Knowledge Refinement via Rule Selection

In several different applications, including data transformation and entity resolution, rules are used to capture aspects of knowledge about the application at hand. Often, a large set of such rules is generated automatically or semi-automatically, and the challenge is to refine the encapsulated knowledge by selecting a subset of rules based on the expected operational behavior of the rules on available data. In this paper, we carry out a systematic complexity-theoretic investigation of the following rule selection problem: given a set of rules specified by Horn formulas, and a pair of an input database and an output database, find a subset of the rules that minimizes the total error, that is, the number of false positive and false negative errors arising from the selected rules. We first establish computational hardness results for the decision problems underlying this minimization problem, as well as upper and lower bounds for its approximability. We then investigate a bi-objective optimization version of the rule selection problem in which both the total error and the size of the selected rules are taken into account. We show that testing for membership in the Pareto front of this bi-objective optimization problem is DP-complete. Finally, we show that a similar DP-completeness result holds for a bi-level optimization version of the rule selection problem, where one minimizes first the total error and then the size

Balanced Combinations of Solutions in Multi-Objective Optimization

For every list of integers x_1, ..., x_m there is some j such that x_1 + ... + x_j - x_{j+1} - ... - x_m \approx 0. So the list can be nearly balanced and for this we only need one alternation between addition and subtraction. But what if the x_i are k-dimensional integer vectors? Using results from topological degree theory we show that balancing is still possible, now with k alternations. This result is useful in multi-objective optimization, as it allows a polynomial-time computable balance of two alternatives with conflicting costs. The application to two multi-objective optimization problems yields the following results: - A randomized 1/2-approximation for multi-objective maximum asymmetric traveling salesman, which improves and simplifies the best known approximation for this problem. - A deterministic 1/2-approximation for multi-objective maximum weighted satisfiability

Inapproximability of Combinatorial Optimization Problems

We survey results on the hardness of approximating combinatorial optimization problems

Scheduling over Scenarios on Two Machines

We consider scheduling problems over scenarios where the goal is to find a single assignment of the jobs to the machines which performs well over all possible scenarios. Each scenario is a subset of jobs that must be executed in that scenario and all scenarios are given explicitly. The two objectives that we consider are minimizing the maximum makespan over all scenarios and minimizing the sum of the makespans of all scenarios. For both versions, we give several approximation algorithms and lower bounds on their approximability. With this research into optimization problems over scenarios, we have opened a new and rich field of interesting problems.Comment: To appear in COCOON 2014. The final publication is available at link.springer.co

Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder

Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat). The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem

Approximate kernel clustering

In the kernel clustering problem we are given a large $n\times n$ positive semi-definite matrix $A=(a_{ij})$ with $\sum_{i,j=1}^na_{ij}=0$ and a small $k\times k$ positive semi-definite matrix $B=(b_{ij})$. The goal is to find a partition $S_1,...,S_k$ of $\{1,... n\}$ which maximizes the quantity $$\sum_{i,j=1}^k (\sum_{(i,j)\in S_i\times S_j}a_{ij})b_{ij}.$$ We study the computational complexity of this generic clustering problem which originates in the theory of machine learning. We design a constant factor polynomial time approximation algorithm for this problem, answering a question posed by Song, Smola, Gretton and Borgwardt. In some cases we manage to compute the sharp approximation threshold for this problem assuming the Unique Games Conjecture (UGC). In particular, when $B$ is the $3\times 3$ identity matrix the UGC hardness threshold of this problem is exactly $\frac{16\pi}{27}$. We present and study a geometric conjecture of independent interest which we show would imply that the UGC threshold when $B$ is the $k\times k$ identity matrix is $\frac{8\pi}{9}(1-\frac{1}{k})$ for every $k\ge 3$

Non-approximability and Polylogarithmic Approximations of the Single-Sink Unsplittable and Confluent Dynamic Flow Problems

Dynamic Flows were introduced by Ford and Fulkerson in 1958 to model flows over time. They define edge capacities to be the total amount of flow that can enter an edge in one time unit. Each edge also has a length, representing the time needed to traverse it. Dynamic Flows have been used to model many problems including traffic congestion, hop-routing of packets and evacuation protocols in buildings. While the basic problem of moving the maximal amount of supplies from sources to sinks is polynomial time solvable, natural minor modifications can make it NP-hard. One such modification is that flows be confluent, i.e., all flows leaving a vertex must leave along the same edge. This corresponds to natural conditions in, e.g., evacuation planning and hop routing. We investigate the single-sink Confluent Quickest Flow problem. The input is a graph with edge capacities and lengths, sources with supplies and a sink. The problem is to find a confluent flow minimizing the time required to send supplies to the sink. Our main results include: a) Logarithmic Non-Approximability: Directed Confluent Quickest Flows cannot be approximated in polynomial time with an O(log n) approximation factor, unless P=NP. b) Polylogarithmic Bicriteria Approximations: Polynomial time (O(log^8 n), O(log^2 kappa)) bicritera approximation algorithms for the Confluent Quickest Flow problem where kappa is the number of sinks, in both directed and undirected graphs. Corresponding results are also developed for the Confluent Maximum Flow over time problem. The techniques developed also improve recent approximation algorithms for static confluent flows

Optimal Approximation Algorithms for Multi-agent Combinatorial Problems with Discounted Price Functions

Submodular functions are an important class of functions in combinatorial optimization which satisfy the natural properties of decreasing marginal costs. The study of these functions has led to strong structural properties with applications in many areas. Recently, there has been significant interest in extending the theory of algorithms for optimizing combinatorial problems (such as network design problem of spanning tree) over submodular functions. Unfortunately, the lower bounds under the general class of submodular functions are known to be very high for many of the classical problems. In this paper, we introduce and study an important subclass of submodular functions, which we call discounted price functions. These functions are succinctly representable and generalize linear cost functions. In this paper we study the following fundamental combinatorial optimization problems: Edge Cover, Spanning Tree, Perfect Matching and Shortest Path, and obtain tight upper and lower bounds for these problems. The main technical contribution of this paper is designing novel adaptive greedy algorithms for the above problems. These algorithms greedily build the solution whist rectifying mistakes made in the previous steps