Alantha Newman

Aggregating inconsistent information: Ranking and clustering

We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the extent of disagreement with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc set problem on tournaments, and correlation and consensus clustering. We show that for all these problems (and various weighted versions of them), we can obtain improved approximation factors using essentially the same remarkably simple algorithm. Additionally, we almost settle a long-standing conjecture of Bang-Jensen and Thomassen and show that unless NP⊆BPP, there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments.

Decision-making based on approximate and smoothed Pareto curves

We consider bicriteria optimization problems and investigate the relationship between two standard approaches to solving them: (i) computing the Pareto curve and (ii) the so-called decision makers approach in which both criteria are combined into a single (usually nonlinear) objective function. Previous work by Papadimitriou and Yannakakis showed how to efficiently approximate the Pareto curve for problems like Shortest Path, Spanning Tree, and Perfect Matching. We wish to determine for which classes of combined objective functions the approximate Pareto curve also yields an approximate solution to the decision makers problem. We show that an FPTAS for the Pareto curve also gives an FPTAS for the decision-makers problem if the combined objective function is growth bounded like a quasi-polynomial function. If the objective function, however, shows exponential growth then the decision-makers problem is NP-hard to approximate within any polynomial factor. In order to bypass these limitations of approximate decision making, we turn our attention to Pareto curves in the probabilistic framework of smoothed analysis. We show that in a smoothed model, we can efficiently generate the (complete and exact) Pareto curve with a small failure probability if there exists an algorithm for generating the Pareto curve whose worst-case running time is pseudopolynomial. This way, we can solve the decision-makers problem w.r.t. any non-decreasing objective function for randomly perturbed instances of, e.g. Shortest Path, Spanning Tree, and Perfect Matching.

Fences Are Futile: On Relaxations for the Linear Ordering Problem

We study polyhedral relaxations for the linear ordering problem. The integrality gap for the standard linear programming relaxation is 2. Our main result is that the integrality gap remains 2 even when the standard relaxations are augmented with k-fence constraints for any k, and with k-Mobius ladder constraints for k up to 7; when augmented with k-Mobius ladder constraints for general k, the gap is at least 33/17 ≅ 1:94. Our proof is non-constructive-we obtain an extremal example via the probabilistic method. Finally, we show that no relaxation that is solvable in polynomial time can have an integrality gap less than 66/65 unless P=NP.

Traveling salesman path problems

In the traveling salesman path problem, we are given a set of cities, traveling costs between city pairs and fixed source and destination cities. The objective is to find a minimum cost path from the source to destination visiting all cities exactly once. In this paper, we study polyhedral and combinatorial properties of a variant we call the traveling salesman walk problem, in which the objective is to find a minimum cost walk from the source to destination visiting all cities at least once. We first characterize traveling salesman walk perfect graphs, graphs for which the convex hull of incidence vectors of traveling salesman walks can be described by linear inequalities. We show these graphs have a description by way of forbidden minors and also characterize them constructively. We also address the asymmetric traveling salesman path problem (ATSPP) and give a factor

Combinatorial Problems on Strings with Applications to Protein Folding

O(\sqrt{n})

Cuts and Orderings: On Semidefinite Relaxations for the Linear Ordering Problem

-approximation algorithm for this problem.

Beck's Three Permutations Conjecture: A Counterexample and Some Consequences

We study the problem of finding a maximum acyclic subgraph of a given directed graph in which the maximum total degree (in plus out) is 3. For these graphs, we present: (i) a simple combinatorial algorithm that achieves an 11/12-approximation (the previous best factor was 2/3 [1]), (ii) a lower bound of 125/126 on approximability, and (iii) an approximation-preserving reduction from the general case: if for any Ɛ > 0, there exists a (17/18 + Ɛ)-approximation algorithm for the maximum acyclic subgraph problem in graphs with maximum degree 3, then there is a (1/2+δ)-approximation algorithm for general graphs for some δ > 0. The problem of finding a better-than-half approximation for general graphs is open.

Graph-TSP from Steiner Cycles

We consider the problem of protein folding in the HP model on the 3D square lattice. This problem is combinatorially equivalent to folding a string of 0’s and 1’s so that the string forms a self-avoiding walk on the lattice and the number of adjacent pairs of 1’s is maximized. The previously best-known approximation algorithm for this problem has a guarantee of \(\frac{3}{8}=.375\) [HI95]. In this paper, we first present a new \(\frac{3}{8}\)-approximation algorithm for the 3D folding problem that improves on the absolute approximation guarantee of the previous algorithm. We then show a connection between the 3D folding problem and a basic combinatorial problem on binary strings, which may be of independent interest. Given a binary string in { a,b }*, we want to find a long subsequence of the string in which every sequence of consecutive a’s is followed by at least as many consecutive b’s. We show a non-trivial lower-bound on the existence of such subsequences. Using this result, we obtain an algorithm with a slightly improved approximation ratio of at least .37501 for the 3D folding problem. All of our algorithms run in linear time.