity of
our algorithm. Furthermore, we experimentally compare its performance with
other existing methods
New methods for 3-SAT decision and worst-case analysis
We prove the worst-case upper bound 1:5045 n for the time complexity of 3-SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2- and 3-clauses, called "blocked clauses", generalizing the extension rule of "Extended Resolution." Our methods for estimating the size of trees lead to a refined measure of formula complexity of 3-clause-sets and can be applied also to arbitrary trees. Keywords: 3-SAT, worst-case upper bounds, analysis of algorithms, Extended Resolution, blocked clauses, generalized autarkness. 1 Introduction In this paper we study the exponential part of time complexity for 3-SAT decision and prove the worst-case upper bound 1:5044:: n for n the number of variables in the input formula, using new algorithmic methods as well as new methods for the analysis. These methods also deepen the already existing approaches in a systematically manner. The following results..
How Good Is Local Search for Capacitated Facility Location Problem: An Experimental Study
Facility location problems have been widely studied since 1960’s. These problems are known to be strongly NP-hard. In capacitated variant of the problem, a capacity constraint is associated with each facility. Capacitated facility location problem (CFLP) instances can be solved exactly using existing MILP solvers but only for small instance sizes. As the size of the problem instance increases beyond few hundred facilities and few hundred clients, it becomes prohibitive to solve these instances exactly. For large problem instances, therefore, other solution methods are used. One approach is to use heuristic methods. These methods usually give good solutions in reasonable time but they do not provide any guarantee about the quality of the solution. Somewhere between these two extremes exist another class of algorithms called approximation algorithms. They also provide only suboptimal solutions to the problem, like heuristic algorithms, in polynomial time. How ever they guarantee worst case upper bounds on the cost of the solution. So, a solution obtained using an approximation algorithm is guaranteed to have its cost between the optimal cost and the upper bound. We present experimental studies done with a local search based approximation algorithm for CFLP given by Bansal et al. [1] to show that this algorithm performs well in practice
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