Simpler and Better Algorithms for Minimum-Norm Load Balancing

Recently, Chakrabarty and Swamy (STOC 2019) introduced the minimum-norm load-balancing problem on unrelated machines, wherein we are given a set J of jobs that need to be scheduled on a set of m unrelated machines, and a monotone, symmetric norm; We seek an assignment sigma: J -> [m] that minimizes the norm of the resulting load vector load_{sigma} in R_+^m, where load_{sigma}(i) is the load on machine i under the assignment sigma. Besides capturing all l_p norms, symmetric norms also capture other norms of interest including top-l norms, and ordered norms. Chakrabarty and Swamy (STOC 2019) give a (38+epsilon)-approximation algorithm for this problem via a general framework they develop for minimum-norm optimization that proceeds by first carefully reducing this problem (in a series of steps) to a problem called min-max ordered load balancing, and then devising a so-called deterministic oblivious LP-rounding algorithm for ordered load balancing. We give a direct, and simple 4+epsilon-approximation algorithm for the minimum-norm load balancing based on rounding a (near-optimal) solution to a novel convex-programming relaxation for the problem. Whereas the natural convex program encoding minimum-norm load balancing problem has a large non-constant integrality gap, we show that this issue can be remedied by including a key constraint that bounds the "norm of the job-cost vector." Our techniques also yield a (essentially) 4-approximation for: (a) multi-norm load balancing, wherein we are given multiple monotone symmetric norms, and we seek an assignment respecting a given budget for each norm; (b) the best simultaneous approximation factor achievable for all symmetric norms for a given instance

Integrality gap analysis for bin packing games

A cooperative bin packing game is an $N$-person game, where the player set $N$ consists of $k$ bins of capacity 1 each and $n$ items of sizes $a_1,\dots,a_n$. The value $v(S)$ of a coalition $S$ of players is defined to be the maximum total size of items in $S$ that can be packed into the bins of $S$. We analyze the integrality gap of the corresponding 0–1 integer program of the value $v(N)$, thereby presenting an alternative proof for the non-emptiness of the 1/3-core for all bin packing games. Further, we show how to improve this bound $\epsilon\leq1/3$ (slightly) and point out that the conclusion in Matsui (2000) [9] is wrong (claiming that the bound 1/3 was tight). We conjecture that the true best possible value is $\epsilon=1/7$. The results are obtained using a new “rounding technique” that we develop to derive good (integral) packings from given fractional ones

Minimizing Flow-Time on Unrelated Machines

We consider some flow-time minimization problems in the unrelated machines setting. In this setting, there is a set of $m$ machines and a set of $n$ jobs, and each job $j$ has a machine dependent processing time of $p_{ij}$ on machine $i$. The flow-time of a job is the total time the job spends in the system (completion time minus its arrival time), and is one of the most natural quality of service measure. We show the following two results: an $O(\min(\log^2 n,\log n \log P))$ approximation algorithm for minimizing the total-flow time, and an $O(\log n)$ approximation for minimizing the maximum flow-time. Here $P$ is the ratio of maximum to minimum job size. These are the first known poly-logarithmic guarantees for both the problems.Comment: The new version fixes some typos in the previous version. The paper is accepted for publication in STOC 201