Fast Algorithms for Parameterized Problems with Relaxed Disjointness Constraints

In parameterized complexity, it is a natural idea to consider different generalizations of classic problems. Usually, such generalization are obtained by introducing a "relaxation" variable, where the original problem corresponds to setting this variable to a constant value. For instance, the problem of packing sets of size at most $p$ into a given universe generalizes the Maximum Matching problem, which is recovered by taking $p=2$. Most often, the complexity of the problem increases with the relaxation variable, but very recently Abasi et al. have given a surprising example of a problem --- $r$-Simple $k$-Path --- that can be solved by a randomized algorithm with running time $O^*(2^{O(k \frac{\log r}{r})})$. That is, the complexity of the problem decreases with $r$. In this paper we pursue further the direction sketched by Abasi et al. Our main contribution is a derandomization tool that provides a deterministic counterpart of the main technical result of Abasi et al.: the $O^*(2^{O(k \frac{\log r}{r})})$ algorithm for $(r,k)$-Monomial Detection, which is the problem of finding a monomial of total degree $k$ and individual degrees at most $r$ in a polynomial given as an arithmetic circuit. Our technique works for a large class of circuits, and in particular it can be used to derandomize the result of Abasi et al. for $r$-Simple $k$-Path. On our way to this result we introduce the notion of representative sets for multisets, which may be of independent interest. Finally, we give two more examples of problems that were already studied in the literature, where the same relaxation phenomenon happens. The first one is a natural relaxation of the Set Packing problem, where we allow the packed sets to overlap at each element at most $r$ times. The second one is Degree Bounded Spanning Tree, where we seek for a spanning tree of the graph with a small maximum degree

Developments in the theory of randomized shortest paths with a comparison of graph node distances

There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into account the global structure of the graph. In this article, we develop the theory of one family of graph node distances, known as the randomized shortest path dissimilarity, which has its foundation in statistical physics. We show that the randomized shortest path dissimilarity can be easily computed in closed form for all pairs of nodes of a graph. Moreover, we come up with a new definition of a distance measure that we call the free energy distance. The free energy distance can be seen as an upgrade of the randomized shortest path dissimilarity as it defines a metric, in addition to which it satisfies the graph-geodetic property. The derivation and computation of the free energy distance are also straightforward. We then make a comparison between a set of generalized distances that interpolate between the shortest path distance and the commute time, or resistance distance. This comparison focuses on the applicability of the distances in graph node clustering and classification. The comparison, in general, shows that the parametrized distances perform well in the tasks. In particular, we see that the results obtained with the free energy distance are among the best in all the experiments.Comment: 30 pages, 4 figures, 3 table

The Moser-Tardos Framework with Partial Resampling

The resampling algorithm of Moser \& Tardos is a powerful approach to develop constructive versions of the Lov\'{a}sz Local Lemma (LLL). We generalize this to partial resampling: when a bad event holds, we resample an appropriately-random subset of the variables that define this event, rather than the entire set as in Moser & Tardos. This is particularly useful when the bad events are determined by sums of random variables. This leads to several improved algorithmic applications in scheduling, graph transversals, packet routing etc. For instance, we settle a conjecture of Szab\'{o} & Tardos (2006) on graph transversals asymptotically, and obtain improved approximation ratios for a packet routing problem of Leighton, Maggs, & Rao (1994)

Conditional Hardness of Earth Mover Distance

The Earth Mover Distance (EMD) between two sets of points A, B subseteq R^d with |A| = |B| is the minimum total Euclidean distance of any perfect matching between A and B. One of its generalizations is asymmetric EMD, which is the minimum total Euclidean distance of any matching of size |A| between sets of points A,B subseteq R^d with |A| <= |B|. The problems of computing EMD and asymmetric EMD are well-studied and have many applications in computer science, some of which also ask for the EMD-optimal matching itself. Unfortunately, all known algorithms require at least quadratic time to compute EMD exactly. Approximation algorithms with nearly linear time complexity in n are known (even for finding approximately optimal matchings), but suffer from exponential dependence on the dimension. In this paper we show that significant improvements in exact and approximate algorithms for EMD would contradict conjectures in fine-grained complexity. In particular, we prove the following results: - Under the Orthogonal Vectors Conjecture, there is some c>0 such that EMD in Omega(c^{log^* n}) dimensions cannot be computed in truly subquadratic time. - Under the Hitting Set Conjecture, for every delta>0, no truly subquadratic time algorithm can find a (1 + 1/n^delta)-approximate EMD matching in omega(log n) dimensions. - Under the Hitting Set Conjecture, for every eta = 1/omega(log n), no truly subquadratic time algorithm can find a (1 + eta)-approximate asymmetric EMD matching in omega(log n) dimensions