Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut

The multiway-cut problem is, given a weighted graph and k >= 2 terminal nodes, to find a minimum-weight set of edges whose removal separates all the terminals. The problem is NP-hard, and even NP-hard to approximate within 1+delta for some small delta > 0. Calinescu, Karloff, and Rabani (1998) gave an algorithm with performance guarantee 3/2-1/k, based on a geometric relaxation of the problem. In this paper, we give improved randomized rounding schemes for their relaxation, yielding a 12/11-approximation algorithm for k=3 and a 1.3438-approximation algorithm in general. Our approach hinges on the observation that the problem of designing a randomized rounding scheme for a geometric relaxation is itself a linear programming problem. The paper explores computational solutions to this problem, and gives a proof that for a general class of geometric relaxations, there are always randomized rounding schemes that match the integrality gap.Comment: Conference version in ACM Symposium on Theory of Computing (1999). To appear in Mathematics of Operations Researc

Parameterized Algorithms for Min-Max Multiway Cut and List Digraph Homomorphism

In this paper we design {\sf FPT}-algorithms for two parameterized problems. The first is \textsc{List Digraph Homomorphism}: given two digraphs $G$ and $H$ and a list of allowed vertices of $H$ for every vertex of $G$, the question is whether there exists a homomorphism from $G$ to $H$ respecting the list constraints. The second problem is a variant of \textsc{Multiway Cut}, namely \textsc{Min-Max Multiway Cut}: given a graph $G$, a non-negative integer $\ell$, and a set $T$ of $r$ terminals, the question is whether we can partition the vertices of $G$ into $r$ parts such that (a) each part contains one terminal and (b) there are at most $\ell$ edges with only one endpoint in this part. We parameterize \textsc{List Digraph Homomorphism} by the number $w$ of edges of $G$ that are mapped to non-loop edges of $H$ and we give a time $2^{O(\ell\cdot\log h+\ell^2\cdot \log \ell)}\cdot n^{4}\cdot \log n$ algorithm, where $h$ is the order of the host graph $H$. We also prove that \textsc{Min-Max Multiway Cut} can be solved in time $2^{O((\ell r)^2\log \ell r)}\cdot n^{4}\cdot \log n$. Our approach introduces a general problem, called {\sc List Allocation}, whose expressive power permits the design of parameterized reductions of both aforementioned problems to it. Then our results are based on an {\sf FPT}-algorithm for the {\sc List Allocation} problem that is designed using a suitable adaptation of the {\em randomized contractions} technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk, and Pilipczuk, FOCS 2012]).Comment: An extended abstract of this work will appear in the Proceedings of the 10th International Symposium on Parameterized and Exact Computation (IPEC), Patras, Greece, September 201