Exact Exponential Algorithms for Clustering Problems

In this paper we initiate a systematic study of exact algorithms for some of the well known clustering problems, namely k-MEDIAN and k-MEANS. In k-MEDIAN, the input consists of a set X of n points belonging to a metric space, and the task is to select a subset C ? X of k points as centers, such that the sum of the distances of every point to its nearest center is minimized. In k-MEANS, the objective is to minimize the sum of squares of the distances instead. It is easy to design an algorithm running in time max_{k ? n} {n choose k} n^?(1) = ?^*(2?) (here, ?^*(?) notation hides polynomial factors in n). In this paper we design first non-trivial exact algorithms for these problems. In particular, we obtain an ?^*((1.89)?) time exact algorithm for k-MEDIAN that works for any value of k. Our algorithm is quite general in that it does not use any properties of the underlying (metric) space - it does not even require the distances to satisfy the triangle inequality. In particular, the same algorithm also works for k-Means. We complement this result by showing that the running time of our algorithm is asymptotically optimal, up to the base of the exponent. That is, unless the Exponential Time Hypothesis fails, there is no algorithm for these problems running in time 2^o(n)?n^?(1). Finally, we consider the "facility location" or "supplier" versions of these clustering problems, where, in addition to the set X we are additionally given a set of m candidate centers (or facilities) F, and objective is to find a subset of k centers from F. The goal is still to minimize the k-Median/k-Means/k-Center objective. For these versions we give a ?(2? (mn)^?(1)) time algorithms using subset convolution. We complement this result by showing that, under the Set Cover Conjecture, the "supplier" versions of these problems do not admit an exact algorithm running in time 2^{(1-?) n} (mn)^?(1)

Exact Algorithms for 0-1 Integer Programs with Linear Equality Constraints

In this paper, we show $O(1.415^n)$-time and $O(1.190^n)$-space exact algorithms for 0-1 integer programs where constraints are linear equalities and coefficients are arbitrary real numbers. Our algorithms are quadratically faster than exhaustive search and almost quadratically faster than an algorithm for an inequality version of the problem by Impagliazzo, Lovett, Paturi and Schneider (arXiv:1401.5512), which motivated our work. Rather than improving the time and space complexity, we advance to a simple direction as inclusion of many NP-hard problems in terms of exact exponential algorithms. Specifically, we extend our algorithms to linear optimization problems

Matching Cut: Kernelization, Single-Exponential Time FPT, and Exact Exponential Algorithms

In a graph, a matching cut is an edge cut that is a matching. Matching Cut, which is known to be NP-complete, is the problem of deciding whether or not a given graph G has a matching cut. In this paper we show that Matching Cut admits a quadratic-vertex kernel for the parameter distance to cluster and a linear-vertex kernel for the parameter distance to clique. We further provide an O^*(2^{dc(G)}) time and an O^*(2^{dc^-}(G)}) time FPT algorithm for Matching Cut, where dc(G) and dc^-(G) are the distance to cluster and distance to co-cluster, respectively. We also improve the running time of the best known branching algorithm to solve Matching Cut from O^*(1.4143^n) to O^*(1.3803^n). Moreover, we point out that, unless NP subseteq coNP/poly, Matching Cut does not admit a polynomial kernel when parameterized by treewidth

Algorithmic Issues in some Disjoint Clustering Problems in Combinatorial Circuits

As the modern integrated circuit continues to grow in complexity, the design of very large-scale integrated (VLSI) circuits involves massive teams employing state-of-the-art computer-aided design (CAD) tools. An old, yet significant CAD problem for VLSI circuits is physical design automation. In this problem, one needs to compute the best physical layout of millions to billions of circuit components on a tiny silicon surface. The process of mapping an electronic design to a chip involves several physical design stages, one of which is clustering. Even for combinatorial circuits, there exist several models for the clustering problem. In particular, we consider the problem of disjoint clustering in combinatorial circuits for delay minimization (CN). The problem of clustering with replication for delay minimization has been well-studied and known to be solvable in polynomial time. However, replication can become expensive when it is unbounded. Consequently, CN is a problem worth investigating. In this dissertation, we establish the computational complexities of several variants of CN. We also present approximation and exact exponential algorithms for some variants of CN. In some cases, we even obtain an approximation factor of strictly less than two. Furthermore, our exact exponential algorithms beat brute force

Computation of Hadwiger Number and Related Contraction Problems: Tight Lower Bounds

We prove that the Hadwiger number of an n-vertex graph G (the maximum size of a clique minor in G) cannot be computed in time n^o(n), unless the Exponential Time Hypothesis (ETH) fails. This resolves a well-known open question in the area of exact exponential algorithms. The technique developed for resolving the Hadwiger number problem has a wider applicability. We use it to rule out the existence of n^o(n)-time algorithms (up to ETH) for a large class of computational problems concerning edge contractions in graphs

Large induced subgraphs via triangulations and CMSO

We obtain an algorithmic meta-theorem for the following optimization problem. Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models \phi. Some special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies of cycles of length 0 modulo m", where for fixed nonnegative integers m and l, the task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\ containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from \Gamma\ as a minor. 3) "Independent \Pi-packing", where for a fixed finite set of connected graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G with the maximum number of connected components, such that each connected component of G[F] is isomorphic to some graph from \Pi. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential maximal cliques in G and f is a function depending of t and \phi\ only. We also show how a similar running time can be obtained for the weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we deduce that our optimization problem can be solved in time O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with polynomial number of minimal separators