A Tight Algorithm for Strongly Connected Steiner Subgraph On Two Terminals With Demands

Given an edge-weighted directed graph $G=(V,E)$ on $n$ vertices and a set $T=\{t_1, t_2, \ldots, t_p\}$ of $p$ terminals, the objective of the \scss ($p$-SCSS) problem is to find an edge set $H\subseteq E$ of minimum weight such that $G[H]$ contains an $t_{i}\rightarrow t_j$ path for each $1\leq i\neq j\leq p$. In this paper, we investigate the computational complexity of a variant of $2$-SCSS where we have demands for the number of paths between each terminal pair. Formally, the \sharinggeneral problem is defined as follows: given an edge-weighted directed graph $G=(V,E)$ with weight function $\omega: E\rightarrow \mathbb{R}^{\geq 0}$, two terminal vertices $s, t$, and integers $k_1, k_2$ ; the objective is to find a set of $k_1$ paths $F_1, F_2, \ldots, F_{k_1}$ from $s\leadsto t$ and $k_2$ paths $B_1, B_2, \ldots, B_{k_2}$ from $t\leadsto s$ such that $\sum_{e\in E} \omega(e)\cdot \phi(e)$ is minimized, where $\phi(e)= \max \Big\{|\{i\in [k_1] : e\in F_i\}|\ ,\ |\{j\in [k_2] : e\in B_j\}|\Big\}$. For each $k\geq 1$, we show the following: The \sharing problem can be solved in $n^{O(k)}$ time. A matching lower bound for our algorithm: the \sharing problem does not have an $f(k)\cdot n^{o(k)}$ algorithm for any computable function $f$, unless the Exponential Time Hypothesis (ETH) fails. Our algorithm for \sharing relies on a structural result regarding an optimal solution followed by using the idea of a "token game" similar to that of Feldman and Ruhl. We show with an example that the structural result does not hold for the \sharinggeneral problem if $\min\{k_1, k_2\}\geq 2$. Therefore \sharing is the most general problem one can attempt to solve with our techniques.Comment: To appear in Algorithmica. An extended abstract appeared in IPEC '1

Scheduling with Setup Costs and Monotone Penalties

We consider single processor preemptive scheduling with job-dependent setup times. In this model, a job-dependent setup time is incurred when a job is started for the first time, and each time it is restarted after preemption. This model is a common generalization of preemptive scheduling, and actually of non-preemptive scheduling as well. The objective is to minimize the sum of any general non-negative, non-decreasing cost functions of the completion times of the jobs -- this generalizes objectives of minimizing weighted flow time, flow-time squared, tardiness or the number of tardy jobs among many others. Our main result is a randomized polynomial time O(1)-speed O(1)-approximation algorithm for this problem. Without speedup, no polynomial time finite multiplicative approximation is possible unless P=NP. We extend the approach of Bansal et al. (FOCS 2007) of rounding a linear programming relaxation which accounts for costs incurred due to the non-preemptive nature of the schedule. A key new idea used in the rounding is that a point in the intersection polytope of two matroids can be decomposed as a convex combination of incidence vectors of sets that are independent in both matroids. In fact, we use this for the intersection of a partition matroid and a laminar matroid, in which case the decomposition can be found efficiently using network flows. Our approach gives a randomized polynomial time offline O(1)-speed O(1)-approximation algorithm for the broadcast scheduling problem with general cost functions as well

A logarithmic approximation for unsplittable flow on line graphs

We consider the unsplittable flow problem on a line. In this problem, we are given a set of n tasks, each specified by a start time s_i, an end time t_i, a demand d_i > 0, and a profit p_i > 0. A task, if accepted, requires di units of bandwidth from time s_i to t_i and accrues a profit of p_i. For every time t, we are also specified the available bandwidth c_t, and the goal is to find a subset of tasks with maximum profit subject to the bandwidth constraints. We present the first polynomial time O(log n) approximation algorithm for this problem. This significantly advances the state of the art, as no polynomial time o(n) approximation was known previously. Previous results for this problem were known only in more restrictive settings; in particular, either the instance satisfies the so-called no-bottleneck assumption: max_i d_i = min_t c_t, or the ratio of both maximum to minimum demands and maximum to minimum capacities are polynomially (or quasi-polynomially) bounded in n. Our result, on the other hand, does not require these assumptions. Our algorithm is based on a combination of dynamic programming and rounding a natural linear programming relaxation for the problem. While there is an O(n) integrality gap known for this LP relaxation, our key idea is to exploit certain structural properties of the problem to show that instances that are bad for the LP can in fact be handled using dynamic programming

Lagrangian Relaxation Based Algorithms for Convex Programming Problems

This thesis deals with a class of Lagrangian relaxation based algorithms developed in the computer science community in last couple of decades. We present a unified framework for designing such algorithms for a large family of convex programming problems. Our algorithms are based on exponential potential functions and given any 2 (0; 1), compute (1 + )-approximate solutions in number of iterations proportional to

General Terms

We show that the sparsest cut in graphs can be approximated withi