Discrete Optimization最新文献

In this paper we present the notion of greyscale of a graph as a colouring of its vertices that uses colours from the real interval [0,1]. Any greyscale induces another colouring by assigning to each edge the non-negative difference between the colours of its vertices. These edge colours are ordered in lexicographical decreasing ordering and give rise to a new element of the graph: the gradation vector. We introduce the notion of minimum gradation vector as a new invariant for the graph and give polynomial algorithms to obtain it. These algorithms also output all greyscales that produce the minimum gradation vector. This way we tackle and solve a novel vectorial optimization problem in graphs that may generate more satisfactory solutions than those generated by known scalar optimization approaches.

We present a fully polynomial-time approximation scheme (FPTAS) for a very general version of the well-known knapsack problem. This generalization covers, with few exceptions, all versions of knapsack problems that have been studied in the literature so far and allows for an objective function consisting of sums or products of possibly nonlinear, separable item profits, while the knapsack constraint states an upper bound on the sum of possibly nonlinear, separable item weights. Moreover, we extend our FPTAS to a multi-objective fully polynomial-time approximation scheme (MFPTAS) for the multi-objective version of the problem.

As applications of our general algorithms, we obtain the first FPTAS for the recently-introduced 0–1 time-bomb knapsack problem as well as FPTASs for a variety of robust knapsack problems. Moreover, we extend our FPTAS to the minimization version of our general problem, which, in particular, allows us to explicitly state an FPTAS for the classical minimization knapsack problem, which has been missing in the literature so far.

Most practical scheduling applications involve some uncertainty about the arriving times and lengths of the jobs. Stochastic online scheduling is a well-established model capturing this. Here the arrivals occur online, while the processing times are random. For this model, Gupta, Moseley, Uetz, and Xie recently devised an efficient policy for non-preemptive scheduling on unrelated machines with the objective to minimize the expected total weighted completion time. We improve upon this policy by adroitly combining greedy job assignment with ${\alpha }_j$-point scheduling on each machine. In this way we obtain a $(3+\sqrt{5})(2+\Delta )$-competitive deterministic and an $(8+4\Delta )$-competitive randomized stochastic online scheduling policy, where $\Delta$ is an upper bound on the squared coefficients of variation of the processing times. We also give constant performance guarantees for these policies within the class of all fixed-assignment policies. The ${\alpha }_j$-point scheduling on a single machine can be enhanced when the upper bound $\Delta$ is known a priori or the processing times are known to be $\delta$-NBUE for some $\delta \ge 1$. This implies improved competitive ratios for unrelated machines but may also be of independent interest.

In this paper, we introduce the concept of saver variables in Max Sat and demonstrate their contribution to the performance of solvers for this problem. We present two types of saver variables: high-rank savers and consensual savers. We show how to incorporate them in various ways into an iterated algorithm, CHAMP, for Max Sat. We conduct an extensive empirical evaluation on two collections of instances — instances from a past Max Sat competition and random instances. It turns out that, by using savers, the number of unsatisfied clauses may be reduced by more than 70% in some families. Moreover, a refined version CHAMP＋ of CHAMP improves the results even further. We show that by combining CHAMP＋ with CCLS, a state-of-the-art solver, we obtain better solutions for many Max Sat instances.

Fundamental to many applications in data analysis are the decompositions of a graph, i.e. partitions of the node set into component-inducing subsets. One way of encoding decompositions is by multicuts, the subsets of those edges that straddle distinct components. Recently, a lifting of multicuts from a graph $G=(V,E)$ to an augmented graph $\hat{G}=(V,E\cup F)$ has been proposed in the field of image analysis, with the goal of obtaining a more expressive characterization of graph decompositions in which it is made explicit also for pairs $F\subseteq \left(\frac{V}{2}\right)\setminus E$ of non-neighboring nodes whether these are in the same or distinct components. In this work, we study in detail the polytope in $R^{E\cup F}$ whose vertices are precisely the characteristic vectors of multicuts of $\hat{G}$ lifted from $G$, connecting it, in particular, to the rich body of prior work on the clique partitioning and multilinear polytope.

Consider a vertex-weighted graph $G$ with a source $s$ and a target $t$. Tracking Paths requires finding a minimum weight set of vertices (trackers) such that the sequence of trackers in each path from $s$ to $t$ is unique. In this work, we derive a factor 6-approximation algorithm for Tracking Paths in weighted graphs and a factor 4-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely related r-Fault Tolerant Feedback Vertex Set problem. There, for a fixed integer $r$ and a given vertex-weighted graph $G$, the task is to find a minimum weight set of vertices intersecting every cycle of $G$ in at least $r+1$ vertices. We give a factor $O\left(r\right)$ approximation algorithm for r-Fault Tolerant Feedback Vertex Set if $r$ is a constant.

We present a new algorithm, Fractional Decomposition Tree (FDT), for finding a feasible solution for an integer program (IP) where all variables are binary. FDT runs in polynomial time and is guaranteed to find a feasible integer solution provided the integrality gap of an instance’s polyhedron, independent of objective function, is bounded. The algorithm gives a construction for Carr and Vempala’s theorem that any feasible solution to the IP’s linear-programming relaxation, when scaled by the instance integrality gap, dominates a convex combination of feasible solutions. FDT is also a tool for studying the integrality gap of IP formulations. The upper bound on the integrality gap of an FDT solution can be exponentially large. However our experiments demonstrate that FDT can be effective in practice. We study the integrality gap of two problems: optimally augmenting a tree to a 2-edge-connected graph and finding a minimum-cost 2-edge-connected multi-subgraph (2EC). We also give a simplified algorithm, DomToIP, that finds a feasible solution to an IP instance, or concludes that it has unbounded integrality gap. We show that FDT’s speed and approximation quality compare well to that of the original feasibility pump heuristic on moderate-sized instances of the vertex cover problem. For a particular set of hard-to-decompose fractional 2EC solutions, FDT always gave a better integer solution than the Best-of-Many Christofides Algorithm (BOMC).

Leaky-forcing is a recently introduced variant of zero forcing that has been studied for families of graphs including paths, cycles, wheels, grids, and trees. In this paper, we extend previous results on the leaky forcing number of the $d$-dimensional hypercube, $Q_d$, to show that the $(d-2)$-leaky forcing number of $Q_d$ is $2^{d-1}$. We also examine a question about the relationship between the size of a minimum $\ell$-leaky-forcing set and a minimum zero forcing set for a graph $G$.